Average Error: 29.3 → 0.1
Time: 2.6s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{z}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r705597 = x;
        double r705598 = r705597 * r705597;
        double r705599 = y;
        double r705600 = r705599 * r705599;
        double r705601 = r705598 + r705600;
        double r705602 = z;
        double r705603 = r705602 * r705602;
        double r705604 = r705601 - r705603;
        double r705605 = 2.0;
        double r705606 = r705599 * r705605;
        double r705607 = r705604 / r705606;
        return r705607;
}
↓
double f(double x, double y, double z) {
        double r705608 = 0.5;
        double r705609 = y;
        double r705610 = x;
        double r705611 = r705609 / r705610;
        double r705612 = r705610 / r705611;
        double r705613 = r705609 + r705612;
        double r705614 = z;
        double r705615 = r705609 / r705614;
        double r705616 = r705614 / r705615;
        double r705617 = r705613 - r705616;
        double r705618 = r705608 * r705617;
        return r705618;
}

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

Original29.3
Target0.2
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 29.3

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

  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 unpow26.9

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

  9. Applied associate-/l*0.1

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

  10. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 
(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)))