Average Error: 29.3 → 0.1
Time: 4.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{{z}^{1}}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r691630 = x;
        double r691631 = r691630 * r691630;
        double r691632 = y;
        double r691633 = r691632 * r691632;
        double r691634 = r691631 + r691633;
        double r691635 = z;
        double r691636 = r691635 * r691635;
        double r691637 = r691634 - r691636;
        double r691638 = 2.0;
        double r691639 = r691632 * r691638;
        double r691640 = r691637 / r691639;
        return r691640;
}
↓
double f(double x, double y, double z) {
        double r691641 = 0.5;
        double r691642 = y;
        double r691643 = x;
        double r691644 = 1.0;
        double r691645 = pow(r691643, r691644);
        double r691646 = r691642 / r691643;
        double r691647 = r691645 / r691646;
        double r691648 = r691642 + r691647;
        double r691649 = z;
        double r691650 = pow(r691649, r691644);
        double r691651 = r691642 / r691649;
        double r691652 = r691650 / r691651;
        double r691653 = r691648 - r691652;
        double r691654 = r691641 * r691653;
        return r691654;
}

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 sqr-pow12.8

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

  6. Applied associate-/l*6.9

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

  7. Simplified6.9

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

  9. Applied sqr-pow6.9

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

  10. Applied associate-/l*0.1

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020036 +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)))