Average Error: 28.7 → 0.2
Time: 5.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{\frac{{x}^{1}}{y}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{\frac{{x}^{1}}{y}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r638078 = x;
        double r638079 = r638078 * r638078;
        double r638080 = y;
        double r638081 = r638080 * r638080;
        double r638082 = r638079 + r638081;
        double r638083 = z;
        double r638084 = r638083 * r638083;
        double r638085 = r638082 - r638084;
        double r638086 = 2.0;
        double r638087 = r638080 * r638086;
        double r638088 = r638085 / r638087;
        return r638088;
}
↓
double f(double x, double y, double z) {
        double r638089 = 0.5;
        double r638090 = y;
        double r638091 = x;
        double r638092 = 1.0;
        double r638093 = pow(r638091, r638092);
        double r638094 = r638093 / r638090;
        double r638095 = r638092 / r638091;
        double r638096 = r638094 / r638095;
        double r638097 = r638090 + r638096;
        double r638098 = z;
        double r638099 = r638090 / r638098;
        double r638100 = r638098 / r638099;
        double r638101 = r638097 - r638100;
        double r638102 = r638089 * r638101;
        return r638102;
}

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

Original28.7
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 28.7

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

  2. Taylor expanded around 0 12.3

    \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

  3. Simplified12.3

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied unpow212.3

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{\color{blue}{z \cdot z}}{y}\right)\]

  6. Applied associate-/l*6.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \color{blue}{\frac{z}{\frac{y}{z}}}\right)\]
  7. Using strategy rm

  8. Applied sqr-pow6.7

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{y}\right) - \frac{z}{\frac{y}{z}}\right)\]

  9. Applied associate-/l*0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{{x}^{\left(\frac{2}{2}\right)}}}}\right) - \frac{z}{\frac{y}{z}}\right)\]

  10. Simplified0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{y}{x}}}\right) - \frac{z}{\frac{y}{z}}\right)\]
  11. Using strategy rm

  12. Applied div-inv0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\color{blue}{y \cdot \frac{1}{x}}}\right) - \frac{z}{\frac{y}{z}}\right)\]

  13. Applied associate-/r*0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{\frac{{x}^{\left(\frac{2}{2}\right)}}{y}}{\frac{1}{x}}}\right) - \frac{z}{\frac{y}{z}}\right)\]

  14. Simplified0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{\frac{{x}^{1}}{y}}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]

  15. Final simplification0.2

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\frac{{x}^{1}}{y}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]

Reproduce

herbie shell --seed 2020089 +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)))