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

Average Error: 4.0 → 4.0

Time: 19.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le 2.63919292179037968 \cdot 10^{171} \lor \neg \left(z \le 5.28755611609775836 \cdot 10^{215}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\frac{z}{t} \cdot \sqrt{t + a} + \left(b - c\right) \cdot \left(\left(\frac{2}{t \cdot 3} - \frac{5}{6}\right) - a\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) + t \cdot \left(3 \cdot \left(\frac{5}{6} \cdot \frac{5}{6} - a \cdot a\right)\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}}}\\ \end{array}\]

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 <= 2.6391929217903797e+171) || !(z <= 5.287556116097758e+215))) {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) log(((double) exp(((double) (((double) (((double) (z / t)) * ((double) sqrt(((double) (t + a)))))) + ((double) (((double) (b - c)) * ((double) (((double) (((double) (2.0 / ((double) (t * 3.0)))) - ((double) (5.0 / 6.0)))) - a))))))))))))))))))));
	} else {
		VAR = ((double) (x / ((double) (x + ((double) (y * ((double) exp(((double) (2.0 * ((double) (((double) (((double) (z * ((double) (((double) sqrt(((double) (t + a)))) * ((double) (t * ((double) (3.0 * ((double) (a - ((double) (5.0 / 6.0)))))))))))) + ((double) (t * ((double) (((double) (b - c)) * ((double) (((double) (2.0 * ((double) (a - ((double) (5.0 / 6.0)))))) + ((double) (t * ((double) (3.0 * ((double) (((double) (((double) (5.0 / 6.0)) * ((double) (5.0 / 6.0)))) - ((double) (a * a)))))))))))))))) / ((double) (t * ((double) (t * ((double) (3.0 * ((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	4.0
Target	3.4
Herbie	4.0

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

Derivation

1. Split input into 2 regimes

2. if z < 2.63919292179037968e171 or 5.28755611609775836e215 < z

   1. Initial program 3.7

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

   3. Applied add-log-exp 7.8

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

   4. Applied add-log-exp 16.4

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

   5. Applied diff-log 16.4

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

   6. Simplified 3.1

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

   if 2.63919292179037968e171 < z < 5.28755611609775836e215

   1. Initial program 10.3

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

   3. Applied flip-+ 11.5

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

   4. Applied frac-sub 16.1

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

   5. Applied associate-*r/ 16.1

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

   6. Applied frac-sub 26.2

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

   7. Simplified 27.2

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

   8. Simplified 27.2

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

4. Final simplification 4.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le 2.63919292179037968 \cdot 10^{171} \lor \neg \left(z \le 5.28755611609775836 \cdot 10^{215}\right):\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \log \left(e^{\frac{z}{t} \cdot \sqrt{t + a} + \left(b - c\right) \cdot \left(\left(\frac{2}{t \cdot 3} - \frac{5}{6}\right) - a\right)}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + y \cdot e^{2 \cdot \frac{z \cdot \left(\sqrt{t + a} \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)\right) + t \cdot \left(\left(b - c\right) \cdot \left(2 \cdot \left(a - \frac{5}{6}\right) + t \cdot \left(3 \cdot \left(\frac{5}{6} \cdot \frac{5}{6} - a \cdot a\right)\right)\right)\right)}{t \cdot \left(t \cdot \left(3 \cdot \left(a - \frac{5}{6}\right)\right)\right)}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020185 
(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)))))))))))