Average Error: 28.4 → 0.2
Time: 5.9s
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 r775539 = x;
        double r775540 = r775539 * r775539;
        double r775541 = y;
        double r775542 = r775541 * r775541;
        double r775543 = r775540 + r775542;
        double r775544 = z;
        double r775545 = r775544 * r775544;
        double r775546 = r775543 - r775545;
        double r775547 = 2.0;
        double r775548 = r775541 * r775547;
        double r775549 = r775546 / r775548;
        return r775549;
}
↓
double f(double x, double y, double z) {
        double r775550 = 0.5;
        double r775551 = y;
        double r775552 = x;
        double r775553 = 1.0;
        double r775554 = pow(r775552, r775553);
        double r775555 = r775554 / r775551;
        double r775556 = r775553 / r775552;
        double r775557 = r775555 / r775556;
        double r775558 = r775551 + r775557;
        double r775559 = z;
        double r775560 = r775551 / r775559;
        double r775561 = r775559 / r775560;
        double r775562 = r775558 - r775561;
        double r775563 = r775550 * r775562;
        return r775563;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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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. Simplified28.4

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

  3. Taylor expanded around 0 12.8

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

  4. Simplified12.8

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

  6. Applied unpow212.8

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

  7. Applied associate-/l*6.9

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

  9. Applied sqr-pow6.9

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

  10. Applied associate-/l*0.2

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

  11. Simplified0.2

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

  13. Applied div-inv0.2

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

  14. Applied associate-/r*0.2

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

  15. Simplified0.2

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

  16. 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 2020057 +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)))