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

Average Error: 4.9 → 0.4

Time: 23.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -inf.0:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -7.8076128104113177 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.08430556300017537 \cdot 10^{-179}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.519702392511672 \cdot 10^{294}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

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) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -inf.0)) {
		VAR = ((double) (((double) (x * ((double) (((double) (y * ((double) (1.0 - z)))) - ((double) (z * t)))))) / ((double) (z * ((double) (1.0 - z))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -7.807612810411318e-307)) {
			VAR_1 = ((double) (x * ((double) (((double) (y / z)) - ((double) (t * ((double) (1.0 / ((double) (1.0 - z))))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 8.084305563000175e-179)) {
				VAR_2 = ((double) (((double) (((double) (x * y)) / z)) + ((double) (((double) (1.0 * ((double) (((double) (t * x)) / ((double) pow(z, 2.0)))))) + ((double) (((double) (t * x)) / z))))));
			} else {
				double VAR_3;
				if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 8.519702392511672e+294)) {
					VAR_3 = ((double) (x * ((double) (((double) (y / z)) - ((double) (t * ((double) (1.0 / ((double) (1.0 - z))))))))));
				} else {
					VAR_3 = ((double) (((double) (x * ((double) (((double) (y * ((double) (1.0 - z)))) - ((double) (z * t)))))) / ((double) (z * ((double) (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	4.9
Target	4.6
Herbie	0.4

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

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 8.519702392511672e294 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 58.4

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

   3. Applied frac-sub 58.4

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

   4. Applied associate-*r/ 0.2

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

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -7.8076128104113177e-307 or 8.08430556300017537e-179 < (- (/ y z) (/ t (- 1.0 z))) < 8.519702392511672e294

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   if -7.8076128104113177e-307 < (- (/ y z) (/ t (- 1.0 z))) < 8.08430556300017537e-179

   1. Initial program 10.7

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

   3. Applied sub-neg 10.7

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

   4. Applied distribute-lft-in 10.7

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

   5. Taylor expanded around inf 1.9

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -inf.0:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -7.8076128104113177 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.08430556300017537 \cdot 10^{-179}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.519702392511672 \cdot 10^{294}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020163 
(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 (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (neg (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))