Average Error: 29.0 → 0.2
Time: 6.8s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{\frac{z}{y}}{\frac{1}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{\frac{z}{y}}{\frac{1}{z}}\right)
double f(double x, double y, double z) {
        double r719314 = x;
        double r719315 = r719314 * r719314;
        double r719316 = y;
        double r719317 = r719316 * r719316;
        double r719318 = r719315 + r719317;
        double r719319 = z;
        double r719320 = r719319 * r719319;
        double r719321 = r719318 - r719320;
        double r719322 = 2.0;
        double r719323 = r719316 * r719322;
        double r719324 = r719321 / r719323;
        return r719324;
}
↓
double f(double x, double y, double z) {
        double r719325 = 0.5;
        double r719326 = y;
        double r719327 = x;
        double r719328 = 1.0;
        double r719329 = pow(r719327, r719328);
        double r719330 = r719326 / r719327;
        double r719331 = r719329 / r719330;
        double r719332 = r719326 + r719331;
        double r719333 = z;
        double r719334 = r719333 / r719326;
        double r719335 = r719328 / r719333;
        double r719336 = r719334 / r719335;
        double r719337 = r719332 - r719336;
        double r719338 = r719325 * r719337;
        return r719338;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original29.0
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 29.0

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

  2. Taylor expanded around 0 12.8

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

  3. Simplified12.8

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

  5. Applied unpow212.8

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

  6. Applied associate-/l*7.0

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

  8. Applied sqr-pow7.0

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

  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

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

  13. Applied associate-/r*0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2020002 +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)))