Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.2%

Time: 21.6s

Alternatives: 19

Speedup: 1.8×

• could not determine a ground truth

  1. z = -2713067599914.753

Specification

\[z \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (* (* (sqrt (* (PI) 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Initial Program: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := t\_0 + 7\\ t_2 := t\_1 + 0.5\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {t\_2}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-t\_2}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{t\_0 + 1}\right) + \frac{-1259.1392167224028}{t\_0 + 2}\right) + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ t_0 7.0)) (t_2 (+ t_1 0.5)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (* (* (sqrt (* (PI) 2.0)) (pow t_2 (+ t_0 0.5))) (exp (- t_2)))
     (+
      (+
       (+
        (+
         (+
          (+
           (+
            (+ 0.9999999999998099 (/ 676.5203681218851 (+ t_0 1.0)))
            (/ -1259.1392167224028 (+ t_0 2.0)))
           (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 (+ t_0 4.0)))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 t_1))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -1\\ t_1 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \frac{2}{\sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, t\_0, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot t\_0} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{t\_1 + 5}\right) + \frac{-0.13857109526572012}{t\_1 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_1 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_1 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -1.0)) (t_1 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (*
      (* (sqrt (PI)) (pow (- 7.5 z) (- 0.5 z)))
      (* (exp (- z 7.5)) (/ 2.0 (sqrt 2.0))))
     (+
      (+
       (+
        (+
         (+
          0.9999999999998099
          (+
           (/
            (fma 676.5203681218851 t_0 (* (- 1.0 z) -1259.1392167224028))
            (* (- 1.0 z) t_0))
           (+
            (/ -176.6150291621406 (- (- 1.0 z) -3.0))
            (/ 771.3234287776531 (- (- 1.0 z) -2.0)))))
         (/ 12.507343278686905 (+ t_1 5.0)))
        (/ -0.13857109526572012 (+ t_1 6.0)))
       (/ 9.984369578019572e-6 (+ t_1 7.0)))
      (/ 1.5056327351493116e-7 (+ t_1 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. exp-to-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. lower-sqrt.f64 98.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000}}{1 - z}} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \color{blue}{\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. frac-add N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000} \cdot \left(\left(1 - z\right) - -1\right) + \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000} \cdot \left(\left(1 - z\right) - -1\right) + \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}\right)}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \color{blue}{\left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 98.1

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\color{blue}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\color{blue}{\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
10. Step-by-step derivation

11. Applied rewrites 98.1%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \frac{2}{\color{blue}{\sqrt{2}}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \left(1 - z\right) - -1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, t\_1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot t\_1} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(-z\right) + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (- (- 1.0 z) -1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (*
      (* (sqrt (PI)) (pow (- 7.5 z) (- 0.5 z)))
      (* (exp (- z 7.5)) (sqrt 2.0)))
     (+
      (+
       (+
        (+
         (+
          0.9999999999998099
          (+
           (/
            (fma 676.5203681218851 t_1 (* (- 1.0 z) -1259.1392167224028))
            (* (- 1.0 z) t_1))
           (+
            (/ -176.6150291621406 (- (- 1.0 z) -3.0))
            (/ 771.3234287776531 (- (- 1.0 z) -2.0)))))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ (- z) 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. exp-to-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. lower-sqrt.f64 98.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000}}{1 - z}} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \color{blue}{\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. frac-add N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000} \cdot \left(\left(1 - z\right) - -1\right) + \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\color{blue}{\frac{\frac{6765203681218851}{10000000000000} \cdot \left(\left(1 - z\right) - -1\right) + \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\color{blue}{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}\right)}}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \color{blue}{\left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}}\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 98.1

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\color{blue}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\color{blue}{\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)}} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\color{blue}{-1 \cdot z} + 8}\right)\right) \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\mathsf{fma}\left(\frac{6765203681218851}{10000000000000}, \left(1 - z\right) - -1, \left(1 - z\right) \cdot \frac{-3147848041806007}{2500000000000}\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + 8}\right)\right) \]

