Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 96.6% → 98.2%

Time: 2.4min

Alternatives: 12

Speedup: 1.4×

• could not determine a ground truth

  1. z = -4886333707927.591

Specification

\[z \leq 0.5\]

Math

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

FPCore

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

Initial Program: 96.6% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \mathsf{fma}\left({z}^{3} + -274.625, \frac{1}{\left(z \cdot z - -6.5 \cdot z\right) + 42.25}, -1\right)\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (+
     (+
      (+
       (+
        (fma (fma 544.9358906000987 z 436.3997278161676) z 260.9048120626994)
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (exp
     (fma
      (- (- 1.0 z) 0.5)
      (log1p (- (- z) -6.5))
      (fma
       (+ (pow z 3.0) -274.625)
       (/ 1.0 (+ (- (* z z) (* -6.5 z)) 42.25))
       -1.0))))
   (sqrt (* 2.0 (PI))))
  (/ (PI) (sin (* z (PI))))))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{-1 + \left(\frac{-13}{2} + z\right)}\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{\left(\frac{-13}{2} + z\right) + -1}\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. lift-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{\left(\frac{-13}{2} + z\right)} + -1\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. flip3-+ N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{\frac{{\frac{-13}{2}}^{3} + {z}^{3}}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}} + -1\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. div-inv N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{\left({\frac{-13}{2}}^{3} + {z}^{3}\right) \cdot \frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}} + -1\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \color{blue}{\mathsf{fma}\left({\frac{-13}{2}}^{3} + {z}^{3}, \frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)}\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. lower-+.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\color{blue}{{\frac{-13}{2}}^{3} + {z}^{3}}, \frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\color{blue}{\frac{-2197}{8}} + {z}^{3}, \frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. lower-pow.f64 N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + \color{blue}{{z}^{3}}, \frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \color{blue}{\frac{1}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. lower-+.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\color{blue}{\frac{-13}{2} \cdot \frac{-13}{2} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\color{blue}{\frac{169}{4}} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. lower--.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \color{blue}{\left(z \cdot z - \frac{-13}{2} \cdot z\right)}}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(\color{blue}{z \cdot z} - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   15. lower-*.f64 99.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \mathsf{fma}\left(-274.625 + {z}^{3}, \frac{1}{42.25 + \left(z \cdot z - \color{blue}{-6.5 \cdot z}\right)}, -1\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

9. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \color{blue}{\mathsf{fma}\left(-274.625 + {z}^{3}, \frac{1}{42.25 + \left(z \cdot z - -6.5 \cdot z\right)}, -1\right)}\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Taylor expanded in z around 0

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)}\right)\right)\right)\right)\right)\right) \]
11. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) + \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right)\right)\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\color{blue}{\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) \cdot z} + \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right)\right)\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z, z, \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right)\right)\right) \]

    4. +-commutative N/A

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), \mathsf{fma}\left(\frac{-2197}{8} + {z}^{3}, \frac{1}{\frac{169}{4} + \left(z \cdot z - \frac{-13}{2} \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\frac{4708246094784852251}{8640000000000000} \cdot z + \frac{314207804027640689}{720000000000000}}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right)\right)\right) \]

    5. lower-fma.f64 99.1

       \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \mathsf{fma}\left(-274.625 + {z}^{3}, \frac{1}{42.25 + \left(z \cdot z - -6.5 \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right)}, z, 260.9048120626994\right)\right)\right)\right)\right)\right)\right) \]

12. Applied rewrites 99.1%

    \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \mathsf{fma}\left(-274.625 + {z}^{3}, \frac{1}{42.25 + \left(z \cdot z - -6.5 \cdot z\right)}, -1\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right)}\right)\right)\right)\right)\right)\right) \]

13. Final simplification 99.1%

    \[\leadsto \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \mathsf{fma}\left({z}^{3} + -274.625, \frac{1}{\left(z \cdot z - -6.5 \cdot z\right) + 42.25}, -1\right)\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
14. Add Preprocessing

Alternative 2: 98.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (+
     (+
      (+
       (+
        (+
         (+
          (fma (fma 519.1279660315847 z 361.7355639412844) z 47.95075976068351)
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- 5.0 (- z -1.0))))
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (exp (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0))))
   (sqrt (* 2.0 (PI))))
  (/ (PI) (sin (* z (PI))))))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

8. Final simplification 99.1%

   \[\leadsto \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \]
9. Add Preprocessing

