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

Average Error: 2.0 → 0.2

Time: 27.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 1.1454557351735624 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{{\left({a}^{1}\right)}^{1} \cdot \frac{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{a}\right)}^{1} \cdot \frac{\frac{1}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\right) \cdot x\\ \end{array}\]

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 temp;
	if ((a <= 1.1454557351735624e-09)) {
		temp = (1.0 / (pow(pow(a, 1.0), 1.0) * ((exp(((log((1.0 / z)) * y) + ((log((1.0 / a)) * t) + b))) * y) / x)));
	} else {
		temp = ((pow((1.0 / a), 1.0) * ((1.0 / exp(fma(y, log((1.0 / z)), fma(log((1.0 / a)), t, b)))) / y)) * x);
	}
	return temp;
}

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	11.8
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 < 1.1454557351735624e-09

   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

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

   3. Simplified 2.8

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

   5. Applied associate-*l/ 0.2

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

   6. Simplified 0.2

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

   8. Applied clear-num 0.2

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

   9. Taylor expanded around inf 0.1

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

   if 1.1454557351735624e-09 < a

   1. Initial program 3.0

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

   2. Taylor expanded around inf 3.0

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

   3. Simplified 0.2

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

   5. Applied *-un-lft-identity 0.2

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

   6. Applied div-inv 0.2

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

   7. Applied times-frac 0.2

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

   8. Simplified 0.2

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 1.1454557351735624 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{{\left({a}^{1}\right)}^{1} \cdot \frac{e^{\log \left(\frac{1}{z}\right) \cdot y + \left(\log \left(\frac{1}{a}\right) \cdot t + b\right)} \cdot y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\left({\left(\frac{1}{a}\right)}^{1} \cdot \frac{\frac{1}{e^{\mathsf{fma}\left(y, \log \left(\frac{1}{z}\right), \mathsf{fma}\left(\log \left(\frac{1}{a}\right), t, b\right)\right)}}}{y}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 +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))