    2. lower-neg.f64 98.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\color{blue}{\left(-z\right)} + 8}\right)\right) \]

12. Applied rewrites 98.1%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{\mathsf{fma}\left(676.5203681218851, \left(1 - z\right) - -1, \left(1 - z\right) \cdot -1259.1392167224028\right)}{\left(1 - z\right) \cdot \left(\left(1 - z\right) - -1\right)} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\color{blue}{\left(-z\right)} + 8}\right)\right) \]
13. Add Preprocessing

Alternative 3: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (*
      (* (sqrt (PI)) (pow (- 7.5 z) (- 0.5 z)))
      (* (exp (- z 7.5)) (sqrt 2.0)))
     (+
      (+
       (+
        (+
         (+
          0.9999999999998099
          (+
           (+
            (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
            (/ 676.5203681218851 (- 1.0 z)))
           (+
            (/ -176.6150291621406 (- (- 1.0 z) -3.0))
            (/ 771.3234287776531 (- (- 1.0 z) -2.0)))))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. exp-to-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. lower-sqrt.f64 98.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
8. Add Preprocessing

Alternative 4: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := 0.5 + t\_0\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_0} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-t\_1} \cdot \left({t\_1}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ 0.5 t_0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (+
      (+
       (+
        (+
         0.9999999999998099
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
        (+
         (/ 771.3234287776531 (- (- 1.0 z) -2.0))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
       (+
        (/ 12.507343278686905 (- (- 1.0 z) -4.0))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
      (+
       (/ 9.984369578019572e-6 t_0)
       (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
     (* (exp (- t_1)) (* (pow t_1 (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]
7. Add Preprocessing

Alternative 5: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(\left(1 - z\right) - -6\right)\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \mathsf{fma}\left(\mathsf{fma}\left(2.9403018100637997 \cdot 10^{-8}, z, 2.0611519559804982 \cdot 10^{-7}\right), z, 1.4451589203350195 \cdot 10^{-6}\right)\right) \cdot \left(e^{-t\_0} \cdot \left({t\_0}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 0.5 (- (- 1.0 z) -6.0))))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (+
      (+
       (+
        (+
         0.9999999999998099
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
        (+
         (/ 771.3234287776531 (- (- 1.0 z) -2.0))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
       (+
        (/ 12.507343278686905 (- (- 1.0 z) -4.0))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
      (fma
       (fma 2.9403018100637997e-8 z 2.0611519559804982e-7)
       z
       1.4451589203350195e-6))
     (* (exp (- t_0)) (* (pow t_0 (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\left(\frac{2023222488469027353}{1400000000000000000000000} + z \cdot \left(\frac{16159431334887105871}{78400000000000000000000000} + \frac{129091010669041056297}{4390400000000000000000000000} \cdot z\right)\right)}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\left(z \cdot \left(\frac{16159431334887105871}{78400000000000000000000000} + \frac{129091010669041056297}{4390400000000000000000000000} \cdot z\right) + \frac{2023222488469027353}{1400000000000000000000000}\right)}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\color{blue}{\left(\frac{16159431334887105871}{78400000000000000000000000} + \frac{129091010669041056297}{4390400000000000000000000000} \cdot z\right) \cdot z} + \frac{2023222488469027353}{1400000000000000000000000}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\mathsf{fma}\left(\frac{16159431334887105871}{78400000000000000000000000} + \frac{129091010669041056297}{4390400000000000000000000000} \cdot z, z, \frac{2023222488469027353}{1400000000000000000000000}\right)}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \mathsf{fma}\left(\color{blue}{\frac{129091010669041056297}{4390400000000000000000000000} \cdot z + \frac{16159431334887105871}{78400000000000000000000000}}, z, \frac{2023222488469027353}{1400000000000000000000000}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. lower-fma.f64 97.6

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2.9403018100637997 \cdot 10^{-8}, z, 2.0611519559804982 \cdot 10^{-7}\right)}, z, 1.4451589203350195 \cdot 10^{-6}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

