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

Average Error: 4.4 → 1.8

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.67571914999741096 \cdot 10^{225}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.78593329367950345 \cdot 10^{112}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(x \cdot \left(-t\right)\right) \cdot \frac{1}{1 - z}\\ \end{array}\]

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 ((((y / z) - (t / (1.0 - z))) <= -1.675719149997411e+225)) {
		VAR = fma((x * y), (1.0 / z), (x * -(t / (1.0 - z))));
	} else {
		double VAR_1;
		if ((((y / z) - (t / (1.0 - z))) <= 1.7859332936795035e+112)) {
			VAR_1 = (x * ((y / z) - (t * (1.0 / (1.0 - z)))));
		} else {
			VAR_1 = (((x * y) / z) + ((x * -t) * (1.0 / (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	4.4
Target	4.1
Herbie	1.8

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

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -1.675719149997411e+225

   1. Initial program 22.3

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

   3. Applied div-inv 22.3

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

   4. Applied fma-neg 22.3

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

   6. Applied fma-udef 22.3

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

   7. Applied distribute-lft-in 22.3

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

   8. Simplified 0.5

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

   10. Applied div-inv 0.5

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

   11. Applied fma-def 0.5

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

   if -1.675719149997411e+225 < (- (/ y z) (/ t (- 1.0 z))) < 1.7859332936795035e+112

   1. Initial program 1.6

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

   3. Applied div-inv 1.6

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

   if 1.7859332936795035e+112 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 10.2

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

   3. Applied div-inv 10.3

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

   4. Applied fma-neg 10.3

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

   6. Applied fma-udef 10.3

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

   7. Applied distribute-lft-in 10.3

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

   8. Simplified 2.4

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

   10. Applied div-inv 2.4

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

   11. Applied distribute-lft-neg-in 2.4

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

   12. Applied associate-*r* 3.7

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

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.67571914999741096 \cdot 10^{225}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot y, \frac{1}{z}, x \cdot \left(-\frac{t}{1 - z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.78593329367950345 \cdot 10^{112}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(x \cdot \left(-t\right)\right) \cdot \frac{1}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020092 +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)))))