Average Error: 28.5 → 0.2
Time: 3.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - \frac{1}{\frac{\frac{y}{z}}{z}}\right)\]
\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) - \frac{1}{\frac{\frac{y}{z}}{z}}\right)
double f(double x, double y, double z) {
        double r734436 = x;
        double r734437 = r734436 * r734436;
        double r734438 = y;
        double r734439 = r734438 * r734438;
        double r734440 = r734437 + r734439;
        double r734441 = z;
        double r734442 = r734441 * r734441;
        double r734443 = r734440 - r734442;
        double r734444 = 2.0;
        double r734445 = r734438 * r734444;
        double r734446 = r734443 / r734445;
        return r734446;
}
↓
double f(double x, double y, double z) {
        double r734447 = 0.5;
        double r734448 = y;
        double r734449 = x;
        double r734450 = fabs(r734449);
        double r734451 = r734450 / r734448;
        double r734452 = r734450 * r734451;
        double r734453 = r734448 + r734452;
        double r734454 = 1.0;
        double r734455 = z;
        double r734456 = r734448 / r734455;
        double r734457 = r734456 / r734455;
        double r734458 = r734454 / r734457;
        double r734459 = r734453 - r734458;
        double r734460 = r734447 * r734459;
        return r734460;
}

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

Original28.5
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 28.5

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

  2. Taylor expanded around 0 12.8

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

  3. Simplified12.8

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

  5. Applied unpow212.8

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

  6. Applied associate-/l*6.9

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

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

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

  9. Applied add-sqr-sqrt6.9

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

  10. Applied times-frac6.9

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

  11. Simplified6.9

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

  12. Simplified0.2

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

  14. Applied clear-num0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2020060 +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)))