9. Applied rewrites 97.6%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2.9403018100637997 \cdot 10^{-8}, z, 2.0611519559804982 \cdot 10^{-7}\right), z, 1.4451589203350195 \cdot 10^{-6}\right)}\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 6: 98.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(\left(1 - z\right) - -6\right)\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \mathsf{fma}\left(2.0611519559804982 \cdot 10^{-7}, z, 1.4451589203350195 \cdot 10^{-6}\right)\right) \cdot \left(e^{-t\_0} \cdot \left({t\_0}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 0.5 (- (- 1.0 z) -6.0))))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (+
      (+
       (+
        (+
         0.9999999999998099
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
        (+
         (/ 771.3234287776531 (- (- 1.0 z) -2.0))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
       (+
        (/ 12.507343278686905 (- (- 1.0 z) -4.0))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
      (fma 2.0611519559804982e-7 z 1.4451589203350195e-6))
     (* (exp (- t_0)) (* (pow t_0 (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\left(\frac{2023222488469027353}{1400000000000000000000000} + \frac{16159431334887105871}{78400000000000000000000000} \cdot z\right)}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\left(\frac{16159431334887105871}{78400000000000000000000000} \cdot z + \frac{2023222488469027353}{1400000000000000000000000}\right)}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   2. lower-fma.f64 97.5

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\mathsf{fma}\left(2.0611519559804982 \cdot 10^{-7}, z, 1.4451589203350195 \cdot 10^{-6}\right)}\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

9. Applied rewrites 97.5%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\mathsf{fma}\left(2.0611519559804982 \cdot 10^{-7}, z, 1.4451589203350195 \cdot 10^{-6}\right)}\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 7: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 + \left(\left(1 - z\right) - -6\right)\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + 1.4451589203350195 \cdot 10^{-6}\right) \cdot \left(e^{-t\_0} \cdot \left({t\_0}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (+ 0.5 (- (- 1.0 z) -6.0))))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (+
      (+
       (+
        (+
         0.9999999999998099
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
        (+
         (/ 771.3234287776531 (- (- 1.0 z) -2.0))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
       (+
        (/ 12.507343278686905 (- (- 1.0 z) -4.0))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
      1.4451589203350195e-6)
     (* (exp (- t_0)) (* (pow t_0 (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{\frac{2023222488469027353}{1400000000000000000000000}}\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
8. Step-by-step derivation

9. Applied rewrites 97.3%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \color{blue}{1.4451589203350195 \cdot 10^{-6}}\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 8: 97.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - -6\\ t_1 := 0.5 + t\_0\\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_0} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-t\_1} \cdot \left({t\_1}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) -6.0)) (t_1 (+ 0.5 t_0)))
   (*
    (/ (fma (* 0.16666666666666666 (* z z)) (* (PI) (PI)) 1.0) z)
    (*
     (+
      (+
       (+
        (+
         0.9999999999998099
         (+
          (/ 676.5203681218851 (- 1.0 z))
          (/ -1259.1392167224028 (- (- 1.0 z) -1.0))))
        (+
         (/ 771.3234287776531 (- (- 1.0 z) -2.0))
         (/ -176.6150291621406 (- (- 1.0 z) -3.0))))
       (+
        (/ 12.507343278686905 (- (- 1.0 z) -4.0))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0))))
      (+
       (/ 9.984369578019572e-6 t_0)
       (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0))))
     (* (exp (- t_1)) (* (pow t_1 (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right)} \]

7. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot {z}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{1 - z} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(\frac{1}{2} + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

   11. lower-PI.f64 96.9

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]

9. Applied rewrites 96.9%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z}} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\frac{676.5203681218851}{1 - z} + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right)\right) + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right)\right) + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right)\right) \cdot \left(e^{-\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)} \cdot \left({\left(0.5 + \left(\left(1 - z\right) - -6\right)\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \]
10. Add Preprocessing

