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

Average Error: 4.7 → 4.7

Time: 2.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.05799137819263262 \cdot 10^{189}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

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 <= -4.0579913781926326e+189)) {
		VAR = ((y / (z / x)) + (x * -(t / (1.0 - z))));
	} else {
		VAR = (((x * y) * (1.0 / z)) + (x * -(t / (1.0 - z))));
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.4
Herbie	4.7

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

Derivation

1. Split input into 2 regimes

2. if x < -4.0579913781926326e+189

   1. Initial program 4.4

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

   3. Applied clear-num 4.7

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

   5. Applied div-inv 4.7

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

   6. Applied fma-neg 4.7

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

   7. Simplified 4.4

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

   9. Applied fma-udef 4.4

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

   10. Applied distribute-lft-in 4.4

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

   11. Simplified 17.6

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

   13. Applied *-commutative 17.6

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

   14. Applied associate-/l* 9.8

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

   if -4.0579913781926326e+189 < x

   1. Initial program 4.7

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

   3. Applied clear-num 4.8

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

   5. Applied div-inv 4.9

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

   6. Applied fma-neg 4.8

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

   7. Simplified 4.7

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

   9. Applied fma-udef 4.7

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

   10. Applied distribute-lft-in 4.7

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

   11. Simplified 4.3

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

   13. Applied div-inv 4.4

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

4. Final simplification 4.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.05799137819263262 \cdot 10^{189}:\\ \;\;\;\;\frac{y}{\frac{z}{x}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020071 +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)))))