Average Error: 28.2 → 0.2
Time: 19.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y - \frac{z - x}{y} \cdot \left(z + x\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y - \frac{z - x}{y} \cdot \left(z + x\right)}{2}
double f(double x, double y, double z) {
        double r36825065 = x;
        double r36825066 = r36825065 * r36825065;
        double r36825067 = y;
        double r36825068 = r36825067 * r36825067;
        double r36825069 = r36825066 + r36825068;
        double r36825070 = z;
        double r36825071 = r36825070 * r36825070;
        double r36825072 = r36825069 - r36825071;
        double r36825073 = 2.0;
        double r36825074 = r36825067 * r36825073;
        double r36825075 = r36825072 / r36825074;
        return r36825075;
}
↓
double f(double x, double y, double z) {
        double r36825076 = y;
        double r36825077 = z;
        double r36825078 = x;
        double r36825079 = r36825077 - r36825078;
        double r36825080 = r36825079 / r36825076;
        double r36825081 = r36825077 + r36825078;
        double r36825082 = r36825080 * r36825081;
        double r36825083 = r36825076 - r36825082;
        double r36825084 = 2.0;
        double r36825085 = r36825083 / r36825084;
        return r36825085;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

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Target

Original28.2
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 28.2

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

  2. Simplified0.1

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

  4. Applied associate-/r/0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019171 
(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)))