Alternative 9: 97.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{1}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(z + 1\right) \cdot \left(\sqrt{2} \cdot e^{-7.5}\right)\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ 1.0 z)
    (*
     (*
      (* (sqrt (PI)) (pow (- 7.5 z) (- 0.5 z)))
      (* (+ z 1.0) (* (sqrt 2.0) (exp -7.5))))
     (+
      (+
       (+
        (+
         (+
          0.9999999999998099
          (+
           (+
            (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
            (/ 676.5203681218851 (- 1.0 z)))
           (+
            (/ -176.6150291621406 (- (- 1.0 z) -3.0))
            (/ 771.3234287776531 (- (- 1.0 z) -2.0)))))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right)} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1}\right)} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right)\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. associate-+l+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{6765203681218851}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 1} + \frac{\frac{-3147848041806007}{2500000000000}}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3} + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right)\right)\right)} + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

4. Applied rewrites 98.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\color{blue}{\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right)} + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot e^{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. exp-to-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{{\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower--.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \color{blue}{\left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)}\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-exp.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\color{blue}{e^{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{\color{blue}{z - \frac{15}{2}}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. lower-sqrt.f64 98.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \color{blue}{\sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

7. Applied rewrites 98.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right)} \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

   1. lower-/.f64 96.3

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

10. Applied rewrites 96.3%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(e^{z - 7.5} \cdot \sqrt{2}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(z \cdot \left(e^{\frac{-15}{2}} \cdot \sqrt{2}\right) + \color{blue}{e^{\frac{-15}{2}} \cdot \sqrt{2}}\right)\right) \cdot \left(\left(\left(\left(\left(\frac{9999999999998099}{10000000000000000} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right) + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Step-by-step derivation

13. Applied rewrites 96.8%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(z + 1\right) \cdot \color{blue}{\left(\sqrt{2} \cdot e^{-7.5}\right)}\right)\right) \cdot \left(\left(\left(\left(\left(0.9999999999998099 + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right)\right)\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
14. Add Preprocessing

Alternative 10: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{fma}\left(1, z, -7.5\right)}\right) \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (PI) (sin (* (PI) z)))
  (*
   (*
    (* (sqrt (PI)) (exp (fma 1.0 z -7.5)))
    (* (pow (fma -1.0 z 7.5) (fma -1.0 z 0.5)) (sqrt 2.0)))
   (+
    (fma
     (fma (fma 606.676680916724 z 545.0353078425886) z 436.896172553987)
     z
     263.383186962231)
    (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z} + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right)}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right)}, z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around -inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-PI.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\color{blue}{\mathsf{PI}\left(\right)}} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-exp.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \color{blue}{e^{\mathsf{neg}\left(\left(\frac{15}{2} + -1 \cdot z\right)\right)}}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{neg}\left(\left(\frac{15}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{15}{2}\right)}\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. distribute-neg-in N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(\frac{15}{2}\right)\right)}}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\left(\mathsf{neg}\left(\color{blue}{-1 \cdot z}\right)\right) + \left(\mathsf{neg}\left(\frac{15}{2}\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. distribute-lft-neg-in N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\color{blue}{\left(\mathsf{neg}\left(-1\right)\right) \cdot z} + \left(\mathsf{neg}\left(\frac{15}{2}\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\color{blue}{1} \cdot z + \left(\mathsf{neg}\left(\frac{15}{2}\right)\right)}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{1 \cdot z + \color{blue}{\frac{-15}{2}}}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\color{blue}{\mathsf{fma}\left(1, z, \frac{-15}{2}\right)}}\right) \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{fma}\left(1, z, \frac{-15}{2}\right)}\right) \cdot \color{blue}{\left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 96.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\mathsf{fma}\left(1, z, -7.5\right)}\right) \cdot \left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{2}\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Add Preprocessing

Alternative 11: 97.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (PI) (sin (* (PI) z)))
  (*
   (*
    (* (* (pow (fma -1.0 z 7.5) (fma -1.0 z 0.5)) (sqrt (PI))) (sqrt 2.0))
    (exp (- (- (- (+ -1.0 z) -1.0) 7.0) 0.5)))
   (fma
    (fma (fma 606.6766809167608 z 545.0353078428827) z 436.8961725563396)
    z
    263.3831869810514))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) + \frac{102757979785251069442117317613}{235200000000000000000000000}}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) \cdot z} + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right)}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z + \frac{64608921419941589693928044520019}{118540800000000000000000000000}}, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right)}, z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)}\right) \]

