Average Error: 28.8 → 0.2
Time: 5.3s
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 r503573 = x;
        double r503574 = r503573 * r503573;
        double r503575 = y;
        double r503576 = r503575 * r503575;
        double r503577 = r503574 + r503576;
        double r503578 = z;
        double r503579 = r503578 * r503578;
        double r503580 = r503577 - r503579;
        double r503581 = 2.0;
        double r503582 = r503575 * r503581;
        double r503583 = r503580 / r503582;
        return r503583;
}
↓
double f(double x, double y, double z) {
        double r503584 = 0.5;
        double r503585 = y;
        double r503586 = x;
        double r503587 = r503586 / r503585;
        double r503588 = 1.0;
        double r503589 = r503588 / r503586;
        double r503590 = r503587 / r503589;
        double r503591 = r503585 + r503590;
        double r503592 = z;
        double r503593 = fabs(r503592);
        double r503594 = r503593 / r503585;
        double r503595 = r503593 * r503594;
        double r503596 = r503591 - r503595;
        double r503597 = r503584 * r503596;
        return r503597;
}

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

Original28.8
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 28.8

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

  2. Taylor expanded around 0 13.0

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

  3. Simplified13.0

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

  5. Applied unpow213.0

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

  6. Applied associate-/l*7.2

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

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

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

  9. Applied add-sqr-sqrt7.2

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

  10. Applied times-frac7.2

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

  11. Simplified7.2

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

  12. Simplified0.1

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \left|z\right| \cdot \color{blue}{\frac{\left|z\right|}{y}}\right)\]
  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 2020034 +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)))