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

Average Error: 1.9 → 0.2

Time: 16.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 2.36502293090958373 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}} \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}{y}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t, double a, double b) {
	double VAR;
	if ((a <= 2.3650229309095837e-89)) {
		VAR = ((double) (((double) (((double) (((double) cbrt(((double) (x * ((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) exp(((double) (b + ((double) (((double) (((double) log(((double) (1.0 / z)))) * y)) + ((double) (((double) log(((double) (1.0 / a)))) * t)))))))))))))) * ((double) cbrt(((double) (x * ((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) exp(((double) (b + ((double) (((double) (((double) log(((double) (1.0 / z)))) * y)) + ((double) (((double) log(((double) (1.0 / a)))) * t)))))))))))))))) * ((double) cbrt(((double) (x * ((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) exp(((double) (b + ((double) (((double) (((double) log(((double) (1.0 / z)))) * y)) + ((double) (((double) log(((double) (1.0 / a)))) * t)))))))))))))))) / y));
	} else {
		VAR = ((double) (x * ((double) (((double) (((double) pow(((double) (1.0 / a)), 1.0)) / ((double) exp(((double) (b + ((double) (((double) (((double) log(((double) (1.0 / z)))) * y)) + ((double) (((double) log(((double) (1.0 / a)))) * t)))))))))) / 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	1.9
Target	10.4
Herbie	0.2

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

Derivation

1. Split input into 2 regimes

2. if a < 2.36502293090958373e-89

   1. Initial program 0.7

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

   2. Taylor expanded around inf 0.8

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

   3. Simplified 0.1

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

   5. Applied add-cube-cbrt 0.3

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}} \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}}}{y}\]

   if 2.36502293090958373e-89 < a

   1. Initial program 2.5

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

   2. Taylor expanded around inf 2.5

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

   3. Simplified 1.8

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

   5. Applied *-un-lft-identity 1.8

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

   6. Applied times-frac 0.2

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

   7. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 2.36502293090958373 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(\sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}} \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}\right) \cdot \sqrt[3]{x \cdot \frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\frac{{\left(\frac{1}{a}\right)}^{1}}{e^{b + \left(\log \left(\frac{1}{z}\right) \cdot y + \log \left(\frac{1}{a}\right) \cdot t\right)}}}{y}\\ \end{array}\]

Reproduce

herbie shell --seed 2020177 
(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 1.0)) y)) (- (+ b 1.0) (* y (log z)))) (if (< t 852031.2288374073) (/ (* (/ x y) (pow a (- t 1.0))) (exp (- b (* (log z) y)))) (/ (* x (/ (pow a (- t 1.0)) y)) (- (+ b 1.0) (* y (log z))))))

  (/ (* x (exp (- (+ (* y (log z)) (* (- t 1.0) (log a))) b))) y))