Average Error: 28.4 → 0.2
Time: 35.4s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}
double f(double x, double y, double z) {
        double r436581 = x;
        double r436582 = r436581 * r436581;
        double r436583 = y;
        double r436584 = r436583 * r436583;
        double r436585 = r436582 + r436584;
        double r436586 = z;
        double r436587 = r436586 * r436586;
        double r436588 = r436585 - r436587;
        double r436589 = 2.0;
        double r436590 = r436583 * r436589;
        double r436591 = r436588 / r436590;
        return r436591;
}
↓
double f(double x, double y, double z) {
        double r436592 = x;
        double r436593 = z;
        double r436594 = r436592 + r436593;
        double r436595 = y;
        double r436596 = r436594 / r436595;
        double r436597 = r436592 - r436593;
        double r436598 = r436596 * r436597;
        double r436599 = r436598 + r436595;
        double r436600 = 2.0;
        double r436601 = r436599 / r436600;
        return r436601;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.4
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 28.4

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

  2. Simplified0.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}}\]
  3. Using strategy rm

  4. Applied fma-udef0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

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

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2)))