Average Error: 29.0 → 0.1
Time: 10.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(y + x \cdot \frac{x}{y}\right) - z \cdot \left(\frac{z}{y} \cdot 0.5\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(y + x \cdot \frac{x}{y}\right) - z \cdot \left(\frac{z}{y} \cdot 0.5\right)
double f(double x, double y, double z) {
        double r470187 = x;
        double r470188 = r470187 * r470187;
        double r470189 = y;
        double r470190 = r470189 * r470189;
        double r470191 = r470188 + r470190;
        double r470192 = z;
        double r470193 = r470192 * r470192;
        double r470194 = r470191 - r470193;
        double r470195 = 2.0;
        double r470196 = r470189 * r470195;
        double r470197 = r470194 / r470196;
        return r470197;
}
↓
double f(double x, double y, double z) {
        double r470198 = 0.5;
        double r470199 = y;
        double r470200 = x;
        double r470201 = r470200 / r470199;
        double r470202 = r470200 * r470201;
        double r470203 = r470199 + r470202;
        double r470204 = r470198 * r470203;
        double r470205 = z;
        double r470206 = r470205 / r470199;
        double r470207 = r470206 * r470198;
        double r470208 = r470205 * r470207;
        double r470209 = r470204 - r470208;
        return r470209;
}

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
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 29.0

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

  2. Taylor expanded around 0 12.9

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

  3. Simplified12.9

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

  5. Applied *-un-lft-identity12.9

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

  6. Applied add-sqr-sqrt38.5

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

  7. Applied unpow-prod-down38.5

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

  8. Applied times-frac35.8

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

  9. Simplified35.8

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

  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-sqrt34.8

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

  14. Applied unpow-prod-down34.8

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

  15. Applied times-frac31.3

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

  16. Simplified31.3

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

  17. Simplified0.2

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

  19. Applied add-cube-cbrt0.4

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

  20. Applied associate-*l*0.4

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

  21. Final simplification0.1

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

Reproduce

herbie shell --seed 2019291 
(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)))