Average Error: 28.6 → 0.2
Time: 16.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \left(x + z\right) \cdot \left(\left(x - z\right) \cdot \frac{1}{y}\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \left(x + z\right) \cdot \left(\left(x - z\right) \cdot \frac{1}{y}\right)}{2}
double f(double x, double y, double z) {
        double r31621498 = x;
        double r31621499 = r31621498 * r31621498;
        double r31621500 = y;
        double r31621501 = r31621500 * r31621500;
        double r31621502 = r31621499 + r31621501;
        double r31621503 = z;
        double r31621504 = r31621503 * r31621503;
        double r31621505 = r31621502 - r31621504;
        double r31621506 = 2.0;
        double r31621507 = r31621500 * r31621506;
        double r31621508 = r31621505 / r31621507;
        return r31621508;
}
↓
double f(double x, double y, double z) {
        double r31621509 = y;
        double r31621510 = x;
        double r31621511 = z;
        double r31621512 = r31621510 + r31621511;
        double r31621513 = r31621510 - r31621511;
        double r31621514 = 1.0;
        double r31621515 = r31621514 / r31621509;
        double r31621516 = r31621513 * r31621515;
        double r31621517 = r31621512 * r31621516;
        double r31621518 = r31621509 + r31621517;
        double r31621519 = 2.0;
        double r31621520 = r31621518 / r31621519;
        return r31621520;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.6
Target0.2
Herbie0.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.6

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{y + \frac{x - z}{\frac{y}{z + x}}}{2}}\]
  3. Using strategy rm

  4. Applied associate-/r/0.1

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

  6. Applied div-inv0.2

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

  7. Final simplification0.2

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

Reproduce

herbie shell --seed 2019170 
(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)))