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

Average Error: 4.5 → 2.7

Time: 4.0s

Precision: 64

Math

\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.00334370782838383 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.6204358638850966 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.37903974306552555 \cdot 10^{137}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\right) \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return (x * ((y / z) - (t / (1.0 - z))));
}

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if ((((y / z) - (t / (1.0 - z))) <= -inf.0)) {
		VAR = (((x * y) / z) + ((x / (1.0 - z)) * -t));
	} else {
		double VAR_1;
		if ((((y / z) - (t / (1.0 - z))) <= -1.0033437078283838e-202)) {
			VAR_1 = ((x * (y / z)) + ((x * -t) / (1.0 - z)));
		} else {
			double VAR_2;
			if ((((y / z) - (t / (1.0 - z))) <= 8.620435863885097e-216)) {
				VAR_2 = (((x * y) / z) + ((x / (1.0 - z)) * -t));
			} else {
				double VAR_3;
				if ((((y / z) - (t / (1.0 - z))) <= 3.3790397430655255e+137)) {
					VAR_3 = ((cbrt((x * ((y / z) - (t / (1.0 - z))))) * cbrt((x * ((y / z) - (t / (1.0 - z)))))) * cbrt((x * ((y / z) - (t / (1.0 - z))))));
				} else {
					VAR_3 = (((x / (cbrt(z) * cbrt(z))) * (y / cbrt(z))) + ((x / (1.0 - z)) * -t));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.1
Herbie	2.7

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or -1.0033437078283838e-202 < (- (/ y z) (/ t (- 1.0 z))) < 8.620435863885097e-216

   1. Initial program 19.9

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied clear-num 20.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\frac{1 - z}{t}}}\right)\]
   4. Using strategy rm

   5. Applied sub-neg 20.5

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

   6. Applied distribute-lft-in 20.5

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

   7. Simplified 16.4

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{\frac{x}{1 - z} \cdot \left(-t\right)}\]
   8. Using strategy rm

   9. Applied associate-*r/ 0.8

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

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -1.0033437078283838e-202

   1. Initial program 0.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied clear-num 0.3

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\frac{1 - z}{t}}}\right)\]
   4. Using strategy rm

   5. Applied sub-neg 0.3

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

   6. Applied distribute-lft-in 0.3

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

   7. Simplified 3.1

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{\frac{x}{1 - z} \cdot \left(-t\right)}\]
   8. Using strategy rm

   9. Applied associate-*l/ 3.7

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

   if 8.620435863885097e-216 < (- (/ y z) (/ t (- 1.0 z))) < 3.3790397430655255e+137

   1. Initial program 0.2

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 1.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\right) \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}}\]

   if 3.3790397430655255e+137 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 11.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
   2. Using strategy rm

   3. Applied clear-num 11.1

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\frac{1 - z}{t}}}\right)\]
   4. Using strategy rm

   5. Applied sub-neg 11.1

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

   6. Applied distribute-lft-in 11.1

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

   7. Simplified 11.7

      \[\leadsto x \cdot \frac{y}{z} + \color{blue}{\frac{x}{1 - z} \cdot \left(-t\right)}\]
   8. Using strategy rm

   9. Applied add-cube-cbrt 12.2

      \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}} + \frac{x}{1 - z} \cdot \left(-t\right)\]

   10. Applied *-un-lft-identity 12.2

       \[\leadsto x \cdot \frac{\color{blue}{1 \cdot y}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} + \frac{x}{1 - z} \cdot \left(-t\right)\]

   11. Applied times-frac 12.2

       \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}}\right)} + \frac{x}{1 - z} \cdot \left(-t\right)\]

   12. Applied associate-*r* 4.7

       \[\leadsto \color{blue}{\left(x \cdot \frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{y}{\sqrt[3]{z}}} + \frac{x}{1 - z} \cdot \left(-t\right)\]

   13. Simplified 4.7

       \[\leadsto \color{blue}{\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}}} \cdot \frac{y}{\sqrt[3]{z}} + \frac{x}{1 - z} \cdot \left(-t\right)\]
3. Recombined 4 regimes into one program.

4. Final simplification 2.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.00334370782838383 \cdot 10^{-202}:\\ \;\;\;\;x \cdot \frac{y}{z} + \frac{x \cdot \left(-t\right)}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.6204358638850966 \cdot 10^{-216}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.37903974306552555 \cdot 10^{137}:\\ \;\;\;\;\left(\sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\right) \cdot \sqrt[3]{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + \frac{x}{1 - z} \cdot \left(-t\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020105 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))