Average Error: 28.9 → 0.2
Time: 11.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + {x}^{1} \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + {x}^{1} \cdot \frac{x}{y}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r494353 = x;
        double r494354 = r494353 * r494353;
        double r494355 = y;
        double r494356 = r494355 * r494355;
        double r494357 = r494354 + r494356;
        double r494358 = z;
        double r494359 = r494358 * r494358;
        double r494360 = r494357 - r494359;
        double r494361 = 2.0;
        double r494362 = r494355 * r494361;
        double r494363 = r494360 / r494362;
        return r494363;
}
↓
double f(double x, double y, double z) {
        double r494364 = 0.5;
        double r494365 = y;
        double r494366 = x;
        double r494367 = 1.0;
        double r494368 = pow(r494366, r494367);
        double r494369 = r494366 / r494365;
        double r494370 = r494368 * r494369;
        double r494371 = r494365 + r494370;
        double r494372 = z;
        double r494373 = fabs(r494372);
        double r494374 = r494373 / r494365;
        double r494375 = r494373 * r494374;
        double r494376 = r494371 - r494375;
        double r494377 = r494364 * r494376;
        return r494377;
}

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

Original28.9
Target0.1
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 28.9

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

  2. Taylor expanded around 0 12.7

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

  3. Simplified12.7

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

  5. Applied *-un-lft-identity12.7

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

  6. Applied add-sqr-sqrt12.7

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

  7. Applied times-frac12.7

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

  8. Simplified12.7

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

  9. Simplified7.0

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

  11. Applied sqr-pow7.0

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

  12. Applied associate-/l*0.2

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

  13. Simplified0.2

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

  15. Applied div-inv0.2

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2019297 
(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)))