Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 99.0%

Time: 24.0s

Alternatives: 12

Speedup: 1.5×

• could not determine a ground truth

  1. z = -3783754464.5449233

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: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\\ \mathsf{fma}\left(\left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}, t\_0, \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0)))
          (* (sqrt (* 2.0 (PI))) (/ (PI) (sin (* (PI) z)))))))
   (fma
    (+
     (+
      (+
       (+
        (+
         (/ 676.5203681218851 (- 1.0 z))
         (+ 0.9999999999998099 (/ -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)))
    t_0
    (*
     (+
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0))
      (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
     t_0))))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right), \left(\frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}, \left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) \cdot \left(\left(\frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}\right)\right)} \]

7. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\left(\left(\left(\left(\frac{676.5203681218851}{1 - z} + \left(0.9999999999998099 + \frac{-1259.1392167224028}{\left(1 - z\right) - -1}\right)\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{\left(1 - z\right) - -3}\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}, e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right), \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\right)\right)\right) \]
8. Add Preprocessing

Alternative 2: 98.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Taylor expanded in z around inf

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. lower-exp.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + z\right)} - \frac{15}{2}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} - z\right) \cdot \log \left(\frac{15}{2} - z\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + \color{blue}{-1 \cdot z}\right) \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \color{blue}{\left(\frac{15}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \left(\frac{15}{2} + \color{blue}{-1 \cdot z}\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + -1 \cdot z, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   11. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   13. lower-log.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \color{blue}{\log \left(\frac{15}{2} + -1 \cdot z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   15. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   16. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   17. lower--.f64 98.5

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), \color{blue}{z - 7.5}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 98.5%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

10. Taylor expanded in z around 0

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

    1. mul-1-neg N/A

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

    2. unsub-neg N/A

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

    3. lower--.f64 98.5

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

12. Applied rewrites 98.5%

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

13. Final simplification 98.5%

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

Alternative 3: 98.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (*
    (*
     (+
      (+
       (+
        (+
         (fma (fma 544.9358906000987 z 436.3997278161676) z 260.9048120626994)
         (/ 12.507343278686905 (- (- 1.0 z) -4.0)))
        (/ -0.13857109526572012 (- (- 1.0 z) -5.0)))
       (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
      (/ 1.5056327351493116e-7 (- (- 1.0 z) -7.0)))
     (sqrt (* 2.0 (PI))))
    (exp (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0))))
   (sin (* (PI) z)))
  (PI)))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)}\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \color{blue}{\left(z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) + \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\color{blue}{\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) \cdot z} + \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \color{blue}{\mathsf{fma}\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z, z, \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \mathsf{fma}\left(\color{blue}{\frac{4708246094784852251}{8640000000000000} \cdot z + \frac{314207804027640689}{720000000000000}}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   5. lower-fma.f64 98.2

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right)}, z, 260.9048120626994\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 98.2%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right)}\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

10. Final simplification 98.2%

    \[\leadsto \frac{\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{\left(1 - z\right) - -4}\right) + \frac{-0.13857109526572012}{\left(1 - z\right) - -5}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \]
11. Add Preprocessing

Alternative 4: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(545.0353078428827 \cdot z, \sqrt{2} \cdot t\_0, \left(436.8961725563396 \cdot \sqrt{2}\right) \cdot t\_0\right), z, \left(263.3831869810514 \cdot \sqrt{2}\right) \cdot t\_0\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (*
    (/
     (*
      (fma
       (fma
        (* 545.0353078428827 z)
        (* (sqrt 2.0) t_0)
        (* (* 436.8961725563396 (sqrt 2.0)) t_0))
       z
       (* (* 263.3831869810514 (sqrt 2.0)) t_0))
      (exp (fma (- 0.5 z) (log (- 7.5 z)) (- z 7.5))))
     (sin (* (PI) z)))
    (PI))))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Taylor expanded in z around inf

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. lower-exp.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + z\right)} - \frac{15}{2}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} - z\right) \cdot \log \left(\frac{15}{2} - z\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + \color{blue}{-1 \cdot z}\right) \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \color{blue}{\left(\frac{15}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \left(\frac{15}{2} + \color{blue}{-1 \cdot z}\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + -1 \cdot z, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   11. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   13. lower-log.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \color{blue}{\log \left(\frac{15}{2} + -1 \cdot z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   15. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   16. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   17. lower--.f64 98.5

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), \color{blue}{z - 7.5}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 98.5%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

10. Taylor expanded in z around 0

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

12. Applied rewrites 98.1%

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

13. Final simplification 98.1%

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

Alternative 5: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{PI}\left(\right)}\\ \frac{\mathsf{fma}\left(\left(436.8961725563396 \cdot t\_0\right) \cdot z, \sqrt{2}, \left(263.3831869810514 \cdot \sqrt{2}\right) \cdot t\_0\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (PI))))
   (*
    (/
     (*
      (fma
       (* (* 436.8961725563396 t_0) z)
       (sqrt 2.0)
       (* (* 263.3831869810514 (sqrt 2.0)) t_0))
      (exp (fma (- 0.5 z) (log (- 7.5 z)) (- z 7.5))))
     (sin (* (PI) z)))
    (PI))))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Taylor expanded in z around inf

