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

Average Error: 4.4 → 0.4

Time: 4.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -1.88748376235 \cdot 10^{-313}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 8.4522645039473741 \cdot 10^{305}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

C

double code(double x, double y, double z, double t) {
	return (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)))) <= -inf.0)) {
		VAR = ((x * ((y * (1.0 - z)) - (z * t))) / (z * (1.0 - z)));
	} else {
		double VAR_1;
		if (((x * ((y / z) - (t / (1.0 - z)))) <= -1.887483762347e-313)) {
			VAR_1 = (x * ((y / z) - (t * (1.0 / (1.0 - z)))));
		} else {
			double VAR_2;
			if (((x * ((y / z) - (t / (1.0 - z)))) <= 0.0)) {
				VAR_2 = (((x * y) / z) + ((1.0 * ((t * x) / pow(z, 2.0))) + ((t * x) / z)));
			} else {
				double VAR_3;
				if (((x * ((y / z) - (t / (1.0 - z)))) <= 8.452264503947374e+305)) {
					VAR_3 = (x * ((y / z) - (t * (1.0 / (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	4.1
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 (* x (- (/ y z) (/ t (- 1.0 z)))) < -inf.0 or 8.452264503947374e+305 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 62.1

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

   3. Applied frac-sub 62.8

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

   4. Applied associate-*r/ 0.9

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

   if -inf.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < -1.887483762347e-313 or 0.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 8.452264503947374e+305

   1. Initial program 0.3

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

   3. Applied div-inv 0.4

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

   if -1.887483762347e-313 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 0.0

   1. Initial program 8.4

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

   3. Applied div-inv 8.4

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

   4. Taylor expanded around inf 0.6

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le -1.88748376235 \cdot 10^{-313}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 0.0:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 8.4522645039473741 \cdot 10^{305}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 
(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)))))