Average Error: 28.8 → 0.1
Time: 4.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)\]
\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}^{1}}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r751669 = x;
        double r751670 = r751669 * r751669;
        double r751671 = y;
        double r751672 = r751671 * r751671;
        double r751673 = r751670 + r751672;
        double r751674 = z;
        double r751675 = r751674 * r751674;
        double r751676 = r751673 - r751675;
        double r751677 = 2.0;
        double r751678 = r751671 * r751677;
        double r751679 = r751676 / r751678;
        return r751679;
}
↓
double f(double x, double y, double z) {
        double r751680 = 0.5;
        double r751681 = y;
        double r751682 = x;
        double r751683 = r751681 / r751682;
        double r751684 = r751682 / r751683;
        double r751685 = r751681 + r751684;
        double r751686 = z;
        double r751687 = 1.0;
        double r751688 = pow(r751686, r751687);
        double r751689 = r751681 / r751686;
        double r751690 = r751688 / r751689;
        double r751691 = r751685 - r751690;
        double r751692 = r751680 * r751691;
        return r751692;
}

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

Derivation

  1. Initial program 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 sqr-pow7.2

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

  9. Applied associate-/l*0.1

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

  10. Simplified0.1

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

  11. Final simplification0.1

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

Reproduce

herbie shell --seed 2020034 
(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)))