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

Average Error: 4.5 → 1.5

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right)\\ \end{array}\]

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 (((((y / z) - (t / (1.0 - z))) <= -inf.0) || !(((y / z) - (t / (1.0 - z))) <= 1.1070795943172787e+304))) {
		temp = ((x * y) / z);
	} else {
		temp = (x * (fma(1.0, (y / z), -((t / cbrt((1.0 - z))) * (1.0 / (cbrt((1.0 - z)) * cbrt((1.0 - z)))))) + ((1.0 / (cbrt((1.0 - z)) * cbrt((1.0 - z)))) * (-(t / cbrt((1.0 - z))) + (t / cbrt((1.0 - z)))))));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.2
Herbie	1.5

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 1.1070795943172787e+304 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 61.8

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

   3. Applied add-cube-cbrt 61.8

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

   4. Applied *-un-lft-identity 61.8

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

   5. Applied times-frac 61.8

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]

   6. Applied add-sqr-sqrt 62.1

      \[\leadsto x \cdot \left(\color{blue}{\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}} - \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\]

   7. Applied prod-diff 62.1

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{y}{z}}, \sqrt{\frac{y}{z}}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right)}\]

   8. Simplified 61.8

      \[\leadsto x \cdot \left(\color{blue}{\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)} + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right)\]

   9. Simplified 61.8

      \[\leadsto x \cdot \left(\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)}\right)\]

   10. Taylor expanded around 0 1.3

       \[\leadsto \color{blue}{\frac{x \cdot y}{z}}\]

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < 1.1070795943172787e+304

   1. Initial program 1.2

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

   3. Applied add-cube-cbrt 1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

   4. Applied *-un-lft-identity 1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

   5. Applied times-frac 1.5

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]

   6. Applied add-sqr-sqrt 28.9

      \[\leadsto x \cdot \left(\color{blue}{\sqrt{\frac{y}{z}} \cdot \sqrt{\frac{y}{z}}} - \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\]

   7. Applied prod-diff 28.9

      \[\leadsto x \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{\frac{y}{z}}, \sqrt{\frac{y}{z}}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right)}\]

   8. Simplified 1.5

      \[\leadsto x \cdot \left(\color{blue}{\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)} + \mathsf{fma}\left(-\frac{t}{\sqrt[3]{1 - z}}, \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}, \frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right)\right)\]

   9. Simplified 1.5

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

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 1.10707959431727866 \cdot 10^{304}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\mathsf{fma}\left(1, \frac{y}{z}, -\frac{t}{\sqrt[3]{1 - z}} \cdot \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}\right) + \frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \left(\left(-\frac{t}{\sqrt[3]{1 - z}}\right) + \frac{t}{\sqrt[3]{1 - z}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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)))))