Average Error: 28.6 → 0.1
Time: 3.6s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r673057 = x;
        double r673058 = r673057 * r673057;
        double r673059 = y;
        double r673060 = r673059 * r673059;
        double r673061 = r673058 + r673060;
        double r673062 = z;
        double r673063 = r673062 * r673062;
        double r673064 = r673061 - r673063;
        double r673065 = 2.0;
        double r673066 = r673059 * r673065;
        double r673067 = r673064 / r673066;
        return r673067;
}
↓
double f(double x, double y, double z) {
        double r673068 = 0.5;
        double r673069 = y;
        double r673070 = x;
        double r673071 = r673070 / r673069;
        double r673072 = r673070 * r673071;
        double r673073 = r673069 + r673072;
        double r673074 = z;
        double r673075 = r673074 / r673069;
        double r673076 = r673074 * r673075;
        double r673077 = r673073 - r673076;
        double r673078 = r673068 * r673077;
        return r673078;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.6
Target0.2
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 28.6

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

  2. Taylor expanded around 0 12.2

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

  3. Simplified12.2

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

  5. Applied *-un-lft-identity12.2

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

  6. Applied add-sqr-sqrt38.5

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

  7. Applied unpow-prod-down38.5

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

  8. Applied times-frac35.9

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

  9. Simplified35.9

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

  10. Simplified6.7

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

  12. Applied *-un-lft-identity6.7

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

  13. Applied add-sqr-sqrt35.8

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

  14. Applied unpow-prod-down35.8

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

  15. Applied times-frac32.5

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

  16. Simplified32.4

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

  17. Simplified0.1

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

  18. Final simplification0.1

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

Reproduce

herbie shell --seed 2020021 +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)))