Average Error: 27.1 → 0.1
Time: 15.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}\]
↓
\[\frac{y - \left(z - x\right) \cdot \frac{z + x}{y}}{2.0}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}
↓
\frac{y - \left(z - x\right) \cdot \frac{z + x}{y}}{2.0}
double f(double x, double y, double z) {
        double r33945429 = x;
        double r33945430 = r33945429 * r33945429;
        double r33945431 = y;
        double r33945432 = r33945431 * r33945431;
        double r33945433 = r33945430 + r33945432;
        double r33945434 = z;
        double r33945435 = r33945434 * r33945434;
        double r33945436 = r33945433 - r33945435;
        double r33945437 = 2.0;
        double r33945438 = r33945431 * r33945437;
        double r33945439 = r33945436 / r33945438;
        return r33945439;
}
↓
double f(double x, double y, double z) {
        double r33945440 = y;
        double r33945441 = z;
        double r33945442 = x;
        double r33945443 = r33945441 - r33945442;
        double r33945444 = r33945441 + r33945442;
        double r33945445 = r33945444 / r33945440;
        double r33945446 = r33945443 * r33945445;
        double r33945447 = r33945440 - r33945446;
        double r33945448 = 2.0;
        double r33945449 = r33945447 / r33945448;
        return r33945449;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original27.1
Target0.1
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 27.1

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2.0}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{y - \frac{z - x}{\frac{y}{z + x}}}{2.0}}\]
  3. Using strategy rm

  4. Applied associate-/r/0.1

    \[\leadsto \frac{y - \color{blue}{\frac{z - x}{y} \cdot \left(z + x\right)}}{2.0}\]
  5. Using strategy rm

  6. Applied div-inv0.2

    \[\leadsto \frac{y - \color{blue}{\left(\left(z - x\right) \cdot \frac{1}{y}\right)} \cdot \left(z + x\right)}{2.0}\]

  7. Applied associate-*l*0.2

    \[\leadsto \frac{y - \color{blue}{\left(z - x\right) \cdot \left(\frac{1}{y} \cdot \left(z + x\right)\right)}}{2.0}\]

  8. Simplified0.1

    \[\leadsto \frac{y - \left(z - x\right) \cdot \color{blue}{\frac{z + x}{y}}}{2.0}\]

  9. Final simplification0.1

    \[\leadsto \frac{y - \left(z - x\right) \cdot \frac{z + x}{y}}{2.0}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))