Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 99.2%

Time: 21.9s

Alternatives: 9

Speedup: 1.1×

• could not determine a ground truth

  1. z = -1315167612672.0715

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: 96.6% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{771.3234287776531}{t\_0 + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_3 := \frac{12.507343278686905}{t\_0 + 5}\\ t_4 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_5 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_6 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + t\_1\right) + t\_6\right) + t\_3\right) + t\_4\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + 0.9999999999998099\right) + t\_1\right) + t\_6\right) + t\_3\right) + t\_4\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7 - z}\right) + t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 771.3234287776531 (+ t_0 3.0)))
        (t_2 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_3 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_4 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_5 (/ (PI) (sin (* (PI) z))))
        (t_6 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -1000.0)
     (*
      t_5
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+
        (+
         (+ (+ (+ (+ 0.9999999999998099 t_1) t_6) t_3) t_4)
         (/ 9.984369578019572e-6 (+ t_0 7.0)))
        t_2)))
     (*
      t_5
      (*
       (*
        (* (* (sqrt (PI)) (sqrt 2.0)) (exp (- z 7.5)))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+
           (+
            (+
             (+
              (+
               (/ -1259.1392167224028 (- (- 1.0 z) -1.0))
               (/ 676.5203681218851 (- 1.0 z)))
              0.9999999999998099)
             t_1)
            t_6)
           t_3)
          t_4)
         (/ 9.984369578019572e-6 (- 7.0 z)))
        t_2))))))

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   6. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate-+l+ N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

   7. Applied rewrites 99.3%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + 0.9999999999998099\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 99.3%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + 0.9999999999998099\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. Taylor expanded in z around 0

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

       1. *-commutative N/A

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

       2. fp-cancel-sign-sub-inv N/A

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

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

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

       4. *-commutative N/A

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

       5. mul-1-neg N/A

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

       6. remove-double-neg N/A

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

       7. lower--.f64 99.3

          \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + 0.9999999999998099\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\color{blue}{7 - z}}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   13. Applied rewrites 99.3%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(\frac{-1259.1392167224028}{\left(1 - z\right) - -1} + \frac{676.5203681218851}{1 - z}\right) + 0.9999999999998099\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\color{blue}{7 - z}}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_2 := \frac{12.507343278686905}{t\_0 + 5}\\ t_3 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_4 := \frac{771.3234287776531}{t\_0 + 3}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\\ t_6 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_7 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -0.65:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(597.824167076735, z, 519.1279660315847\right), z, 361.7355639412844\right), z, 47.95075976068351\right) + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_2 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_3 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_4 (/ 771.3234287776531 (+ t_0 3.0)))
        (t_5 (/ 9.984369578019572e-6 (+ t_0 7.0)))
        (t_6 (/ (PI) (sin (* (PI) z))))
        (t_7 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -0.65)
     (*
      t_6
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+ (+ (+ (+ (+ (+ 0.9999999999998099 t_4) t_7) t_2) t_3) t_5) t_1)))
     (*
      t_6
      (*
       (*
        (* (* (exp (+ -7.5 z)) (sqrt (PI))) (sqrt 2.0))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+
           (+
            (+
             (fma
              (fma
               (fma 597.824167076735 z 519.1279660315847)
               z
               361.7355639412844)
              z
              47.95075976068351)
             t_4)
            t_7)
           t_2)
          t_3)
         t_5)
        t_1))))))

Derivation

1. Split input into 2 regimes

2. if z < -0.650000000000000022

   1. Initial program 19.1%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 83.6%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -0.650000000000000022 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{479507597606835099}{10000000000000000} + z \cdot \left(\frac{904338909853211}{2500000000000} + z \cdot \left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right)\right)\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(z \cdot \left(\frac{904338909853211}{2500000000000} + z \cdot \left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right)\right) + \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{904338909853211}{2500000000000} + z \cdot \left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right)\right) \cdot z} + \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{904338909853211}{2500000000000} + z \cdot \left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right), z, \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right) + \frac{904338909853211}{2500000000000}}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\color{blue}{\left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z\right) \cdot z} + \frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2076511864126339}{4000000000000} + \frac{23912966683069397}{40000000000000} \cdot z, z, \frac{904338909853211}{2500000000000}\right)}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{23912966683069397}{40000000000000} \cdot z + \frac{2076511864126339}{4000000000000}}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      8. lower-fma.f64 98.8