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. lower-exp.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\left(z + \log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right)\right) - \frac{15}{2}}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + z\right)} - \frac{15}{2}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\log \left(\frac{15}{2} - z\right) \cdot \left(\frac{1}{2} - z\right) + \left(z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} - z\right) \cdot \log \left(\frac{15}{2} - z\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\left(\frac{1}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + \color{blue}{-1 \cdot z}\right) \cdot \log \left(\frac{15}{2} - z\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   7. sub-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \color{blue}{\left(\frac{15}{2} + \left(\mathsf{neg}\left(z\right)\right)\right)} + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\left(\frac{1}{2} + -1 \cdot z\right) \cdot \log \left(\frac{15}{2} + \color{blue}{-1 \cdot z}\right) + \left(z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + -1 \cdot z, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   11. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - z}, \log \left(\frac{15}{2} + -1 \cdot z\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   13. lower-log.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \color{blue}{\log \left(\frac{15}{2} + -1 \cdot z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   14. mul-1-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \left(\frac{15}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right), z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   15. sub-neg N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   16. lower--.f64 N/A

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{\left(1 - z\right) - -7} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{\left(1 - z\right) - -5} + \left(\frac{\frac{2501468655737381}{200000000000000}}{\left(1 - z\right) - -4} + \left(\frac{\frac{-883075145810703}{5000000000000}}{\left(1 - z\right) - -3} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \left(\left(\frac{\frac{-3147848041806007}{2500000000000}}{\left(1 - z\right) - -1} + \frac{9999999999998099}{10000000000000000}\right) + \frac{\frac{6765203681218851}{10000000000000}}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\frac{1}{2} - z, \log \color{blue}{\left(\frac{15}{2} - z\right)}, z - \frac{15}{2}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   17. lower--.f64 98.5

       \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), \color{blue}{z - 7.5}\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 98.5%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \color{blue}{e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

10. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*l* N/A

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

    4. *-commutative N/A

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

    5. associate-*r* N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. associate-*r* N/A

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

    9. lower-fma.f64 N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lower-sqrt.f64 N/A

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

    13. lower-PI.f64 N/A

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

    14. lower-sqrt.f64 N/A

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

12. Applied rewrites 97.6%

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

13. Final simplification 97.6%

    \[\leadsto \frac{\mathsf{fma}\left(\left(436.8961725563396 \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot z, \sqrt{2}, \left(263.3831869810514 \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(0.5 - z, \log \left(7.5 - z\right), z - 7.5\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \]
14. Add Preprocessing

Alternative 6: 97.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(263.3831869810514 \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/
   (*
    (* (* 263.3831869810514 (sqrt 2.0)) (sqrt (PI)))
    (exp (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0))))
   (sin (* (PI) z)))
  (PI)))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sqrt.f64 N/A

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

   5. pow1/2 N/A

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

   6. pow-to-exp N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. prod-exp N/A

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

4. Applied rewrites 97.7%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{PI}\left(\right) \cdot 2\right), 0.5, \mathsf{fma}\left(\log \left(\left(1 - z\right) - -6.5\right), \left(1 - z\right) - 0.5, -\left(\left(1 - z\right) - -6.5\right)\right)\right)}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{\left(1 - z\right) - -7} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{\left(1 - z\right) - -5} + \left(\frac{12.507343278686905}{\left(1 - z\right) - -4} + \left(\frac{-176.6150291621406}{\left(1 - z\right) - -3} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + 0.9999999999998099\right) + \frac{676.5203681218851}{1 - z}\right)\right)\right)\right)\right)\right)\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)}} \]

7. Taylor expanded in z around 0

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right)\right)} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\color{blue}{\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{2}\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(\frac{1106209385320415913103082059}{4200000000000000000000000} \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

   7. lower-PI.f64 97.4

      \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\left(\left(263.3831869810514 \cdot \sqrt{2}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

9. Applied rewrites 97.4%

   \[\leadsto \mathsf{PI}\left(\right) \cdot \frac{\color{blue}{\left(\left(263.3831869810514 \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)}}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]

10. Final simplification 97.4%

    \[\leadsto \frac{\left(\left(263.3831869810514 \cdot \sqrt{2}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \mathsf{PI}\left(\right) \]
11. Add Preprocessing

Alternative 7: 96.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\\ \frac{\sqrt{7.5} \cdot \sqrt{2}}{e^{7.5} \cdot z} \cdot \left(\left(t\_0 \cdot t\_0\right) \cdot 263.3831869810514\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt (PI)))))
   (*
    (/ (* (sqrt 7.5) (sqrt 2.0)) (* (exp 7.5) z))
    (* (* t_0 t_0) 263.3831869810514))))

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 97.1%

   \[\leadsto \left(\left(\sqrt{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot 263.3831869810514\right) \cdot \frac{\color{blue}{\sqrt{7.5}} \cdot \sqrt{2}}{e^{7.5} \cdot z} \]

10. Final simplification 97.1%

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

Alternative 8: 96.3% accurate, 3.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 97.1%

   \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \left(\frac{\sqrt{7.5}}{e^{7.5}} \cdot \color{blue}{\frac{\sqrt{2}}{z}}\right) \]

10. Final simplification 97.1%

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

Alternative 9: 96.3% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 96.9%

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

10. Final simplification 96.9%

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

Alternative 10: 95.8% accurate, 3.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 96.5%

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

11. Applied rewrites 96.7%

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

12. Final simplification 96.7%

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

Alternative 11: 95.7% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 96.6%

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

Alternative 12: 95.7% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 97.0%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-neg.f64 N/A

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

   4. exp-neg N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

4. Applied rewrites 96.4%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\frac{{\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{\mathsf{PI}\left(\right) \cdot 2}}{e^{\left(1 - z\right) - -6.5}}} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 96.6

       \[\leadsto \left(\sqrt{\mathsf{PI}\left(\right)} \cdot 263.3831869810514\right) \cdot \frac{\sqrt{7.5} \cdot \sqrt{2}}{\color{blue}{e^{7.5}} \cdot z} \]

7. Applied rewrites 96.6%

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

9. Applied rewrites 96.5%

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

10. Final simplification 96.5%

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

Reproduce

herbie shell --seed 2024276 
(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))))))