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

Average Error: 4.4 → 1.9

Time: 4.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + \frac{-x}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.3464474438162144 \cdot 10^{295}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

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 ((((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))))) <= 0.0)) {
		VAR = ((double) (((double) (((double) (x * ((double) (((double) (((double) cbrt(y)) * ((double) cbrt(y)))) / ((double) (((double) cbrt(z)) * ((double) cbrt(z)))))))) * ((double) (((double) cbrt(y)) / ((double) cbrt(z)))))) + ((double) (((double) (((double) -(x)) / ((double) (((double) cbrt(((double) (1.0 - z)))) * ((double) cbrt(((double) (1.0 - z)))))))) * ((double) (t / ((double) cbrt(((double) (1.0 - z))))))))));
	} else {
		double VAR_1;
		if ((((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))))) <= 1.3464474438162144e+295)) {
			VAR_1 = ((double) (((double) sqrt(((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))))))) * ((double) sqrt(((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z))))))))))));
		} else {
			VAR_1 = ((double) (((double) (x * ((double) (((double) (y * ((double) (1.0 - z)))) - ((double) (z * t)))))) / ((double) (z * ((double) (1.0 - z))))));
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.1
Herbie	1.9

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

Derivation

1. Split input into 3 regimes

2. if (* x (- (/ y z) (/ t (- 1.0 z)))) < 0.0

   1. Initial program 4.8

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

   3. Applied sub-neg 4.8

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

   4. Applied distribute-lft-in 4.8

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

   6. Applied add-cube-cbrt 5.0

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

   7. Applied *-un-lft-identity 5.0

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

   8. Applied times-frac 5.0

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

   9. Applied distribute-lft-neg-in 5.0

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

   10. Applied associate-*r* 5.4

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

   11. Simplified 5.4

       \[\leadsto x \cdot \frac{y}{z} + \color{blue}{\frac{-x}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}} \cdot \frac{t}{\sqrt[3]{1 - z}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 5.9

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

   14. Applied add-cube-cbrt 6.0

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

   15. Applied times-frac 6.0

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

   16. Applied associate-*r* 2.7

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

   if 0.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 1.3464474438162144e+295

   1. Initial program 0.3

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

   3. Applied add-sqr-sqrt 0.7

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

   if 1.3464474438162144e+295 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 52.9

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

   3. Applied frac-sub 56.6

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

   4. Applied associate-*r/ 5.3

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

4. Final simplification 1.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + \frac{-x}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.3464474438162144 \cdot 10^{295}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020122 
(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)))))