Average Error: 28.7 → 0.1
Time: 11.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\mathsf{fma}\left(\frac{x + z}{y}, x - z, y\right)}{2}
double f(double x, double y, double z) {
        double r760807 = x;
        double r760808 = r760807 * r760807;
        double r760809 = y;
        double r760810 = r760809 * r760809;
        double r760811 = r760808 + r760810;
        double r760812 = z;
        double r760813 = r760812 * r760812;
        double r760814 = r760811 - r760813;
        double r760815 = 2.0;
        double r760816 = r760809 * r760815;
        double r760817 = r760814 / r760816;
        return r760817;
}
↓
double f(double x, double y, double z) {
        double r760818 = x;
        double r760819 = z;
        double r760820 = r760818 + r760819;
        double r760821 = y;
        double r760822 = r760820 / r760821;
        double r760823 = r760818 - r760819;
        double r760824 = fma(r760822, r760823, r760821);
        double r760825 = 2.0;
        double r760826 = r760824 / r760825;
        return r760826;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original28.7
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 28.7

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

  2. Simplified0.1

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

  3. Final simplification0.1

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

Reproduce

herbie shell --seed 2020043 +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)))