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

Average Error: 3.9 → 2.2

Time: 13.2s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -4.3351175562285256 \cdot 10^{156}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\left(b - c\right) \cdot \left(\frac{z}{t} \cdot \frac{\sqrt{t + a}}{b - c} - \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{1}{t} \cdot \left(z \cdot \sqrt{t + a} - t \cdot \frac{a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)}{\frac{1}{b - c}}\right)\right)}}\\ \end{array}\]

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (((double) (z * ((double) sqrt(((double) (t + a)))))) / t)) - ((double) (((double) (b - c)) * ((double) (((double) (a + ((double) (5.0 / 6.0)))) - ((double) (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 ((z <= -4.3351175562285256e+156)) {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), ((double) (((double) (b - c)) * ((double) (((double) (((double) (z / t)) * ((double) (((double) sqrt(((double) (t + a)))) / ((double) (b - c)))))) - ((double) (a + ((double) (((double) (5.0 / 6.0)) - ((double) (2.0 / ((double) (t * 3.0))))))))))))))))))));
	} else {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) pow(((double) exp(2.0)), ((double) (((double) (1.0 / t)) * ((double) (((double) (z * ((double) sqrt(((double) (t + a)))))) - ((double) (t * ((double) (((double) (a + ((double) (((double) (5.0 / 6.0)) - ((double) (2.0 / ((double) (t * 3.0)))))))) / ((double) (1.0 / ((double) (b - c))))))))))))))))))));
	}
	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	3.9
Target	3.2
Herbie	2.2

\[\begin{array}{l} \mathbf{if}\;t \lt -2.1183266448915811 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}\\ \mathbf{elif}\;t \lt 5.19658877065154709 \cdot 10^{-123}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(3 \cdot t\right) \cdot \left(a - \frac{5}{6}\right)\right) - \left(\left(\frac{5}{6} + a\right) \cdot \left(3 \cdot t\right) - 2\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot \left(\left(b - c\right) \cdot t\right)\right)}{\left(\left(t \cdot t\right) \cdot 3\right) \cdot \left(a - \frac{5}{6}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \frac{2}{t \cdot 3}\right)\right)}}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if z < -4.3351175562285256e156

   1. Initial program 11.3

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

   2. Simplified 6.5

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

   4. Applied flip-- 10.9

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

   5. Applied associate-*l/ 12.2

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

   6. Applied associate-*r/ 16.4

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

   7. Applied frac-sub 27.7

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

   8. Simplified 27.7

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

   10. Applied flip-+ 37.8

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

   11. Applied associate-*r/ 39.6

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

   12. Applied associate-/r/ 39.6

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

   13. Simplified 3.7

       \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\color{blue}{\left(\frac{z}{t} \cdot \left(1 \cdot \frac{\sqrt{t + a}}{b - c}\right) - \frac{a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)}{1}\right)} \cdot \left(b - c\right)\right)}}\]

   if -4.3351175562285256e156 < z

   1. Initial program 2.9

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

   2. Simplified 3.0

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

   4. Applied flip-- 6.8

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

   5. Applied associate-*l/ 7.1

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

   6. Applied associate-*r/ 7.0

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

   7. Applied frac-sub 19.9

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

   8. Simplified 20.5

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

   10. Applied *-un-lft-identity 20.5

       \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{\color{blue}{1 \cdot \left(z \cdot \left(\sqrt{t + a} \cdot \left(b + c\right)\right) - t \cdot \left(\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(b \cdot b - c \cdot c\right)\right)\right)}}{t \cdot \left(b + c\right)}\right)}}\]

   11. Applied times-frac 12.6

       \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\color{blue}{\left(\frac{1}{t} \cdot \frac{z \cdot \left(\sqrt{t + a} \cdot \left(b + c\right)\right) - t \cdot \left(\left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right) \cdot \left(b \cdot b - c \cdot c\right)\right)}{b + c}\right)}}}\]

   12. Simplified 2.0

       \[\leadsto \frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{1}{t} \cdot \color{blue}{\left(z \cdot \left(1 \cdot \sqrt{t + a}\right) - t \cdot \frac{a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)}{\frac{1}{b - c}}\right)}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -4.3351175562285256 \cdot 10^{156}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\left(b - c\right) \cdot \left(\frac{z}{t} \cdot \frac{\sqrt{t + a}}{b - c} - \left(a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)\right)\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot {\left(e^{2}\right)}^{\left(\frac{1}{t} \cdot \left(z \cdot \sqrt{t + a} - t \cdot \frac{a + \left(\frac{5}{6} - \frac{2}{t \cdot 3}\right)}{\frac{1}{b - c}}\right)\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020181 
(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)))))))))))