Average Error: 27.9 → 0.4
Time: 3.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \left(z \cdot \left(\sqrt[3]{\frac{z}{y}} \cdot \sqrt[3]{\frac{z}{y}}\right)\right) \cdot \sqrt[3]{\frac{z}{y}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \left(z \cdot \left(\sqrt[3]{\frac{z}{y}} \cdot \sqrt[3]{\frac{z}{y}}\right)\right) \cdot \sqrt[3]{\frac{z}{y}}\right)
double f(double x, double y, double z) {
        double r722649 = x;
        double r722650 = r722649 * r722649;
        double r722651 = y;
        double r722652 = r722651 * r722651;
        double r722653 = r722650 + r722652;
        double r722654 = z;
        double r722655 = r722654 * r722654;
        double r722656 = r722653 - r722655;
        double r722657 = 2.0;
        double r722658 = r722651 * r722657;
        double r722659 = r722656 / r722658;
        return r722659;
}
↓
double f(double x, double y, double z) {
        double r722660 = 0.5;
        double r722661 = y;
        double r722662 = x;
        double r722663 = fabs(r722662);
        double r722664 = r722663 / r722661;
        double r722665 = r722663 * r722664;
        double r722666 = r722661 + r722665;
        double r722667 = z;
        double r722668 = r722667 / r722661;
        double r722669 = cbrt(r722668);
        double r722670 = r722669 * r722669;
        double r722671 = r722667 * r722670;
        double r722672 = r722671 * r722669;
        double r722673 = r722666 - r722672;
        double r722674 = r722660 * r722673;
        return r722674;
}

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

Original27.9
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 27.9

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

  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-sqrt38.2

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

  7. Applied unpow-prod-down38.2

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

  8. Applied times-frac35.8

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

  9. Simplified35.8

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

  10. Simplified6.9

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

  12. Applied *-un-lft-identity6.9

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

  13. Applied add-sqr-sqrt6.9

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

  14. Applied times-frac6.9

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

  15. Simplified6.9

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

  16. Simplified0.2

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

  18. Applied add-cube-cbrt0.4

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

  19. Applied associate-*r*0.4

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

  20. Final simplification0.4

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

Reproduce

herbie shell --seed 2020049 
(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)))