Average Error: 28.2 → 0.2
Time: 6.6s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\left(z + x\right) \cdot \frac{x - z}{y} + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\left(z + x\right) \cdot \frac{x - z}{y} + y}{2}
double f(double x, double y, double z) {
        double r998853 = x;
        double r998854 = r998853 * r998853;
        double r998855 = y;
        double r998856 = r998855 * r998855;
        double r998857 = r998854 + r998856;
        double r998858 = z;
        double r998859 = r998858 * r998858;
        double r998860 = r998857 - r998859;
        double r998861 = 2.0;
        double r998862 = r998855 * r998861;
        double r998863 = r998860 / r998862;
        return r998863;
}
↓
double f(double x, double y, double z) {
        double r998864 = z;
        double r998865 = x;
        double r998866 = r998864 + r998865;
        double r998867 = r998865 - r998864;
        double r998868 = y;
        double r998869 = r998867 / r998868;
        double r998870 = r998866 * r998869;
        double r998871 = r998870 + r998868;
        double r998872 = 2.0;
        double r998873 = r998871 / r998872;
        return r998873;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

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. Simplified12.7

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

  4. Applied *-un-lft-identity12.7

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

  5. Applied difference-of-squares12.7

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

  6. Applied times-frac0.2

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

  7. Simplified0.2

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

  8. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 
(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)))