Average Error: 28.4 → 0.2
Time: 3.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r699434 = x;
        double r699435 = r699434 * r699434;
        double r699436 = y;
        double r699437 = r699436 * r699436;
        double r699438 = r699435 + r699437;
        double r699439 = z;
        double r699440 = r699439 * r699439;
        double r699441 = r699438 - r699440;
        double r699442 = 2.0;
        double r699443 = r699436 * r699442;
        double r699444 = r699441 / r699443;
        return r699444;
}
↓
double f(double x, double y, double z) {
        double r699445 = 0.5;
        double r699446 = y;
        double r699447 = x;
        double r699448 = r699447 / r699446;
        double r699449 = 1.0;
        double r699450 = r699449 / r699447;
        double r699451 = r699448 / r699450;
        double r699452 = r699446 + r699451;
        double r699453 = z;
        double r699454 = fabs(r699453);
        double r699455 = r699454 / r699446;
        double r699456 = r699454 * r699455;
        double r699457 = r699452 - r699456;
        double r699458 = r699445 * r699457;
        return r699458;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

Enter valid numbers for all inputs

Target

Original28.4
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 28.4

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

  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-sqrt12.2

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

  7. Applied times-frac12.2

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

  8. Simplified12.2

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

  9. Simplified6.6

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

  11. Applied unpow26.6

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

  12. Applied associate-/l*0.1

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

  14. Applied div-inv0.2

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

  15. Applied associate-/r*0.2

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

  16. Final simplification0.2

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

Reproduce

herbie shell --seed 2020064 
(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)))