Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.4% → 98.2%

Time: 16.1s

Alternatives: 8

Speedup: 3.9×

• could not determine a ground truth

  1. z = -9091961552.362455

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

Initial Program: 96.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(t\_0 + 7\right) + 0.5\right)}^{\left(t\_0 + 0.5\right)}\right) \cdot \left(\left(z + 1\right) \cdot e^{-7.5}\right)\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)))
   (*
    (/ (PI) (sin (* (PI) z)))
    (*
     (*
      (* (* (sqrt (PI)) (sqrt 2.0)) (pow (+ (+ t_0 7.0) 0.5) (+ t_0 0.5)))
      (* (+ z 1.0) (exp -7.5)))
     (+
      263.3831869810514
      (*
       z
       (+
        436.8961725563396
        (* z (+ 545.0353078428827 (* 606.6766809167608 z))))))))))

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-exp.f64 98.4

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

10. Applied rewrites 98.4%

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

    1. lift-exp.f64 N/A

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

    2. lift-exp.f64 N/A

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

    3. lift-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. distribute-rgt1-in N/A

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

    7. lower-*.f64 N/A

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

    8. lower-+.f64 N/A

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

    9. lift-exp.f64 98.4

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

12. Applied rewrites 98.4%

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

Alternative 2: 97.6% accurate, 1.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{z \cdot \left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)}} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{z}} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(\mathsf{PI}\left(\right) + \frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \cdot \color{blue}{z}} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   3. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(\frac{-1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) + \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\left(\left(\frac{-1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{3} + \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot {z}^{2}, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   7. pow2 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   10. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

   11. lift-PI.f64 97.8

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]

10. Applied rewrites 97.8%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{\mathsf{fma}\left(-0.16666666666666666 \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z}} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. lift-PI.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

    2. lift-pow.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\mathsf{fma}\left(\frac{-1}{6} \cdot \left(z \cdot z\right), {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) \cdot z} \cdot \left(\left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{2}\right) \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}^{\left(\left(\left(1 - z\right) - 1\right) + \frac{1}{2}\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + \frac{1}{2}\right)}\right) \cdot \left(\frac{1106209385320415913103082059}{4200000000000000000000000} + z \cdot \left(\frac{102757979785251069442117317613}{235200000000000000000000000} + z \cdot \left(\frac{64608921419941589693928044520019}{118540800000000000000000000000} + \frac{4027292589444183035165374538123333}{6638284800000000000000000000000} \cdot z\right)\right)\right)\right) \]

    3. unpow3 N/A

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

    4. unpow2 N/A

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

    5. lower-*.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. lift-PI.f64 N/A

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

    9. lift-PI.f64 N/A

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

    10. lift-PI.f64 97.8

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

12. Applied rewrites 97.8%

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

13. Final simplification 97.8%

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

Alternative 3: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 97.5

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

10. Applied rewrites 97.5%

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

11. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f64 97.5

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

13. Applied rewrites 97.5%

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

14. Final simplification 97.5%

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

Alternative 4: 97.2% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 97.5

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

10. Applied rewrites 97.5%

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

11. Taylor expanded in z around inf

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

    1. pow-to-exp N/A

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

    2. lift-pow.f64 N/A

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

    3. lift--.f64 N/A

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

    4. lift--.f64 97.5

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

13. Applied rewrites 97.5%

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

14. Final simplification 97.5%

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

Alternative 5: 97.1% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 97.5

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

10. Applied rewrites 97.5%

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

11. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 97.4

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

13. Applied rewrites 97.4%

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

14. Final simplification 97.4%

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

Alternative 6: 96.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-+.f64 N/A

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

   6. lower-*.f64 96.8

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

5. Applied rewrites 96.8%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\left(\left(\sqrt{\mathsf{PI}\left(\right) \cdot 2} \cdot {\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}^{\left(\left(\left(1 - z\right) - 1\right) + 0.5\right)}\right) \cdot e^{-\left(\left(\left(\left(1 - z\right) - 1\right) + 7\right) + 0.5\right)}\right) \cdot \color{blue}{\left(263.3831869810514 + z \cdot \left(436.8961725563396 + z \cdot \left(545.0353078428827 + 606.6766809167608 \cdot z\right)\right)\right)}\right) \]
6. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lower-sqrt.f64 97.7

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

7. Applied rewrites 97.7%

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

8. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 97.5

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

10. Applied rewrites 97.5%

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

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

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

Alternative 7: 96.5% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in z around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

8. Applied rewrites 96.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lift-*.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. metadata-eval N/A

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

   10. sqrt-unprod N/A

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

   11. *-commutative N/A

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

   12. lower-/.f64 N/A

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

10. Applied rewrites 95.9%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-exp.f64 N/A

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

    4. lift-sqrt.f64 N/A

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

    5. lift-PI.f64 N/A

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

    6. lift-sqrt.f64 N/A

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

    7. *-commutative N/A

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

    8. lift-sqrt.f64 N/A

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

    9. associate-*r* N/A

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

    10. lower-*.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. lift-sqrt.f64 N/A

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

    13. lift-PI.f64 N/A

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

    14. lift-exp.f64 96.6

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

12. Applied rewrites 96.6%

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

Alternative 8: 95.9% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.6%

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

3. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 96.1%

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

6. Taylor expanded in z around 0

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

   1. sqrt-unprod N/A

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

   2. metadata-eval N/A

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

8. Applied rewrites 96.1%

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

Reproduce

herbie shell --seed 2025051 
(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))))))