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

Average Error: 4.7 → 2.8

Time: 4.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

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 * (cbrt((1.0 - z)) * cbrt((1.0 - z)))) - (z * (1.0 * (t / cbrt((1.0 - z))))))) / (z * (cbrt((1.0 - z)) * cbrt((1.0 - z)))));
	} else {
		VAR = (x * ((y / z) - (t / (1.0 - z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.3
Herbie	2.8

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0

   1. Initial program 64.0

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

   3. Applied add-cube-cbrt 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 64.0

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

   7. Applied associate-*l/ 64.0

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

   8. Applied frac-sub 64.0

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

   9. Applied associate-*r/ 0.3

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

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

   1. Initial program 2.9

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020103 
(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)))))