Average Error: 4.8 → 1.4
Time: 4.5s
Precision: 64
\[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
↓
\[\left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\]
x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)
↓
\left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)
double f(double x, double y, double z, double t) {
        double r431312 = x;
        double r431313 = y;
        double r431314 = z;
        double r431315 = r431313 / r431314;
        double r431316 = t;
        double r431317 = 1.0;
        double r431318 = r431317 - r431314;
        double r431319 = r431316 / r431318;
        double r431320 = r431315 - r431319;
        double r431321 = r431312 * r431320;
        return r431321;
}
↓
double f(double x, double y, double z, double t) {
        double r431322 = x;
        double r431323 = y;
        double r431324 = cbrt(r431323);
        double r431325 = z;
        double r431326 = cbrt(r431325);
        double r431327 = r431324 / r431326;
        double r431328 = r431322 * r431327;
        double r431329 = r431328 * r431327;
        double r431330 = r431329 * r431327;
        double r431331 = t;
        double r431332 = 1.0;
        double r431333 = r431332 - r431325;
        double r431334 = r431331 / r431333;
        double r431335 = -r431334;
        double r431336 = r431322 * r431335;
        double r431337 = r431330 + r431336;
        return r431337;
}

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

Original4.8
Target4.4
Herbie1.4
\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

  1. Initial program 4.8

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

  3. Applied sub-neg4.8

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

  4. Applied distribute-lft-in4.8

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

  6. Applied add-cube-cbrt5.3

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

  7. Applied add-cube-cbrt5.4

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

  8. Applied times-frac5.4

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

  9. Applied associate-*r*1.8

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

  11. Applied times-frac1.8

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

  12. Applied associate-*r*1.4

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

  13. Final simplification1.4

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

Reproduce

herbie shell --seed 2019356 
(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)))))