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

Average Error: 4.4 → 2.0

Time: 7.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -1.8620952498364704 \cdot 10^{304}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -7.1082852077845676 \cdot 10^{-296}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, -\frac{t}{1 - z} \cdot 1\right) + \frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 5.18036198201241854 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 2.1182887113062035 \cdot 10^{251}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

C

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

↓

double code(double x, double y, double z, double t) {
	double VAR;
	if (((x * ((y / z) - (t / (1.0 - z)))) <= -1.8620952498364704e+304)) {
		VAR = ((x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z)));
	} else {
		double VAR_1;
		if (((x * ((y / z) - (t / (1.0 - z)))) <= -7.108285207784568e-296)) {
			VAR_1 = (x * (fma((1.0 / (cbrt(z) * cbrt(z))), (y / cbrt(z)), -((t / (1.0 - z)) * 1.0)) + ((t / (1.0 - z)) * (-1.0 + 1.0))));
		} else {
			double VAR_2;
			if (((x * ((y / z) - (t / (1.0 - z)))) <= 5.1803619820124185e-257)) {
				VAR_2 = fma((x / z), y, fma(1.0, ((t * x) / pow(z, 2.0)), ((t * x) / z)));
			} else {
				double VAR_3;
				if (((x * ((y / z) - (t / (1.0 - z)))) <= 2.1182887113062035e+251)) {
					VAR_3 = (sqrt((x * ((y / z) - (t / (1.0 - z))))) * sqrt((x * ((y / z) - (t / (1.0 - z))))));
				} else {
					VAR_3 = ((x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z)));
				}
				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	4.4
Target	4.2
Herbie	2.0

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

Derivation

1. Split input into 4 regimes

2. if (* x (- (/ y z) (/ t (- 1.0 z)))) < -1.8620952498364704e+304 or 2.1182887113062035e+251 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 38.2

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

   3. Applied frac-sub 43.3

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

   4. Applied associate-*r/ 9.0

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

   if -1.8620952498364704e+304 < (* x (- (/ y z) (/ t (- 1.0 z)))) < -7.108285207784568e-296

   1. Initial program 0.3

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

   3. Applied add-cube-cbrt 0.8

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

   4. Applied add-cube-cbrt 1.4

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

   5. Applied *-un-lft-identity 1.4

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

   6. Applied times-frac 1.4

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

   7. Applied prod-diff 1.4

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

   8. Simplified 0.9

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

   9. Simplified 0.9

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

   if -7.108285207784568e-296 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 5.1803619820124185e-257

   1. Initial program 6.5

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

   3. Applied add-cube-cbrt 6.6

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

   4. Applied add-cube-cbrt 6.6

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

   5. Applied *-un-lft-identity 6.6

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

   6. Applied times-frac 6.6

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

   7. Applied prod-diff 6.6

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

   8. Simplified 6.6

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

   9. Simplified 6.6

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

   10. Taylor expanded around inf 1.9

       \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

   11. Simplified 4.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)}\]

   if 5.1803619820124185e-257 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 2.1182887113062035e+251

   1. Initial program 0.4

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

   3. Applied add-sqr-sqrt 0.7

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

4. Final simplification 2.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -1.8620952498364704 \cdot 10^{304}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -7.1082852077845676 \cdot 10^{-296}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(\frac{1}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{y}{\sqrt[3]{z}}, -\frac{t}{1 - z} \cdot 1\right) + \frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 5.18036198201241854 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 2.1182887113062035 \cdot 10^{251}:\\ \;\;\;\;\sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)} \cdot \sqrt{x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020075 +o rules:numerics
(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)))))