Average Error: 9.7 → 0.3
Time: 28.2s
Precision: 64
\[\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t\]
↓
\[\left(\left(\log \left(\sqrt{y}\right) \cdot x + \log \left(\sqrt{y}\right) \cdot x\right) + \left(\left(\log 1 - y \cdot 1\right) \cdot z - \frac{z}{\frac{1}{y}} \cdot \frac{\frac{1}{2}}{\frac{1}{y}}\right)\right) - t\]
\left(x \cdot \log y + z \cdot \log \left(1 - y\right)\right) - t
↓
\left(\left(\log \left(\sqrt{y}\right) \cdot x + \log \left(\sqrt{y}\right) \cdot x\right) + \left(\left(\log 1 - y \cdot 1\right) \cdot z - \frac{z}{\frac{1}{y}} \cdot \frac{\frac{1}{2}}{\frac{1}{y}}\right)\right) - t
double f(double x, double y, double z, double t) {
        double r18976284 = x;
        double r18976285 = y;
        double r18976286 = log(r18976285);
        double r18976287 = r18976284 * r18976286;
        double r18976288 = z;
        double r18976289 = 1.0;
        double r18976290 = r18976289 - r18976285;
        double r18976291 = log(r18976290);
        double r18976292 = r18976288 * r18976291;
        double r18976293 = r18976287 + r18976292;
        double r18976294 = t;
        double r18976295 = r18976293 - r18976294;
        return r18976295;
}
↓
double f(double x, double y, double z, double t) {
        double r18976296 = y;
        double r18976297 = sqrt(r18976296);
        double r18976298 = log(r18976297);
        double r18976299 = x;
        double r18976300 = r18976298 * r18976299;
        double r18976301 = r18976300 + r18976300;
        double r18976302 = 1.0;
        double r18976303 = log(r18976302);
        double r18976304 = r18976296 * r18976302;
        double r18976305 = r18976303 - r18976304;
        double r18976306 = z;
        double r18976307 = r18976305 * r18976306;
        double r18976308 = r18976302 / r18976296;
        double r18976309 = r18976306 / r18976308;
        double r18976310 = 0.5;
        double r18976311 = r18976310 / r18976308;
        double r18976312 = r18976309 * r18976311;
        double r18976313 = r18976307 - r18976312;
        double r18976314 = r18976301 + r18976313;
        double r18976315 = t;
        double r18976316 = r18976314 - r18976315;
        return r18976316;
}

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

Original9.7
Target0.3
Herbie0.3
\[\left(-z\right) \cdot \left(\left(0.5 \cdot \left(y \cdot y\right) + y\right) + \frac{0.3333333333333333148296162562473909929395}{1 \cdot \left(1 \cdot 1\right)} \cdot \left(y \cdot \left(y \cdot y\right)\right)\right) - \left(t - x \cdot \log y\right)\]

Derivation

  1. Initial program 9.7

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

  2. Taylor expanded around 0 0.3

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

  3. Simplified0.3

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

  5. Applied add-sqr-sqrt0.3

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

  6. Applied log-prod0.3

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

  7. Applied distribute-lft-in0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, B"

  :herbie-target
  (- (* (- z) (+ (+ (* 0.5 (* y y)) y) (* (/ 0.3333333333333333 (* 1.0 (* 1.0 1.0))) (* y (* y y))))) (- t (* x (log y))))

  (- (+ (* x (log y)) (* z (log (- 1.0 y)))) t))