6. Taylor expanded in z around -inf

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{2}\right)\right)} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]
7. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)}\right) \cdot \sqrt{2}\right)} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)}\right) \cdot \sqrt{2}\right)} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\color{blue}{\left(e^{\log \left(\frac{15}{2} + -1 \cdot z\right) \cdot \left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   5. exp-to-pow N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{{\left(\frac{15}{2} + -1 \cdot z\right)}^{\left(\frac{1}{2} + -1 \cdot z\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   6. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\color{blue}{{\left(\frac{15}{2} + -1 \cdot z\right)}^{\left(\frac{1}{2} + -1 \cdot z\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   7. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\frac{15}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)}^{\left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   8. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{15}{2}\right)}}^{\left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   9. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\color{blue}{-1 \cdot z} + \frac{15}{2}\right)}^{\left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\color{blue}{\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}}^{\left(\frac{1}{2} + -1 \cdot z\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   11. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   12. +-commutative N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + \frac{1}{2}\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   13. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\left(\color{blue}{-1 \cdot z} + \frac{1}{2}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\color{blue}{\left(\mathsf{fma}\left(-1, z, \frac{1}{2}\right)\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   15. lower-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\left(\mathsf{fma}\left(-1, z, \frac{1}{2}\right)\right)} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   16. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, \frac{15}{2}\right)\right)}^{\left(\mathsf{fma}\left(-1, z, \frac{1}{2}\right)\right)} \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \sqrt{2}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   17. lower-sqrt.f64 96.7

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \color{blue}{\sqrt{2}}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

8. Applied rewrites 96.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\color{blue}{\left(\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right)} \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

9. Final simplification 96.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left({\left(\mathsf{fma}\left(-1, z, 7.5\right)\right)}^{\left(\mathsf{fma}\left(-1, z, 0.5\right)\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]
10. Add Preprocessing

Alternative 12: 96.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(e^{\left(\left(-1 + z\right) + -6\right) - 0.5} \cdot \left({\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (/ (PI) (sin (* z (PI))))
   (*
    (exp (- (+ (+ -1.0 z) -6.0) 0.5))
    (*
     (pow (+ (- (- 1.0 z) -6.0) 0.5) (- (- 1.0 z) 0.5))
     (sqrt (* 2.0 (PI))))))
  (fma
   (fma (fma 606.6766809167608 z 545.0353078428827) z 436.8961725563396)
   z
   263.3831869810514)))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) + \frac{102757979785251069442117317613}{235200000000000000000000000}}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) \cdot z} + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right)}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z + \frac{64608921419941589693928044520019}{118540800000000000000000000000}}, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right)}, z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)}\right) \]

7. Applied rewrites 96.2%

   \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(e^{-\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)} \cdot \left({\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)} \]

8. Final simplification 96.2%

   \[\leadsto \left(\frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(e^{\left(\left(-1 + z\right) + -6\right) - 0.5} \cdot \left({\left(\left(\left(1 - z\right) - -6\right) + 0.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right) \]
9. Add Preprocessing

Alternative 13: 96.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/
     (PI)
     (* (fma (* -0.16666666666666666 (* z z)) (* (* (PI) (PI)) (PI)) (PI)) z))
    (*
     (*
      (* (sqrt (* (PI) 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (exp (- (- (- (+ -1.0 z) -1.0) 7.0) 0.5)))
     (+
      (fma
       (fma (fma 606.676680916724 z 545.0353078425886) z 436.896172553987)
       z
       263.383186962231)
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z} + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right)}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right)}, z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{z \cdot \left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\left(\frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \mathsf{PI}\left(\right)\right)} \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(\color{blue}{\left(\frac{-1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3}} + \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{fma}\left(\frac{-1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right)} \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\color{blue}{\frac{-1}{6} \cdot {z}^{2}}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), \color{blue}{{\mathsf{PI}\left(\right)}^{3}}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\color{blue}{\mathsf{PI}\left(\right)}}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-PI.f64 96.0

