Average Error: 28.2 → 0.2
Time: 4.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\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) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r671059 = x;
        double r671060 = r671059 * r671059;
        double r671061 = y;
        double r671062 = r671061 * r671061;
        double r671063 = r671060 + r671062;
        double r671064 = z;
        double r671065 = r671064 * r671064;
        double r671066 = r671063 - r671065;
        double r671067 = 2.0;
        double r671068 = r671061 * r671067;
        double r671069 = r671066 / r671068;
        return r671069;
}
↓
double f(double x, double y, double z) {
        double r671070 = 0.5;
        double r671071 = y;
        double r671072 = x;
        double r671073 = r671072 / r671071;
        double r671074 = r671072 * r671073;
        double r671075 = r671071 + r671074;
        double r671076 = z;
        double r671077 = fabs(r671076);
        double r671078 = r671077 / r671071;
        double r671079 = r671077 * r671078;
        double r671080 = r671075 - r671079;
        double r671081 = r671070 * r671080;
        return r671081;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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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. Taylor expanded around 0 12.7

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

  3. Simplified12.7

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

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

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

  6. Applied add-sqr-sqrt38.2

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

  7. Applied unpow-prod-down38.2

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

  8. Applied times-frac35.5

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

  9. Simplified35.5

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

  10. Simplified7.3

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

  12. Applied *-un-lft-identity7.3

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

  13. Applied add-sqr-sqrt7.3

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

  14. Applied times-frac7.3

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

  15. Simplified7.3

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020047 +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)))