Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A

Average Error: 2.0 → 2.4

Time: 10.2s

Precision: 64

Math

\[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]

↓

\[\begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le -546.37453138819501 \lor \neg \left(\left(t - 1\right) \cdot \log a \le 86.757485924579555\right):\\ \;\;\;\;\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1}}{\frac{\frac{y}{\frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{{z}^{y}}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) / y);
}

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if (((((t - 1.0) * log(a)) <= -546.374531388195) || !(((t - 1.0) * log(a)) <= 86.75748592457956))) {
		VAR = ((x * exp((((y * log(z)) + ((t - 1.0) * log(a))) - b))) * (1.0 / y));
	} else {
		VAR = ((x / 1.0) / ((y / ((1.0 / pow(a, 1.0)) / exp(fma(log((1.0 / a)), t, b)))) / pow(z, y)));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Bits error versus b

Target

Original	2.0
Target	11.4
Herbie	2.4

\[\begin{array}{l} \mathbf{if}\;t \lt -0.88458485041274715:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \mathbf{elif}\;t \lt 852031.22883740731:\\ \;\;\;\;\frac{\frac{x}{y} \cdot {a}^{\left(t - 1\right)}}{e^{b - \log z \cdot y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \frac{{a}^{\left(t - 1\right)}}{y}}{\left(b + 1\right) - y \cdot \log z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* (- t 1.0) (log a)) < -546.374531388195 or 86.75748592457956 < (* (- t 1.0) (log a))

   1. Initial program 1.1

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied div-inv 1.1

      \[\leadsto \color{blue}{\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}}\]

   if -546.374531388195 < (* (- t 1.0) (log a)) < 86.75748592457956

   1. Initial program 4.8

      \[\frac{x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}}{y}\]
   2. Using strategy rm

   3. Applied div-inv 4.8

      \[\leadsto \color{blue}{\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}}\]

   4. Taylor expanded around inf 4.8

      \[\leadsto \color{blue}{\frac{x \cdot e^{1 \cdot \log \left(\frac{1}{a}\right) - \left(y \cdot \log \left(\frac{1}{z}\right) + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)\right)}}{y}}\]

   5. Simplified 3.6

      \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{a}^{1}}}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}}\]
   6. Using strategy rm

   7. Applied fma-udef 3.6

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{e^{\color{blue}{y \cdot \log \left(\frac{1}{z}\right) + \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{y}\]

   8. Applied exp-sum 9.6

      \[\leadsto \frac{\frac{\frac{x}{{a}^{1}}}{\color{blue}{e^{y \cdot \log \left(\frac{1}{z}\right)} \cdot e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{y}\]

   9. Applied div-inv 9.6

      \[\leadsto \frac{\frac{\color{blue}{x \cdot \frac{1}{{a}^{1}}}}{e^{y \cdot \log \left(\frac{1}{z}\right)} \cdot e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}{y}\]

   10. Applied times-frac 9.5

       \[\leadsto \frac{\color{blue}{\frac{x}{e^{y \cdot \log \left(\frac{1}{z}\right)}} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{y}\]
   11. Using strategy rm

   12. Applied log-rec 9.5

       \[\leadsto \frac{\frac{x}{e^{y \cdot \color{blue}{\left(-\log z\right)}}} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}{y}\]

   13. Applied distribute-rgt-neg-out 9.5

       \[\leadsto \frac{\frac{x}{e^{\color{blue}{-y \cdot \log z}}} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}{y}\]

   14. Applied exp-neg 9.5

       \[\leadsto \frac{\frac{x}{\color{blue}{\frac{1}{e^{y \cdot \log z}}}} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}{y}\]

   15. Applied associate-/r/ 9.5

       \[\leadsto \frac{\color{blue}{\left(\frac{x}{1} \cdot e^{y \cdot \log z}\right)} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}{y}\]

   16. Applied associate-*l* 9.5

       \[\leadsto \frac{\color{blue}{\frac{x}{1} \cdot \left(e^{y \cdot \log z} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}\right)}}{y}\]

   17. Applied associate-/l* 6.4

       \[\leadsto \color{blue}{\frac{\frac{x}{1}}{\frac{y}{e^{y \cdot \log z} \cdot \frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}}\]

   18. Simplified 6.4

       \[\leadsto \frac{\frac{x}{1}}{\color{blue}{\frac{\frac{y}{\frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{{z}^{y}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - 1\right) \cdot \log a \le -546.37453138819501 \lor \neg \left(\left(t - 1\right) \cdot \log a \le 86.757485924579555\right):\\ \;\;\;\;\left(x \cdot e^{\left(y \cdot \log z + \left(t - 1\right) \cdot \log a\right) - b}\right) \cdot \frac{1}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{1}}{\frac{\frac{y}{\frac{\frac{1}{{a}^{1}}}{e^{\mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)}}}}{{z}^{y}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +o rules:numerics
(FPCore (x y z t a b)
  :name "Numeric.SpecFunctions:incompleteBetaWorker from math-functions-0.1.5.2, A"
  :precision binary64

  :herbie-target
  (if (< t -0.8845848504127471) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1)) y)) (- (+ b 1) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1) (log a))) b))) y))