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

Average Error: 4.2 → 3.1

Time: 31.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.9310092579926095 \cdot 10^{194}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - 1 \cdot \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot \left(1 \cdot t\right)\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

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 ((((y / z) - (t / (1.0 - z))) <= 5.93100925799261e+194)) {
		temp = (x * ((y / z) - (1.0 * (t / (1.0 - z)))));
	} else {
		temp = ((x * ((y * (1.0 - z)) - (z * (1.0 * t)))) / (z * (1.0 - z)));
	}
	return temp;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.2
Target	4.1
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < 5.93100925799261e+194

   1. Initial program 2.9

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

   3. Applied clear-num 3.0

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

   5. Applied *-un-lft-identity 3.0

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

   6. Applied *-un-lft-identity 3.0

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

   7. Applied times-frac 3.0

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

   8. Applied add-cube-cbrt 3.0

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

   9. Applied times-frac 3.0

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

   10. Simplified 3.0

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

   11. Simplified 2.9

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

   if 5.93100925799261e+194 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 17.0

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

   3. Applied clear-num 17.0

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

   5. Applied *-un-lft-identity 17.0

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

   6. Applied *-un-lft-identity 17.0

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

   7. Applied times-frac 17.0

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

   8. Applied add-cube-cbrt 17.0

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

   9. Applied times-frac 17.0

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

   10. Simplified 17.0

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

   11. Simplified 17.0

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

   13. Applied associate-*r/ 17.0

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

   14. Applied frac-sub 21.2

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

   15. Applied associate-*r/ 5.2

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 5.9310092579926095 \cdot 10^{194}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - 1 \cdot \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(1 - z\right) - z \cdot \left(1 \cdot t\right)\right)}{z \cdot \left(1 - z\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020049 +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)))))