Alternative 3: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot z\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ (fma (* (* 0.16666666666666666 (* z z)) (PI)) (PI) 1.0) z)
  (*
   (*
    (+
     (+
      (+
       (+
        (+
         (+
          (fma (fma 519.1279660315847 z 361.7355639412844) z 47.95075976068351)
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- 5.0 (- z -1.0))))
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (exp (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0))))
   (sqrt (* 2.0 (PI))))))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1 + \frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\frac{1}{6} \cdot \left({z}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1}}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} + 1}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)} + 1}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{6} \cdot {z}^{2}\right) \cdot \mathsf{PI}\left(\right)}, \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left({z}^{2} \cdot \frac{1}{6}\right)} \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left({z}^{2} \cdot \frac{1}{6}\right)} \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{1}{6}\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\color{blue}{\left(z \cdot z\right)} \cdot \frac{1}{6}\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. lower-PI.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(z \cdot z\right) \cdot \frac{1}{6}\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. lower-PI.f64 98.8

       \[\leadsto \frac{\mathsf{fma}\left(\left(\left(z \cdot z\right) \cdot 0.16666666666666666\right) \cdot \mathsf{PI}\left(\right), \color{blue}{\mathsf{PI}\left(\right)}, 1\right)}{z} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Applied rewrites 98.8%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\left(\left(z \cdot z\right) \cdot 0.16666666666666666\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Final simplification 98.8%

    \[\leadsto \frac{\mathsf{fma}\left(\left(0.16666666666666666 \cdot \left(z \cdot z\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right), 1\right)}{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \]
12. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (/ 1.0 z)
  (*
   (*
    (+
     (+
      (+
       (+
        (+
         (+
          (fma (fma 519.1279660315847 z 361.7355639412844) z 47.95075976068351)
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- 5.0 (- z -1.0))))
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (exp (fma (- (- 1.0 z) 0.5) (log1p (- (- z) -6.5)) (+ (+ -6.5 z) -1.0))))
   (sqrt (* 2.0 (PI))))))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 99.1%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - \frac{1}{2}, \mathsf{log1p}\left(\left(-z\right) - \frac{-13}{2}\right), -1 + \left(\frac{-13}{2} + z\right)\right)} \cdot \left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. lower-/.f64 98.6

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Applied rewrites 98.6%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\sqrt{2 \cdot \mathsf{PI}\left(\right)} \cdot \left(e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), -1 + \left(-6.5 + z\right)\right)} \cdot \left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

11. Final simplification 98.6%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot e^{\mathsf{fma}\left(\left(1 - z\right) - 0.5, \mathsf{log1p}\left(\left(-z\right) - -6.5\right), \left(-6.5 + z\right) + -1\right)}\right) \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \]
12. Add Preprocessing

Alternative 5: 96.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(e^{\left(-6.5 + z\right) + -1} \cdot \left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right)\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (exp (+ (+ -6.5 z) -1.0))
   (*
    (* (pow (- (- 1.0 z) -6.5) (- 0.5 z)) (sqrt (* 2.0 (PI))))
    (+
     (+
      (+
       (+
        (+
         (+
          (fma (fma 519.1279660315847 z 361.7355639412844) z 47.95075976068351)
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- 5.0 (- z -1.0))))
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))))
  (/ 1.0 z)))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\color{blue}{\left(\frac{1}{2} + -1 \cdot z\right)}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
12. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    2. sub-neg N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\color{blue}{\left(\frac{1}{2} - z\right)}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    3. lower--.f64 97.0

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\color{blue}{\left(0.5 - z\right)}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

13. Applied rewrites 97.0%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\color{blue}{\left(0.5 - z\right)}} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

14. Final simplification 97.0%

    \[\leadsto \left(e^{\left(-6.5 + z\right) + -1} \cdot \left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(0.5 - z\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right)\right) \cdot \frac{1}{z} \]
15. Add Preprocessing

Alternative 6: 96.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7} + \left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI))))
    (+
     (+
      (/ 9.984369578019572e-6 7.0)
      (+
       (+
        (+
         (+
          (fma (fma 519.1279660315847 z 361.7355639412844) z 47.95075976068351)
          (/ 771.3234287776531 (- (- 1.0 z) -2.0)))
         (/ -176.6150291621406 (- 5.0 (- z -1.0))))
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0)))))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0)))))
   (exp (+ (+ -6.5 z) -1.0)))
  (/ 1.0 z)))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\color{blue}{7}} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
12. Step-by-step derivation

13. Applied rewrites 97.0%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\color{blue}{7}} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

