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

Average Error: 4.6 → 1.3

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -inf.0 \lor \neg \left(x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.5172692539115746 \cdot 10^{298}\right):\\ \;\;\;\;\frac{1}{\frac{z \cdot \left(1 - z\right)}{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

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)))))))) <= -inf.0) || !(((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))))) <= 1.5172692539115746e+298))) {
		VAR = ((double) (1.0 / ((double) (((double) (z * ((double) (1.0 - z)))) / ((double) (x * ((double) (((double) (y * ((double) (1.0 - z)))) - ((double) (z * t))))))))));
	} else {
		VAR = ((double) (x * ((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z))))))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.4
Herbie	1.3

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

Derivation

1. Split input into 2 regimes

2. if (* x (- (/ y z) (/ t (- 1.0 z)))) < -inf.0 or 1.5172692539115746e298 < (* x (- (/ y z) (/ t (- 1.0 z))))

   1. Initial program 57.9

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

   3. Applied frac-sub 59.4

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

   4. Applied associate-*r/ 2.4

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

   6. Applied clear-num 2.5

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

   if -inf.0 < (* x (- (/ y z) (/ t (- 1.0 z)))) < 1.5172692539115746e298

   1. Initial program 1.2

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

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) = -inf.0 \lor \neg \left(x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \le 1.5172692539115746 \cdot 10^{298}\right):\\ \;\;\;\;\frac{1}{\frac{z \cdot \left(1 - z\right)}{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020153 
(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)))))