Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I

Average Error: 4.0 → 3.7

Time: 12.2s

Precision: binary64

Math

\[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -1.1071516139533105 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + y \cdot \sqrt[3]{{\left({\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}\right)}^{3}}}\\ \mathbf{elif}\;t \leq 1.8925673024019224 \cdot 10^{-306}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(c - b\right) \cdot \left(t \cdot \left(3 \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right) + 2 \cdot \left(\frac{5}{6} - a\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)} \cdot \left(\sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)} \cdot \sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)\right)}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) ((((double) (z * ((double) sqrt(((double) (t + a)))))) / t) - ((double) (((double) (b - c)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))))))))));
}

↓

double code(double x, double y, double z, double t, double a, double b, double c) {
	double VAR;
	if ((t <= -1.1071516139533105e-97)) {
		VAR = (x / ((double) (x + ((double) (y * ((double) cbrt(((double) pow(((double) pow(((double) exp(2.0)), ((double) (((double) (c * 0.8333333333333334)) + ((double) (a * ((double) (c - b)))))))), 3.0)))))))));
	} else {
		double VAR_1;
		if ((t <= 1.8925673024019224e-306)) {
			VAR_1 = (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), (((double) (((double) (z * ((double) (((double) sqrt(((double) (t + a)))) * ((double) (t * ((double) (3.0 * ((double) (a - (5.0 / 6.0))))))))))) + ((double) (t * ((double) (((double) (c - b)) * ((double) (((double) (t * ((double) (3.0 * ((double) (((double) (a * a)) - ((double) ((5.0 / 6.0) * (5.0 / 6.0))))))))) + ((double) (2.0 * ((double) ((5.0 / 6.0) - a)))))))))))) / ((double) (t * ((double) (t * ((double) (3.0 * ((double) (a - (5.0 / 6.0)))))))))))))))));
		} else {
			VAR_1 = (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), ((double) (((double) cbrt(((double) (((double) (z * (((double) sqrt(((double) (t + a)))) / t))) + ((double) (((double) (c - b)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))) * ((double) (((double) cbrt(((double) (((double) (z * (((double) sqrt(((double) (t + a)))) / t))) + ((double) (((double) (c - b)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))) * ((double) cbrt(((double) (((double) (z * (((double) sqrt(((double) (t + a)))) / t))) + ((double) (((double) (c - b)) * ((double) (((double) (a + (5.0 / 6.0))) - (2.0 / ((double) (t * 3.0))))))))))))))))))))));
		}
		VAR = VAR_1;
	}
	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

Bits error versus c

Target

Original	4.0
Target	3.1
Herbie	3.7

\[\begin{array}{l} \mathbf{if}\;t < -2.118326644891581 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t < 5.196588770651547 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if t < -1.10715161395331051e-97

   1. Initial program 2.6

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Simplified 1.4

      \[\leadsto \color{blue}{\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}}\]
   3. Using strategy rm

   4. Applied add-log-exp 5.1

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\color{blue}{\log \left(e^{\frac{2}{t \cdot 3}}\right)} - \left(a + \frac{5}{6}\right)\right)\right)}}\]

   5. Taylor expanded around inf 10.0

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\left(a \cdot c + 0.8333333333333334 \cdot c\right) - a \cdot b\right)}}}\]

   6. Simplified 7.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\color{blue}{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}}}\]
   7. Using strategy rm

   8. Applied add-cbrt-cube 7.4

      \[\leadsto \frac{x}{x + y \cdot \color{blue}{\sqrt[3]{\left({\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)} \cdot {\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}\right) \cdot {\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}}}}\]

   9. Simplified 7.4

      \[\leadsto \frac{x}{x + y \cdot \sqrt[3]{\color{blue}{{\left({\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}\right)}^{3}}}}\]

   if -1.10715161395331051e-97 < t < 1.89256730240192235e-306

   1. Initial program 8.8

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Simplified 10.8

      \[\leadsto \color{blue}{\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}}\]
   3. Using strategy rm

   4. Applied flip-+ 13.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}}\right)\right)}}\]

   5. Applied frac-sub 13.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \color{blue}{\frac{2 \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot 3\right) \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)}{\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}\right)}}\]

   6. Applied associate-*r/ 13.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \color{blue}{\frac{\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot 3\right) \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right)}{\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}\right)}}\]

   7. Applied associate-*r/ 11.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\frac{z \cdot \sqrt{t + a}}{t}} + \frac{\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot 3\right) \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right)}{\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}\right)}}\]

   8. Applied frac-add 7.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) - \left(t \cdot 3\right) \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right)\right)}{t \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right)}\right)}}}\]

   9. Simplified 8.4

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{\color{blue}{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) - t \cdot \left(3 \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right)\right)\right)}}{t \cdot \left(\left(t \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)\right)}\right)}}\]

   10. Simplified 8.4

       \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) - t \cdot \left(3 \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right)\right)\right)}{\color{blue}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}}\right)}}\]

   if 1.89256730240192235e-306 < t

   1. Initial program 3.2

      \[\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\]

   2. Simplified 2.2

      \[\leadsto \color{blue}{\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)\right)}}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 2.2

      \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\left(\sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)} \cdot \sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)}\right) \cdot \sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(b - c\right) \cdot \left(\frac{2}{t \cdot 3} - \left(a + \frac{5}{6}\right)\right)}\right)}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.1071516139533105 \cdot 10^{-97}:\\ \;\;\;\;\frac{x}{x + y \cdot \sqrt[3]{{\left({\left(e^{2}\right)}^{\left(c \cdot 0.8333333333333334 + a \cdot \left(c - b\right)\right)}\right)}^{3}}}\\ \mathbf{elif}\;t \leq 1.8925673024019224 \cdot 10^{-306}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(c - b\right) \cdot \left(t \cdot \left(3 \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right)\right) + 2 \cdot \left(\frac{5}{6} - a\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)} \cdot \left(\sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)} \cdot \sqrt[3]{z \cdot \frac{\sqrt{t + a}}{t} + \left(c - b\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020196 
(FPCore (x y z t a b c)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, I"
  :precision binary64

  :herbie-target
  (if (< t -2.118326644891581e-50) (/ x (+ x (* y (exp (* 2.0 (- (+ (* a c) (* 0.8333333333333334 c)) (* a b))))))) (if (< t 5.196588770651547e-123) (/ x (+ x (* y (exp (* 2.0 (/ (- (* (* z (sqrt (+ t a))) (* (* 3.0 t) (- a (/ 5.0 6.0)))) (* (- (* (+ (/ 5.0 6.0) a) (* 3.0 t)) 2.0) (* (- a (/ 5.0 6.0)) (* (- b c) t)))) (* (* (* t t) 3.0) (- a (/ 5.0 6.0))))))))) (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0))))))))))))

  (/ x (+ x (* y (exp (* 2.0 (- (/ (* z (sqrt (+ t a))) t) (* (- b c) (- (+ a (/ 5.0 6.0)) (/ 2.0 (* t 3.0)))))))))))