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(597.824167076735, z, 519.1279660315847\right)}, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 98.8%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(597.824167076735, z, 519.1279660315847\right), z, 361.7355639412844\right), z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_2 := \frac{12.507343278686905}{t\_0 + 5}\\ t_3 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_4 := \frac{771.3234287776531}{t\_0 + 3}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\\ t_6 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_7 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -1000:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_2 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_3 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_4 (/ 771.3234287776531 (+ t_0 3.0)))
        (t_5 (/ 9.984369578019572e-6 (+ t_0 7.0)))
        (t_6 (/ (PI) (sin (* (PI) z))))
        (t_7 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -1000.0)
     (*
      t_6
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+ (+ (+ (+ (+ (+ 0.9999999999998099 t_4) t_7) t_2) t_3) t_5) t_1)))
     (*
      t_6
      (*
       (*
        (* (* (exp (+ -7.5 z)) (sqrt (PI))) (sqrt 2.0))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+
           (+
            (+
             (fma
              (fma 519.1279660315847 z 361.7355639412844)
              z
              47.95075976068351)
             t_4)
            t_7)
           t_2)
          t_3)
         t_5)
        t_1))))))

Derivation

1. Split input into 2 regimes

2. if z < -1e3

   1. Initial program 0.0%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 100.0%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -1e3 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{479507597606835099}{10000000000000000} + z \cdot \left(\frac{904338909853211}{2500000000000} + \frac{2076511864126339}{4000000000000} \cdot z\right)\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(z \cdot \left(\frac{904338909853211}{2500000000000} + \frac{2076511864126339}{4000000000000} \cdot z\right) + \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{904338909853211}{2500000000000} + \frac{2076511864126339}{4000000000000} \cdot z\right) \cdot z} + \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\frac{904338909853211}{2500000000000} + \frac{2076511864126339}{4000000000000} \cdot z, z, \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\color{blue}{\frac{2076511864126339}{4000000000000} \cdot z + \frac{904338909853211}{2500000000000}}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      5. lower-fma.f64 98.3

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right)}, z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 98.3%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_2 := \frac{12.507343278686905}{t\_0 + 5}\\ t_3 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_4 := \frac{771.3234287776531}{t\_0 + 3}\\ t_5 := \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\\ t_6 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_7 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -0.082:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_6 \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + t\_4\right) + t\_7\right) + t\_2\right) + t\_3\right) + t\_5\right) + t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_2 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_3 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_4 (/ 771.3234287776531 (+ t_0 3.0)))
        (t_5 (/ 9.984369578019572e-6 (+ t_0 7.0)))
        (t_6 (/ (PI) (sin (* (PI) z))))
        (t_7 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -0.082)
     (*
      t_6
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+ (+ (+ (+ (+ (+ 0.9999999999998099 t_4) t_7) t_2) t_3) t_5) t_1)))
     (*
      t_6
      (*
       (*
        (* (* (sqrt (PI)) (sqrt 2.0)) (exp (- z 7.5)))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+ (+ (+ (fma 361.7355639412844 z 47.95075976068351) t_4) t_7) t_2)
          t_3)
         t_5)
        t_1))))))

Derivation

1. Split input into 2 regimes

2. if z < -0.0820000000000000034

   1. Initial program 32.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.4%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -0.0820000000000000034 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{479507597606835099}{10000000000000000} + \frac{904338909853211}{2500000000000} \cdot z\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{904338909853211}{2500000000000} \cdot z + \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-fma.f64 98.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \left(e^{z - \frac{15}{2}} \cdot \sqrt{2}\right)\right) \cdot {\color{blue}{\left(\frac{15}{2} - z\right)}}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot e^{z - 7.5}\right) \cdot {\color{blue}{\left(7.5 - z\right)}}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_3 := \frac{12.507343278686905}{t\_0 + 5}\\ t_4 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_5 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_6 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -0.082:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{t\_0 + 3}\right) + t\_6\right) + t\_3\right) + t\_4\right) + t\_1\right) + t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + \frac{771.3234287776531}{3 - z}\right) + t\_6\right) + t\_3\right) + t\_4\right) + t\_1\right) + t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 9.984369578019572e-6 (+ t_0 7.0)))
        (t_2 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_3 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_4 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_5 (/ (PI) (sin (* (PI) z))))
        (t_6 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -0.082)
     (*
      t_5
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+
        (+
         (+
          (+
           (+ (+ 0.9999999999998099 (/ 771.3234287776531 (+ t_0 3.0))) t_6)
           t_3)
          t_4)
         t_1)
        t_2)))
     (*
      t_5
      (*
       (*
        (* (* (exp (+ -7.5 z)) (sqrt (PI))) (sqrt 2.0))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+
           (+
            (+
             (fma 361.7355639412844 z 47.95075976068351)
             (/ 771.3234287776531 (- 3.0 z)))
            t_6)
           t_3)
          t_4)
         t_1)
        t_2))))))