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 96.0%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Final simplification 96.0%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Add Preprocessing

Alternative 14: 96.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (fma (* 0.16666666666666666 (* z z)) (* (PI) (PI)) 1.0) z)
    (*
     (*
      (* (sqrt (* (PI) 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (exp (- (- (- (+ -1.0 z) -1.0) 7.0) 0.5)))
     (+
      (fma
       (fma (fma 606.676680916724 z 545.0353078425886) z 436.896172553987)
       z
       263.383186962231)
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{1382761731551712743134679}{5250000000000000000000} + z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right)\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(z \cdot \left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) + \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\left(\color{blue}{\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{16055934341359023345617179}{36750000000000000000000} + z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right)} + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) + \frac{16055934341359023345617179}{36750000000000000000000}}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z\right) \cdot z} + \frac{16055934341359023345617179}{36750000000000000000000}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1261892996482553330703662111}{2315250000000000000000000} + \frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z, z, \frac{16055934341359023345617179}{36750000000000000000000}\right)}, z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{19664514596894233501133157847}{32413500000000000000000000} \cdot z + \frac{1261892996482553330703662111}{2315250000000000000000000}}, z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right)}, z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right)} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot {z}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{19664514596894233501133157847}{32413500000000000000000000}, z, \frac{1261892996482553330703662111}{2315250000000000000000000}\right), z, \frac{16055934341359023345617179}{36750000000000000000000}\right), z, \frac{1382761731551712743134679}{5250000000000000000000}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. lower-PI.f64 96.0

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

8. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

9. Final simplification 96.0%

   \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.676680916724, z, 545.0353078425886\right), z, 436.896172553987\right), z, 263.383186962231\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
10. Add Preprocessing

Alternative 15: 96.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (fma (* 0.16666666666666666 (* z z)) (* (PI) (PI)) 1.0) z)
    (*
     (*
      (* (sqrt (* (PI) 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (exp (- (- (- (+ -1.0 z) -1.0) 7.0) 0.5)))
     (fma
      (fma (fma 606.6766809167608 z 545.0353078428827) z 436.8961725563396)
      z
      263.3831869810514)))))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)}\right) \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\left(z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\color{blue}{\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right) \cdot z} + \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)}\right) \]

   4. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) + \frac{102757979785251069442117317613}{235200000000000000000000000}}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right) \cdot z} + \frac{102757979785251069442117317613}{235200000000000000000000000}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right)}, z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z + \frac{64608921419941589693928044520019}{118540800000000000000000000000}}, z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   8. lower-fma.f64 96.0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right)}, z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

5. Applied rewrites 96.0%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)}\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot {z}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\left(z \cdot z\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{6} \cdot \left(z \cdot z\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4027292589444183035165374538123333}{6638284800000000000000000000000}, z, \frac{64608921419941589693928044520019}{118540800000000000000000000000}\right), z, \frac{102757979785251069442117317613}{235200000000000000000000000}\right), z, \frac{1106209385320415913103082059}{4200000000000000000000000}\right)\right) \]

   11. lower-PI.f64 96.0

       \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

8. Applied rewrites 96.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z}} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]

9. Final simplification 96.0%

   \[\leadsto \frac{\mathsf{fma}\left(0.16666666666666666 \cdot \left(z \cdot z\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{\left(\left(\left(-1 + z\right) - -1\right) - 7\right) - 0.5}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.6766809167608, z, 545.0353078428827\right), z, 436.8961725563396\right), z, 263.3831869810514\right)\right) \]
10. Add Preprocessing

Alternative 16: 96.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\mathsf{PI}\left(\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (* (/ (PI) (sqrt (PI))) (* (* (sqrt 7.5) (sqrt 2.0)) (exp -7.5))) z)
  263.3831869810514))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514} \]
6. Step-by-step derivation