14. Final simplification 97.0%

    \[\leadsto \left(\left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\frac{9.984369578019572 \cdot 10^{-6}}{7} + \left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right) + \frac{771.3234287776531}{\left(1 - z\right) - -2}\right) + \frac{-176.6150291621406}{5 - \left(z - -1\right)}\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right)\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \]
15. Add Preprocessing

Alternative 7: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.656776085461, z, 544.9358906000987\right), z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (+
     (+
      (+
       (+
        (fma
         (fma (fma 606.656776085461 z 544.9358906000987) z 436.3997278161676)
         z
         260.9048120626994)
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))
   (exp (+ (+ -6.5 z) -1.0)))
  (/ 1.0 z)))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right)\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(z \cdot \left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right) + \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\color{blue}{\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right)\right) \cdot z} + \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\frac{314207804027640689}{720000000000000} + z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right), z, \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    4. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{z \cdot \left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right) + \frac{314207804027640689}{720000000000000}}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    5. *-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z\right) \cdot z} + \frac{314207804027640689}{720000000000000}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    6. lower-fma.f64 N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{4708246094784852251}{8640000000000000} + \frac{62898174544540606049}{103680000000000000} \cdot z, z, \frac{314207804027640689}{720000000000000}\right)}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    7. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{62898174544540606049}{103680000000000000} \cdot z + \frac{4708246094784852251}{8640000000000000}}, z, \frac{314207804027640689}{720000000000000}\right), z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    8. lower-fma.f64 96.9

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(606.656776085461, z, 544.9358906000987\right)}, z, 436.3997278161676\right), z, 260.9048120626994\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

13. Applied rewrites 96.9%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.656776085461, z, 544.9358906000987\right), z, 436.3997278161676\right), z, 260.9048120626994\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

14. Final simplification 96.9%

    \[\leadsto \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(606.656776085461, z, 544.9358906000987\right), z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \]
15. Add Preprocessing

Alternative 8: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI))))
    (+
     (+
      (+
       (+
        (fma (fma 544.9358906000987 z 436.3997278161676) z 260.9048120626994)
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0)))))
   (exp (+ (+ -6.5 z) -1.0)))
  (/ 1.0 z)))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right)\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(z \cdot \left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) + \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    2. *-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\color{blue}{\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z\right) \cdot z} + \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    3. lower-fma.f64 N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\frac{314207804027640689}{720000000000000} + \frac{4708246094784852251}{8640000000000000} \cdot z, z, \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    4. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\frac{4708246094784852251}{8640000000000000} \cdot z + \frac{314207804027640689}{720000000000000}}, z, \frac{7827144361880981797}{30000000000000000}\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    5. lower-fma.f64 96.9

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right)}, z, 260.9048120626994\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

13. Applied rewrites 96.9%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

14. Final simplification 96.9%

    \[\leadsto \left(\left(\left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right) \cdot \left(\left(\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(544.9358906000987, z, 436.3997278161676\right), z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \]
15. Add Preprocessing

Alternative 9: 96.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(436.3997278161676, z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \end{array} \]

FPCore

(FPCore (z)
 :precision binary64
 (*
  (*
   (*
    (+
     (+
      (+
       (+
        (fma 436.3997278161676 z 260.9048120626994)
        (/ 12.507343278686905 (- 6.0 (- z -1.0))))
       (/ -0.13857109526572012 (- 7.0 (- z -1.0))))
      (/ 9.984369578019572e-6 (- (- 1.0 z) -6.0)))
     (/ 1.5056327351493116e-7 (- 9.0 (- z -1.0))))
    (* (pow (- (- 1.0 z) -6.5) (- (- 1.0 z) 0.5)) (sqrt (* 2.0 (PI)))))
   (exp (+ (+ -6.5 z) -1.0)))
  (/ 1.0 z)))

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

7. Applied rewrites 97.2%

   \[\leadsto \frac{\mathsf{PI}\left(\right)}{\sin \left(\mathsf{PI}\left(\right) \cdot z\right)} \cdot \color{blue}{\left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right)} \]

8. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \left(\frac{\frac{-883075145810703}{5000000000000}}{5 - \left(z - -1\right)} + \left(\frac{\frac{7713234287776531}{10000000000000}}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(\frac{2076511864126339}{4000000000000}, z, \frac{904338909853211}{2500000000000}\right), z, \frac{479507597606835099}{10000000000000000}\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
9. Step-by-step derivation