Derivation

1. Split input into 2 regimes

2. if z < -0.0820000000000000034

   1. Initial program 32.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.4%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -0.0820000000000000034 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{479507597606835099}{10000000000000000} + \frac{904338909853211}{2500000000000} \cdot z\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{904338909853211}{2500000000000} \cdot z + \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-fma.f64 98.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{3 + -1 \cdot z}}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   10. Step-by-step derivation

       1. fp-cancel-sign-sub-inv N/A

          \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\color{blue}{3 - \left(\mathsf{neg}\left(-1\right)\right) \cdot z}}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       2. metadata-eval N/A

          \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{3 - \color{blue}{1} \cdot z}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       3. *-lft-identity N/A

          \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{3 - \color{blue}{z}}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

       4. lower--.f64 98.5

          \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + \frac{771.3234287776531}{\color{blue}{3 - z}}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   11. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + \frac{771.3234287776531}{\color{blue}{3 - z}}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := \frac{771.3234287776531}{t\_0 + 3}\\ t_2 := \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\\ t_3 := \frac{12.507343278686905}{t\_0 + 5}\\ t_4 := \frac{-0.13857109526572012}{t\_0 + 6}\\ t_5 := \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)}\\ t_6 := \frac{-176.6150291621406}{t\_0 + 4}\\ \mathbf{if}\;z \leq -0.082:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {7.5}^{\left(t\_0 + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + t\_1\right) + t\_6\right) + t\_3\right) + t\_4\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5 \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + t\_1\right) + t\_6\right) + t\_3\right) + t\_4\right) + \frac{9.984369578019572 \cdot 10^{-6}}{7}\right) + t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0))
        (t_1 (/ 771.3234287776531 (+ t_0 3.0)))
        (t_2 (/ 1.5056327351493116e-7 (+ t_0 8.0)))
        (t_3 (/ 12.507343278686905 (+ t_0 5.0)))
        (t_4 (/ -0.13857109526572012 (+ t_0 6.0)))
        (t_5 (/ (PI) (sin (* (PI) z))))
        (t_6 (/ -176.6150291621406 (+ t_0 4.0))))
   (if (<= z -0.082)
     (*
      t_5
      (*
       (* (* (sqrt (* (PI) 2.0)) (pow 7.5 (+ t_0 0.5))) (exp (- (+ 7.0 0.5))))
       (+
        (+
         (+ (+ (+ (+ 0.9999999999998099 t_1) t_6) t_3) t_4)
         (/ 9.984369578019572e-6 (+ t_0 7.0)))
        t_2)))
     (*
      t_5
      (*
       (*
        (* (* (exp (+ -7.5 z)) (sqrt (PI))) (sqrt 2.0))
        (pow (- 7.5 z) (- 0.5 z)))
       (+
        (+
         (+
          (+ (+ (+ (fma 361.7355639412844 z 47.95075976068351) t_1) t_6) t_3)
          t_4)
         (/ 9.984369578019572e-6 7.0))
        t_2))))))

Derivation

1. Split input into 2 regimes

2. if z < -0.0820000000000000034

   1. Initial program 32.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 74.4%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\color{blue}{7.5}}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(7 + 0.5\right)}\right) \cdot \left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   if -0.0820000000000000034 < z

   1. Initial program 97.3%

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

   3. Taylor expanded in z around -inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. cancel-sign-sub N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity N/A

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

      11. *-commutative N/A

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

   5. Applied rewrites 98.4%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   6. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{479507597606835099}{10000000000000000} + \frac{904338909853211}{2500000000000} \cdot z\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\left(\frac{904338909853211}{2500000000000} \cdot z + \frac{479507597606835099}{10000000000000000}\right)} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

      2. lower-fma.f64 98.5

         \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   8. Applied rewrites 98.5%

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right)} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

