Jmat.Real.gamma, branch z less than 0.5

Percentage Accurate: 97.2% → 97.2%
Time: 9.4s
Alternatives: 1
Speedup: 1.0×

Specification

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

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 1 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 97.2% accurate, 1.0× speedup?

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

Alternative 1: 97.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - z\right) - 1\\ t_1 := 7 + t\_0\\ \left(\left(\frac{1.5056327351493116 \cdot 10^{-7}}{8 + t\_0} + \left(\frac{9.984369578019572 \cdot 10^{-6}}{t\_1} + \left(\frac{-0.13857109526572012}{6 + t\_0} + \left(\frac{12.507343278686905}{5 + t\_0} + \left(\frac{-176.6150291621406}{4 + t\_0} + \left(\frac{771.3234287776531}{3 + t\_0} + \left(\frac{-1259.1392167224028}{t\_0 + 2} + \left(\frac{676.5203681218851}{t\_0 + 1} + 0.9999999999998099\right)\right)\right)\right)\right)\right)\right)\right) \cdot \left(e^{\left(\left(1 - \left(1 - z\right)\right) - 7\right) - 0.5} \cdot \left({\left(0.5 + t\_1\right)}^{\left(0.5 + t\_0\right)} \cdot \sqrt{2 \cdot \mathsf{PI}\left(\right)}\right)\right)\right) \cdot \frac{\mathsf{PI}\left(\right)}{\sin \left(z \cdot \mathsf{PI}\left(\right)\right)} \end{array} \end{array} \]
(FPCore (z)
 :precision binary64
 (let* ((t_0 (- (- 1.0 z) 1.0)) (t_1 (+ 7.0 t_0)))
   (*
    (*
     (+
      (/ 1.5056327351493116e-7 (+ 8.0 t_0))
      (+
       (/ 9.984369578019572e-6 t_1)
       (+
        (/ -0.13857109526572012 (+ 6.0 t_0))
        (+
         (/ 12.507343278686905 (+ 5.0 t_0))
         (+
          (/ -176.6150291621406 (+ 4.0 t_0))
          (+
           (/ 771.3234287776531 (+ 3.0 t_0))
           (+
            (/ -1259.1392167224028 (+ t_0 2.0))
            (+ (/ 676.5203681218851 (+ t_0 1.0)) 0.9999999999998099))))))))
     (*
      (exp (- (- (- 1.0 (- 1.0 z)) 7.0) 0.5))
      (* (pow (+ 0.5 t_1) (+ 0.5 t_0)) (sqrt (* 2.0 (PI))))))
    (/ (PI) (sin (* z (PI)))))))
\begin{array}{l}

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

  1. Initial program 95.9%

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

  3. Final simplification95.9%

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

Reproduce

?
herbie shell --seed 2024352 
(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))))))