   1. lower-/.f64 97.0

      \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

10. Applied rewrites 97.0%

    \[\leadsto \color{blue}{\frac{1}{z}} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \left(\frac{-176.6150291621406}{5 - \left(z - -1\right)} + \left(\frac{771.3234287776531}{\left(1 - z\right) - -2} + \mathsf{fma}\left(\mathsf{fma}\left(519.1279660315847, z, 361.7355639412844\right), z, 47.95075976068351\right)\right)\right)\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

11. Taylor expanded in z around 0

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(\frac{7827144361880981797}{30000000000000000} + \frac{314207804027640689}{720000000000000} \cdot z\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{\frac{3764081837873279}{25000000000000000000000}}{9 - \left(z - -1\right)} + \left(\frac{\frac{2496092394504893}{250000000000000000000}}{\left(1 - z\right) - -6} + \left(\frac{\frac{-3464277381643003}{25000000000000000}}{7 - \left(z - -1\right)} + \left(\frac{\frac{2501468655737381}{200000000000000}}{6 - \left(z - -1\right)} + \color{blue}{\left(\frac{314207804027640689}{720000000000000} \cdot z + \frac{7827144361880981797}{30000000000000000}\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - \frac{-13}{2}\right)}^{\left(\left(1 - z\right) - \frac{1}{2}\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(\frac{-13}{2} + z\right)}\right) \]

    2. lower-fma.f64 96.9

       \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(436.3997278161676, z, 260.9048120626994\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

13. Applied rewrites 96.9%

    \[\leadsto \frac{1}{z} \cdot \left(\left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6} + \left(\frac{-0.13857109526572012}{7 - \left(z - -1\right)} + \left(\frac{12.507343278686905}{6 - \left(z - -1\right)} + \color{blue}{\mathsf{fma}\left(436.3997278161676, z, 260.9048120626994\right)}\right)\right)\right)\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{-1 + \left(-6.5 + z\right)}\right) \]

14. Final simplification 96.9%

    \[\leadsto \left(\left(\left(\left(\left(\left(\mathsf{fma}\left(436.3997278161676, z, 260.9048120626994\right) + \frac{12.507343278686905}{6 - \left(z - -1\right)}\right) + \frac{-0.13857109526572012}{7 - \left(z - -1\right)}\right) + \frac{9.984369578019572 \cdot 10^{-6}}{\left(1 - z\right) - -6}\right) + \frac{1.5056327351493116 \cdot 10^{-7}}{9 - \left(z - -1\right)}\right) \cdot \left({\left(\left(1 - z\right) - -6.5\right)}^{\left(\left(1 - z\right) - 0.5\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right) \cdot e^{\left(-6.5 + z\right) + -1}\right) \cdot \frac{1}{z} \]
15. Add Preprocessing

Alternative 10: 96.4% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-exp.f64 95.9

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

8. Applied rewrites 95.9%

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

   1. rem-square-sqrt N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lower-*.f64 96.9

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

10. Applied rewrites 96.9%

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

11. Final simplification 96.9%

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

Alternative 11: 96.2% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-exp.f64 95.9

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

8. Applied rewrites 95.9%

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

9. Taylor expanded in z around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower-sqrt.f64 N/A

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

    7. lower-PI.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-/l* N/A

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

    10. lower-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sqrt.f64 N/A

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

    14. lower-sqrt.f64 N/A

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

    15. lower-/.f64 N/A

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

    16. lower-exp.f64 96.6

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

11. Applied rewrites 96.6%

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

13. Applied rewrites 96.6%

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

14. Final simplification 96.6%

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

Alternative 12: 96.0% accurate, 3.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 96.2%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 97.2

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

5. Applied rewrites 97.2%

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

6. Taylor expanded in z around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-exp.f64 95.9

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

8. Applied rewrites 95.9%

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

9. Taylor expanded in z around 0

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

    1. *-commutative N/A

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

    2. associate-*r* N/A

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

    3. lower-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower-sqrt.f64 N/A

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

    7. lower-PI.f64 N/A

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

    8. *-commutative N/A

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

    9. associate-/l* N/A

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

    10. lower-*.f64 N/A

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

    11. *-commutative N/A

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

    12. lower-*.f64 N/A

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

    13. lower-sqrt.f64 N/A

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

    14. lower-sqrt.f64 N/A

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

    15. lower-/.f64 N/A

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

    16. lower-exp.f64 96.6

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

11. Applied rewrites 96.6%

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

13. Applied rewrites 96.5%

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

14. Final simplification 96.5%

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

Reproduce

herbie shell --seed 2024267 
(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))))))