   9. Taylor expanded in z around 0

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{\frac{-15}{2} + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(\frac{15}{2} - z\right)}^{\left(\frac{1}{2} - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\frac{904338909853211}{2500000000000}, z, \frac{479507597606835099}{10000000000000000}\right) + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\color{blue}{7}}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 98.5%

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(e^{-7.5 + z} \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \sqrt{2}\right) \cdot {\left(7.5 - z\right)}^{\left(0.5 - z\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(361.7355639412844, z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\color{blue}{7}}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (* (* (exp -7.5) (sqrt 15.0)) (/ (PI) (sqrt (PI))))
     (+
      (+
       (+
        (+
         (+
          (+ 47.95075976068351 (/ 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_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-PI.f64 93.6

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 93.6%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\frac{15}{2}} \cdot \sqrt{2}\right) \cdot e^{\frac{-15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{479507597606835099}{10000000000000000}} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 94.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{47.95075976068351} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 95.7%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}\right) \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
13. Add Preprocessing

Alternative 8: 95.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (* (* (exp -7.5) (sqrt 15.0)) (sqrt (PI)))
     (+
      (+
       (+
        (+
         (+
          (+ 47.95075976068351 (/ 771.3234287776531 (+ t_0 3.0)))
          (/ -176.6150291621406 4.0))
         (/ 12.507343278686905 (+ t_0 5.0)))
        (/ -0.13857109526572012 (+ t_0 6.0)))
       (/ 9.984369578019572e-6 (+ t_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-PI.f64 93.6

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 93.6%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\frac{15}{2}} \cdot \sqrt{2}\right) \cdot e^{\frac{-15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{479507597606835099}{10000000000000000}} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 94.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{47.95075976068351} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(e^{\frac{-15}{2}} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\frac{479507597606835099}{10000000000000000} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\color{blue}{4}}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
12. Step-by-step derivation

13. Applied rewrites 94.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\color{blue}{4}}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
14. Add Preprocessing

Alternative 9: 95.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot e^{-7.5}\right) \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{t\_0 + 3}\right) + \frac{-176.6150291621406}{t\_0 + 4}\right) + \frac{12.507343278686905}{t\_0 + 5}\right) + \frac{-0.13857109526572012}{t\_0 + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{t\_0 + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{t\_0 + 8}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (* (sqrt (* 15.0 (PI))) (exp -7.5))
     (+
      (+
       (+
        (+
         (+
          (+ 47.95075976068351 (/ 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_0 7.0)))
      (/ 1.5056327351493116e-7 (+ t_0 8.0)))))))

Derivation

1. Initial program 95.8%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-PI.f64 93.6

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\color{blue}{\mathsf{PI}\left(\right)}}\right) \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

5. Applied rewrites 93.6%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(\left(\left(0.9999999999998099 + \frac{676.5203681218851}{\left(\left(1 - z\right) - 1\right) + 1}\right) + \frac{-1259.1392167224028}{\left(\left(1 - z\right) - 1\right) + 2}\right) + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]

6. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\frac{15}{2}} \cdot \sqrt{2}\right) \cdot e^{\frac{-15}{2}}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{\frac{479507597606835099}{10000000000000000}} + \frac{\frac{7713234287776531}{10000000000000}}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{\frac{-883075145810703}{5000000000000}}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{\frac{2501468655737381}{200000000000000}}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{\frac{-3464277381643003}{25000000000000000}}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{\frac{2496092394504893}{250000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{\frac{3764081837873279}{25000000000000000000000}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 94.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{7.5} \cdot \sqrt{2}\right) \cdot e^{-7.5}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\color{blue}{47.95075976068351} + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
9. Step-by-step derivation

10. Applied rewrites 94.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\left(e^{-7.5} \cdot \sqrt{15}\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 94.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\color{blue}{\left(\sqrt{15 \cdot \mathsf{PI}\left(\right)} \cdot e^{-7.5}\right)} \cdot \left(\left(\left(\left(\left(\left(47.95075976068351 + \frac{771.3234287776531}{\left(\left(1 - z\right) - 1\right) + 3}\right) + \frac{-176.6150291621406}{\left(\left(1 - z\right) - 1\right) + 4}\right) + \frac{12.507343278686905}{\left(\left(1 - z\right) - 1\right) + 5}\right) + \frac{-0.13857109526572012}{\left(\left(1 - z\right) - 1\right) + 6}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(\left(1 - z\right) - 1\right) + 7}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{\left(\left(1 - z\right) - 1\right) + 8}\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025016 
(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))))))