Average Error: 4.6 → 1.8
Time: 5.8s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}} + x \cdot \left(-\frac{t}{1 - z}\right)
double f(double x, double y, double z, double t) {
        double r417233 = x;
        double r417234 = y;
        double r417235 = z;
        double r417236 = r417234 / r417235;
        double r417237 = t;
        double r417238 = 1.0;
        double r417239 = r417238 - r417235;
        double r417240 = r417237 / r417239;
        double r417241 = r417236 - r417240;
        double r417242 = r417233 * r417241;
        return r417242;
}
↓
double f(double x, double y, double z, double t) {
        double r417243 = y;
        double r417244 = cbrt(r417243);
        double r417245 = r417244 * r417244;
        double r417246 = z;
        double r417247 = cbrt(r417246);
        double r417248 = r417247 * r417247;
        double r417249 = r417245 / r417248;
        double r417250 = x;
        double r417251 = r417247 / r417244;
        double r417252 = r417250 / r417251;
        double r417253 = r417249 * r417252;
        double r417254 = t;
        double r417255 = 1.0;
        double r417256 = r417255 - r417246;
        double r417257 = r417254 / r417256;
        double r417258 = -r417257;
        double r417259 = r417250 * r417258;
        double r417260 = r417253 + r417259;
        return r417260;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original4.6
Target4.6
Herbie1.8
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Initial program 4.6

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

  3. Applied div-inv4.7

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

  4. Applied fma-neg4.7

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

  6. Applied fma-udef4.7

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

  7. Applied distribute-lft-in4.7

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

  8. Simplified5.2

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

  10. Applied associate-/l*4.4

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

  12. Applied add-cube-cbrt4.9

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

  13. Applied add-cube-cbrt5.0

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

  14. Applied times-frac5.0

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

  15. Applied *-un-lft-identity5.0

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

  16. Applied times-frac1.8

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

  17. Simplified1.8

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

  18. Final simplification1.8

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

Reproduce

herbie shell --seed 2020024 +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)))))