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

Average Error: 4.7 → 0.5

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -9.39688798881046142 \cdot 10^{287}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-x}{\sqrt{1 - z}} \cdot \frac{t}{\sqrt{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.24814217605651677 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.46500928468374139 \cdot 10^{219}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-x}{\sqrt{1 - z}} \cdot \frac{t}{\sqrt{1 - z}}\\ \end{array}\]

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)))))) <= -9.396887988810461e+287)) {
		VAR = ((double) (((double) (((double) (x * y)) / z)) + ((double) (((double) (((double) -(x)) / ((double) sqrt(((double) (1.0 - z)))))) * ((double) (t / ((double) sqrt(((double) (1.0 - z))))))))));
	} else {
		double VAR_1;
		if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= -1.2481421760565168e-278)) {
			VAR_1 = ((double) (((double) (x / ((double) (z / y)))) + ((double) (x * ((double) -(((double) (t / ((double) (1.0 - z))))))))));
		} else {
			double VAR_2;
			if ((((double) (((double) (y / z)) - ((double) (t / ((double) (1.0 - z)))))) <= 0.0)) {
				VAR_2 = ((double) fma(y, ((double) (x / z)), ((double) fma(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)))))) <= 3.4650092846837414e+219)) {
					VAR_3 = ((double) (((double) (x / ((double) (z / y)))) + ((double) (x * ((double) -(((double) (t / ((double) (1.0 - z))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) (x * y)) / z)) + ((double) (((double) (((double) -(x)) / ((double) sqrt(((double) (1.0 - z)))))) * ((double) (t / ((double) sqrt(((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	4.7
Target	4.3
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -9.396887988810461e+287 or 3.4650092846837414e+219 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 30.7

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

   3. Applied div-inv 30.8

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

   4. Applied fma-neg 30.8

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

   6. Applied fma-udef 30.8

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

   7. Applied distribute-lft-in 30.8

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

   8. Simplified 0.4

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

   10. Applied add-sqr-sqrt 2.5

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

   11. Applied *-un-lft-identity 2.5

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

   12. Applied times-frac 2.5

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

   13. Applied distribute-lft-neg-in 2.5

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

   14. Applied associate-*r* 2.6

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

   15. Simplified 2.6

       \[\leadsto \frac{x \cdot y}{z} + \color{blue}{\frac{-x}{\sqrt{1 - z}}} \cdot \frac{t}{\sqrt{1 - z}}\]

   if -9.396887988810461e+287 < (- (/ y z) (/ t (- 1.0 z))) < -1.2481421760565168e-278 or 0.0 < (- (/ y z) (/ t (- 1.0 z))) < 3.4650092846837414e+219

   1. Initial program 0.2

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

   3. Applied div-inv 0.3

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

   4. Applied fma-neg 0.3

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

   6. Applied fma-udef 0.3

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

   7. Applied distribute-lft-in 0.3

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

   8. Simplified 5.8

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

   10. Applied associate-/l* 0.3

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

   if -1.2481421760565168e-278 < (- (/ y z) (/ t (- 1.0 z))) < 0.0

   1. Initial program 17.9

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

   3. Applied div-inv 17.9

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

   4. Applied fma-neg 17.9

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

   5. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

   6. Simplified 0.1

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -9.39688798881046142 \cdot 10^{287}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-x}{\sqrt{1 - z}} \cdot \frac{t}{\sqrt{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.24814217605651677 \cdot 10^{-278}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 3.46500928468374139 \cdot 10^{219}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z} + \frac{-x}{\sqrt{1 - z}} \cdot \frac{t}{\sqrt{1 - z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020120 +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)))))