Average Error: 27.7 → 0.4
Time: 4.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \left(\sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}} \cdot \sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}}\right) \cdot \sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}}\right)\]
\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(\sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}} \cdot \sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}}\right) \cdot \sqrt[3]{\left|z\right| \cdot \frac{\left|z\right|}{y}}\right)
double f(double x, double y, double z) {
        double r641046 = x;
        double r641047 = r641046 * r641046;
        double r641048 = y;
        double r641049 = r641048 * r641048;
        double r641050 = r641047 + r641049;
        double r641051 = z;
        double r641052 = r641051 * r641051;
        double r641053 = r641050 - r641052;
        double r641054 = 2.0;
        double r641055 = r641048 * r641054;
        double r641056 = r641053 / r641055;
        return r641056;
}
↓
double f(double x, double y, double z) {
        double r641057 = 0.5;
        double r641058 = y;
        double r641059 = x;
        double r641060 = r641059 / r641058;
        double r641061 = r641059 * r641060;
        double r641062 = r641058 + r641061;
        double r641063 = z;
        double r641064 = fabs(r641063);
        double r641065 = r641064 / r641058;
        double r641066 = r641064 * r641065;
        double r641067 = cbrt(r641066);
        double r641068 = r641067 * r641067;
        double r641069 = r641068 * r641067;
        double r641070 = r641062 - r641069;
        double r641071 = r641057 * r641070;
        return r641071;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original27.7
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 27.7

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

  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

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

  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-frac35.5

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

  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. Simplified6.4

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

  12. Applied *-un-lft-identity6.4

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

  13. Applied add-sqr-sqrt6.4

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

  14. Applied times-frac6.4

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

  15. Simplified6.4

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

  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. Using strategy rm

  18. Applied add-cube-cbrt0.4

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

  19. Final simplification0.4

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

Reproduce

herbie shell --seed 2020100 
(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)))