Average Error: 28.3 → 0.2
Time: 3.5s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r658060 = x;
        double r658061 = r658060 * r658060;
        double r658062 = y;
        double r658063 = r658062 * r658062;
        double r658064 = r658061 + r658063;
        double r658065 = z;
        double r658066 = r658065 * r658065;
        double r658067 = r658064 - r658066;
        double r658068 = 2.0;
        double r658069 = r658062 * r658068;
        double r658070 = r658067 / r658069;
        return r658070;
}
↓
double f(double x, double y, double z) {
        double r658071 = 0.5;
        double r658072 = y;
        double r658073 = x;
        double r658074 = fabs(r658073);
        double r658075 = r658074 / r658072;
        double r658076 = r658074 * r658075;
        double r658077 = r658072 + r658076;
        double r658078 = z;
        double r658079 = fabs(r658078);
        double r658080 = r658079 / r658072;
        double r658081 = r658079 * r658080;
        double r658082 = r658077 - r658081;
        double r658083 = r658071 * r658082;
        return r658083;
}

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

Original28.3
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 28.3

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

  2. Simplified28.3

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{2}}{y}}\]

  3. Taylor expanded around 0 12.6

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

  4. Simplified12.6

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

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

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

  7. Applied add-sqr-sqrt12.6

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

  8. Applied times-frac12.6

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

  9. Simplified12.6

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

  10. Simplified7.2

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

  12. Applied *-un-lft-identity7.2

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

  13. Applied add-sqr-sqrt7.2

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

  14. Applied times-frac7.2

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

  15. Simplified7.2

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020065 +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)))