7. Applied rewrites 95.7%

   \[\leadsto \frac{\frac{\mathsf{PI}\left(\right)}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514 \]
8. Add Preprocessing

Alternative 17: 95.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot 263.3831869810514 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (* (* (exp -7.5) (/ (sqrt 15.0) z)) (sqrt (PI))) 263.3831869810514))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514} \]
6. Step-by-step derivation

7. Applied rewrites 95.0%

   \[\leadsto \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15} \cdot e^{-7.5}}{z} \cdot \color{blue}{263.3831869810514} \]

8. Taylor expanded in z around 0

   \[\leadsto \left(\frac{e^{\frac{-15}{2}} \cdot \sqrt{15}}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000} \]
9. Step-by-step derivation

10. Applied rewrites 95.4%

    \[\leadsto \left(\left(e^{-7.5} \cdot \frac{\sqrt{15}}{z}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot 263.3831869810514 \]
11. Add Preprocessing

Alternative 18: 95.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15} \cdot e^{-7.5}}{z} \cdot 263.3831869810514 \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (/ (* (sqrt (* (PI) 15.0)) (exp -7.5)) z) 263.3831869810514))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514} \]
6. Step-by-step derivation

7. Applied rewrites 95.0%

   \[\leadsto \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15} \cdot e^{-7.5}}{z} \cdot \color{blue}{263.3831869810514} \]
8. Add Preprocessing

Alternative 19: 95.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot 263.3831869810514\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (* (sqrt (* 15.0 (PI))) (* (/ (exp -7.5) z) 263.3831869810514)))

Derivation

1. Initial program 96.2%

   \[\frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{e^{\frac{-15}{2}} \cdot \left(\sqrt{2} \cdot \sqrt{\frac{15}{2}}\right)}{z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \frac{1106209385320415913103082059}{4200000000000000000000000}} \]

5. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{PI}\left(\right)} \cdot \left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right)}{z} \cdot 263.3831869810514} \]
6. Step-by-step derivation

7. Applied rewrites 95.0%

   \[\leadsto \frac{\sqrt{\mathsf{PI}\left(\right) \cdot 15} \cdot e^{-7.5}}{z} \cdot \color{blue}{263.3831869810514} \]
8. Step-by-step derivation

9. Applied rewrites 94.9%

   \[\leadsto \left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \frac{e^{-7.5}}{z}\right) \cdot \color{blue}{263.3831869810514} \]
10. Step-by-step derivation

11. Applied rewrites 94.9%

    \[\leadsto \color{blue}{\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot \left(\frac{e^{-7.5}}{z} \cdot 263.3831869810514\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(FPCore (z)
  :name "Jmat.Real.gamma, branch z less than 0.5"
  :precision binary64
  :pre (<= z 0.5)
  (* (/ (PI) (sin (* (PI) z))) (* (* (* (sqrt (* (PI) 2.0)) (pow (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5) (+ (- (- 1.0 z) 1.0) 0.5))) (exp (- (+ (+ (- (- 1.0 z) 1.0) 7.0) 0.5)))) (+ (+ (+ (+ (+ (+ (+ (+ 0.9999999999998099 (/ 676.5203681218851 (+ (- (- 1.0 z) 1.0) 1.0))) (/ -1259.1392167224028 (+ (- (- 1.0 z) 1.0) 2.0))) (/ 771.3234287776531 (+ (- (- 1.0 z) 1.0) 3.0))) (/ -176.6150291621406 (+ (- (- 1.0 z) 1.0) 4.0))) (/ 12.507343278686905 (+ (- (- 1.0 z) 1.0) 5.0))) (/ -0.13857109526572012 (+ (- (- 1.0 z) 1.0) 6.0))) (/ 9.984369578019572e-6 (+ (- (- 1.0 z) 1.0) 7.0))) (/ 1.5056327351493116e-7 (+ (- (- 1.0 z) 1.0) 8.0))))))