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

Average Error: 3.9 → 10.2

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -2.48698002866519163 \cdot 10^{-125} \lor \neg \left(t \le 1.1853839002074552 \cdot 10^{139}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{t \cdot \left(z \cdot \left(\sqrt{t + a} \cdot \left(a - \frac{5}{6}\right)\right) - \left(b - c\right) \cdot \left(t \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) - \left(a - \frac{5}{6}\right) \cdot 0.66666666666666663\right)\right)}{t \cdot \left(t \cdot \left(a - \frac{5}{6}\right)\right)}}}\\ \end{array}\]

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 (((t <= -2.4869800286651916e-125) || !(t <= 1.1853839002074552e+139))) {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (c * ((double) (a + 0.8333333333333334)))) - ((double) (a * b))))))))))))));
	} else {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (t * ((double) (((double) (z * ((double) (((double) sqrt(((double) (t + a)))) * ((double) (a - ((double) (5.0 / 6.0)))))))) - ((double) (((double) (b - c)) * ((double) (((double) (t * ((double) (((double) (a * a)) - ((double) (((double) (5.0 / 6.0)) * ((double) (5.0 / 6.0)))))))) - ((double) (((double) (a - ((double) (5.0 / 6.0)))) * 0.6666666666666666)))))))))) / ((double) (t * ((double) (t * ((double) (a - ((double) (5.0 / 6.0))))))))))))))))))));
	}
	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	3.0
Herbie	10.2

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

Derivation

1. Split input into 2 regimes

2. if t < -2.48698002866519163e-125 or 1.1853839002074552e139 < t

   1. Initial program 3.5

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

   2. Taylor expanded around inf 11.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(\left(a \cdot c + 0.83333333333333337 \cdot c\right) - a \cdot b\right)}}}\]

   3. Simplified 11.1

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}}\]

   if -2.48698002866519163e-125 < t < 1.1853839002074552e139

   1. Initial program 4.3

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

   2. Taylor expanded around 0 4.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\left(a + \frac{5}{6}\right) - \color{blue}{\frac{0.66666666666666663}{t}}\right)\right)}}\]
   3. Using strategy rm

   4. Applied flip-+ 6.3

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \left(\color{blue}{\frac{a \cdot a - \frac{5}{6} \cdot \frac{5}{6}}{a - \frac{5}{6}}} - \frac{0.66666666666666663}{t}\right)\right)}}\]

   5. Applied frac-sub 7.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \left(b - c\right) \cdot \color{blue}{\frac{\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot t - \left(a - \frac{5}{6}\right) \cdot 0.66666666666666663}{\left(a - \frac{5}{6}\right) \cdot t}}\right)}}\]

   6. Applied associate-*r/ 7.5

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \left(\frac{z \cdot \sqrt{t + a}}{t} - \color{blue}{\frac{\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot t - \left(a - \frac{5}{6}\right) \cdot 0.66666666666666663\right)}{\left(a - \frac{5}{6}\right) \cdot t}}\right)}}\]

   7. Applied frac-sub 8.4

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \color{blue}{\frac{\left(z \cdot \sqrt{t + a}\right) \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right) - t \cdot \left(\left(b - c\right) \cdot \left(\left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) \cdot t - \left(a - \frac{5}{6}\right) \cdot 0.66666666666666663\right)\right)}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right)}}}}\]

   8. Simplified 9.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{\color{blue}{t \cdot \left(z \cdot \left(\sqrt{t + a} \cdot \left(a - \frac{5}{6}\right)\right) - \left(b - c\right) \cdot \left(t \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) - 0.66666666666666663 \cdot \left(a - \frac{5}{6}\right)\right)\right)}}{t \cdot \left(\left(a - \frac{5}{6}\right) \cdot t\right)}}}\]

   9. Simplified 9.6

      \[\leadsto \frac{x}{x + y \cdot e^{2 \cdot \frac{t \cdot \left(z \cdot \left(\sqrt{t + a} \cdot \left(a - \frac{5}{6}\right)\right) - \left(b - c\right) \cdot \left(t \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) - 0.66666666666666663 \cdot \left(a - \frac{5}{6}\right)\right)\right)}{\color{blue}{t \cdot \left(t \cdot \left(a - \frac{5}{6}\right)\right)}}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -2.48698002866519163 \cdot 10^{-125} \lor \neg \left(t \le 1.1853839002074552 \cdot 10^{139}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \left(c \cdot \left(a + 0.83333333333333337\right) - a \cdot b\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{t \cdot \left(z \cdot \left(\sqrt{t + a} \cdot \left(a - \frac{5}{6}\right)\right) - \left(b - c\right) \cdot \left(t \cdot \left(a \cdot a - \frac{5}{6} \cdot \frac{5}{6}\right) - \left(a - \frac{5}{6}\right) \cdot 0.66666666666666663\right)\right)}{t \cdot \left(t \cdot \left(a - \frac{5}{6}\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020179 
(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)))))))))))