Average Error: 9.0 → 0.4
Time: 24.3s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t
double f(double x, double y, double z, double t) {
        double r244471 = x;
        double r244472 = y;
        double r244473 = log(r244472);
        double r244474 = r244471 * r244473;
        double r244475 = z;
        double r244476 = 1.0;
        double r244477 = r244476 - r244472;
        double r244478 = log(r244477);
        double r244479 = r244475 * r244478;
        double r244480 = r244474 + r244479;
        double r244481 = t;
        double r244482 = r244480 - r244481;
        return r244482;
}
↓
double f(double x, double y, double z, double t) {
        double r244483 = 2.0;
        double r244484 = y;
        double r244485 = sqrt(r244484);
        double r244486 = cbrt(r244485);
        double r244487 = r244486 * r244486;
        double r244488 = log(r244487);
        double r244489 = r244483 * r244488;
        double r244490 = x;
        double r244491 = r244489 * r244490;
        double r244492 = 1.0;
        double r244493 = r244492 / r244484;
        double r244494 = -0.3333333333333333;
        double r244495 = pow(r244493, r244494);
        double r244496 = log(r244495);
        double r244497 = r244496 * r244490;
        double r244498 = r244491 + r244497;
        double r244499 = z;
        double r244500 = 1.0;
        double r244501 = log(r244500);
        double r244502 = 0.5;
        double r244503 = r244500 * r244500;
        double r244504 = r244503 / r244484;
        double r244505 = r244502 / r244504;
        double r244506 = r244500 + r244505;
        double r244507 = r244484 * r244506;
        double r244508 = r244501 - r244507;
        double r244509 = r244499 * r244508;
        double r244510 = r244498 + r244509;
        double r244511 = t;
        double r244512 = r244510 - r244511;
        return r244512;
}

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

Original9.0
Target0.3
Herbie0.4
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.0

    \[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]

  2. Taylor expanded around 0 0.4

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - \left(1 \cdot y + \frac{1}{2} \cdot \frac{{y}^{2}}{{1}^{2}}\right)\right)}\right) - t\]

  3. Simplified0.4

    \[\leadsto \left(x \cdot \log y + z \cdot \color{blue}{\left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)}\right) - t\]
  4. Using strategy rm

  5. Applied add-cube-cbrt0.4

    \[\leadsto \left(x \cdot \log \color{blue}{\left(\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}\right)} + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  6. Applied log-prod0.4

    \[\leadsto \left(x \cdot \color{blue}{\left(\log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  7. Applied distribute-lft-in0.4

    \[\leadsto \left(\color{blue}{\left(x \cdot \log \left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) + x \cdot \log \left(\sqrt[3]{y}\right)\right)} + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  8. Simplified0.4

    \[\leadsto \left(\left(\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot x} + x \cdot \log \left(\sqrt[3]{y}\right)\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  9. Simplified0.4

    \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot x + \color{blue}{\log \left(\sqrt[3]{y}\right) \cdot x}\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  10. Taylor expanded around inf 0.4

    \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{y}\right)\right) \cdot x + \log \color{blue}{\left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right)} \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]
  11. Using strategy rm

  12. Applied add-sqr-sqrt0.4

    \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  13. Applied cbrt-prod0.4

    \[\leadsto \left(\left(\left(2 \cdot \log \color{blue}{\left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)}\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

  14. Final simplification0.4

    \[\leadsto \left(\left(\left(2 \cdot \log \left(\sqrt[3]{\sqrt{y}} \cdot \sqrt[3]{\sqrt{y}}\right)\right) \cdot x + \log \left({\left(\frac{1}{y}\right)}^{\frac{-1}{3}}\right) \cdot x\right) + z \cdot \left(\log 1 - y \cdot \left(1 + \frac{\frac{1}{2}}{\frac{1 \cdot 1}{y}}\right)\right)\right) - t\]

Reproduce

herbie shell --seed 2019326 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1 (* 1 1))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1 y)))) t))