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

Average Error: 4.4 → 4.4

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -1.77373656942475791 \cdot 10^{-180} \lor \neg \left(t \le 6.2185368398920625 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot t\right) \cdot \left(-\frac{1}{1 - z}\right)\\ \end{array}\]

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 temp;
	if (((t <= -1.773736569424758e-180) || !(t <= 6.218536839892062e-38))) {
		temp = (((x * y) * (1.0 / z)) + (x * -(t / (1.0 - z))));
	} else {
		temp = ((x * (y / z)) + ((x * t) * -(1.0 / (1.0 - z))));
	}
	return temp;
}

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	4.4

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

Derivation

1. Split input into 2 regimes

2. if t < -1.773736569424758e-180 or 6.218536839892062e-38 < t

   1. Initial program 3.7

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

   3. Applied sub-neg 3.7

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

   4. Applied distribute-lft-in 3.7

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

   6. Applied div-inv 3.8

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

   7. Applied associate-*r* 4.4

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

   if -1.773736569424758e-180 < t < 6.218536839892062e-38

   1. Initial program 5.7

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

   3. Applied sub-neg 5.7

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

   4. Applied distribute-lft-in 5.7

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

   6. Applied div-inv 5.7

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

   7. Applied distribute-rgt-neg-in 5.7

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

   8. Applied associate-*r* 4.5

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

4. Final simplification 4.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.77373656942475791 \cdot 10^{-180} \lor \neg \left(t \le 6.2185368398920625 \cdot 10^{-38}\right):\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(x \cdot t\right) \cdot \left(-\frac{1}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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)))))