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

Average Error: 4.6 → 3.2

Time: 3.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -3.03854886497741823 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\frac{t}{\sqrt{1 - z}}}{\sqrt{1 - z}}\right)\\ \mathbf{elif}\;z \le 2.8473073316618129 \cdot 10^{-34}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

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 ((z <= -3.038548864977418e-73)) {
		VAR = (x * ((y / z) - ((t / sqrt((1.0 - z))) / sqrt((1.0 - z)))));
	} else {
		double VAR_1;
		if ((z <= 2.847307331661813e-34)) {
			VAR_1 = (((x * y) * (1.0 / z)) + (x * -(t / (1.0 - z))));
		} else {
			VAR_1 = (((x / sqrt(z)) * (y / sqrt(z))) + (x * -(t / (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	4.6
Target	4.3
Herbie	3.2

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

Derivation

1. Split input into 3 regimes

2. if z < -3.038548864977418e-73

   1. Initial program 1.9

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

   3. Applied add-sqr-sqrt 2.0

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

   4. Applied associate-/r* 2.0

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

   if -3.038548864977418e-73 < z < 2.847307331661813e-34

   1. Initial program 10.0

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

   3. Applied sub-neg 10.0

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

   4. Applied distribute-lft-in 10.0

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

   6. Applied div-inv 10.1

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

   7. Applied associate-*r* 4.7

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

   if 2.847307331661813e-34 < z

   1. Initial program 1.9

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

   3. Applied sub-neg 1.9

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

   4. Applied distribute-lft-in 1.9

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

   6. Applied add-sqr-sqrt 2.1

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

   7. Applied *-un-lft-identity 2.1

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

   8. Applied times-frac 2.1

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

   9. Applied associate-*r* 2.8

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

   10. Simplified 2.8

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

4. Final simplification 3.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -3.03854886497741823 \cdot 10^{-73}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{\frac{t}{\sqrt{1 - z}}}{\sqrt{1 - z}}\right)\\ \mathbf{elif}\;z \le 2.8473073316618129 \cdot 10^{-34}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(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)))))