Average Error: 28.4 → 7.2
Time: 6.7s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{1}{2} \cdot \left(\frac{x \cdot x}{y} + \left(y - \frac{z}{\frac{y}{z}}\right)\right)
double f(double x, double y, double z) {
        double r476362 = x;
        double r476363 = r476362 * r476362;
        double r476364 = y;
        double r476365 = r476364 * r476364;
        double r476366 = r476363 + r476365;
        double r476367 = z;
        double r476368 = r476367 * r476367;
        double r476369 = r476366 - r476368;
        double r476370 = 2.0;
        double r476371 = r476364 * r476370;
        double r476372 = r476369 / r476371;
        return r476372;
}
↓
double f(double x, double y, double z) {
        double r476373 = 1.0;
        double r476374 = 2.0;
        double r476375 = r476373 / r476374;
        double r476376 = x;
        double r476377 = r476376 * r476376;
        double r476378 = y;
        double r476379 = r476377 / r476378;
        double r476380 = z;
        double r476381 = r476378 / r476380;
        double r476382 = r476380 / r476381;
        double r476383 = r476378 - r476382;
        double r476384 = r476379 + r476383;
        double r476385 = r476375 * r476384;
        return r476385;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.4
Target0.2
Herbie7.2
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.4

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

  2. Taylor expanded around 0 12.9

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

  3. Simplified12.9

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied unpow212.9

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

  6. Applied associate-/l*7.2

    \[\leadsto \frac{1}{2} \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \color{blue}{\frac{z}{\frac{y}{z}}}\right)\]
  7. Using strategy rm

  8. Applied unpow27.2

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

  9. Applied associate-/l*0.2

    \[\leadsto \frac{1}{2} \cdot \left(\left(y + \color{blue}{\frac{x}{\frac{y}{x}}}\right) - \frac{z}{\frac{y}{z}}\right)\]
  10. Using strategy rm

  11. Applied clear-num0.2

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

  12. Final simplification7.2

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

Reproduce

herbie shell --seed 2019304 
(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)))