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

Average Error: 4.5 → 5.5

Time: 13.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \le 4.442830040929055 \cdot 10^{-308} \lor \neg \left(y \le 2.91684953680309792 \cdot 10^{127}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \frac{x \cdot \left(-\frac{t}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)}{\sqrt[3]{1 - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z} \cdot \left(-y\right) + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((y <= 4.442830040929055e-308) || !(y <= 2.916849536803098e+127))) {
		VAR = ((double) (((double) (x / ((double) (z / y)))) + ((double) (((double) (x * ((double) -(((double) (t / ((double) (((double) cbrt(((double) (1.0 - z)))) * ((double) cbrt(((double) (1.0 - z)))))))))))) / ((double) cbrt(((double) (1.0 - z))))))));
	} else {
		VAR = ((double) (((double) (((double) (x / ((double) -(z)))) * ((double) -(y)))) + ((double) (x * ((double) -(((double) (t / ((double) (1.0 - z))))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.1
Herbie	5.5

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

Derivation

1. Split input into 2 regimes

2. if y < 4.442830040929055e-308 or 2.916849536803098e+127 < y

   1. Initial program 6.0

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

   3. Applied sub-neg 6.0

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

   4. Applied distribute-lft-in 6.0

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

   6. Applied clear-num 6.1

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

   7. Applied un-div-inv 5.6

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

   9. Applied add-cube-cbrt 5.8

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

   10. Applied associate-/r* 5.8

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

   11. Applied distribute-neg-frac 5.8

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

   12. Applied associate-*r/ 6.2

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

   if 4.442830040929055e-308 < y < 2.916849536803098e+127

   1. Initial program 2.3

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

   3. Applied sub-neg 2.3

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

   4. Applied distribute-lft-in 2.3

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

   6. Applied clear-num 2.4

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

   7. Applied un-div-inv 2.2

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

   9. Applied frac-2neg 2.2

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

   10. Applied associate-/r/ 4.3

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

4. Final simplification 5.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le 4.442830040929055 \cdot 10^{-308} \lor \neg \left(y \le 2.91684953680309792 \cdot 10^{127}\right):\\ \;\;\;\;\frac{x}{\frac{z}{y}} + \frac{x \cdot \left(-\frac{t}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)}{\sqrt[3]{1 - z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-z} \cdot \left(-y\right) + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020114 
(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)))))