Average Error: 28.9 → 0.4
Time: 3.8s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left(\sqrt[3]{x \cdot \frac{x}{y}} \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \left(\sqrt[3]{x \cdot \frac{x}{y}} \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r676117 = x;
        double r676118 = r676117 * r676117;
        double r676119 = y;
        double r676120 = r676119 * r676119;
        double r676121 = r676118 + r676120;
        double r676122 = z;
        double r676123 = r676122 * r676122;
        double r676124 = r676121 - r676123;
        double r676125 = 2.0;
        double r676126 = r676119 * r676125;
        double r676127 = r676124 / r676126;
        return r676127;
}
↓
double f(double x, double y, double z) {
        double r676128 = 0.5;
        double r676129 = y;
        double r676130 = x;
        double r676131 = r676130 / r676129;
        double r676132 = r676130 * r676131;
        double r676133 = cbrt(r676132);
        double r676134 = r676133 * r676133;
        double r676135 = r676134 * r676133;
        double r676136 = r676129 + r676135;
        double r676137 = z;
        double r676138 = r676137 / r676129;
        double r676139 = r676137 * r676138;
        double r676140 = r676136 - r676139;
        double r676141 = r676128 * r676140;
        return r676141;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

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Target

Original28.9
Target0.2
Herbie0.4
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.9

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

  2. Taylor expanded around 0 12.6

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

  3. Simplified12.6

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

  5. Applied *-un-lft-identity12.6

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

  6. Applied add-sqr-sqrt38.9

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

  7. Applied unpow-prod-down38.9

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

  8. Applied times-frac36.0

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

  9. Simplified35.9

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

  10. Simplified6.8

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

  12. Applied *-un-lft-identity6.8

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

  13. Applied add-sqr-sqrt35.2

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

  14. Applied unpow-prod-down35.2

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

  15. Applied times-frac31.7

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

  16. Simplified31.6

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

  17. Simplified0.1

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

  19. Applied add-cube-cbrt0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\left(\sqrt[3]{x \cdot \frac{x}{y}} \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{x}{y}}}\right) - z \cdot \frac{z}{y}\right)\]

  20. Final simplification0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \left(\sqrt[3]{x \cdot \frac{x}{y}} \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) \cdot \sqrt[3]{x \cdot \frac{x}{y}}\right) - z \cdot \frac{z}{y}\right)\]

Reproduce

herbie shell --seed 2020083 
(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)))