Average Error: 29.3 → 0.1
Time: 3.8s
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 r685278 = x;
        double r685279 = r685278 * r685278;
        double r685280 = y;
        double r685281 = r685280 * r685280;
        double r685282 = r685279 + r685281;
        double r685283 = z;
        double r685284 = r685283 * r685283;
        double r685285 = r685282 - r685284;
        double r685286 = 2.0;
        double r685287 = r685280 * r685286;
        double r685288 = r685285 / r685287;
        return r685288;
}
↓
double f(double x, double y, double z) {
        double r685289 = 0.5;
        double r685290 = y;
        double r685291 = x;
        double r685292 = 1.0;
        double r685293 = pow(r685291, r685292);
        double r685294 = r685290 / r685291;
        double r685295 = r685293 / r685294;
        double r685296 = r685290 + r685295;
        double r685297 = z;
        double r685298 = pow(r685297, r685292);
        double r685299 = r685290 / r685297;
        double r685300 = r685298 / r685299;
        double r685301 = r685296 - r685300;
        double r685302 = r685289 * r685301;
        return r685302;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Your Program's Arguments

Results

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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 
(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)))