Average Error: 28.4 → 0.2
Time: 4.0s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{x}{y} \cdot x\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{x}{y} \cdot x\right) - \left|z\right| \cdot \frac{\left|z\right|}{y}\right)
double f(double x, double y, double z) {
        double r701240 = x;
        double r701241 = r701240 * r701240;
        double r701242 = y;
        double r701243 = r701242 * r701242;
        double r701244 = r701241 + r701243;
        double r701245 = z;
        double r701246 = r701245 * r701245;
        double r701247 = r701244 - r701246;
        double r701248 = 2.0;
        double r701249 = r701242 * r701248;
        double r701250 = r701247 / r701249;
        return r701250;
}
↓
double f(double x, double y, double z) {
        double r701251 = 0.5;
        double r701252 = y;
        double r701253 = x;
        double r701254 = r701253 / r701252;
        double r701255 = r701254 * r701253;
        double r701256 = r701252 + r701255;
        double r701257 = z;
        double r701258 = fabs(r701257);
        double r701259 = r701258 / r701252;
        double r701260 = r701258 * r701259;
        double r701261 = r701256 - r701260;
        double r701262 = r701251 * r701261;
        return r701262;
}

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

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 12.5

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

  3. Simplified12.5

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

  5. Applied unpow212.5

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

  6. Applied associate-/l*6.9

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

  8. Applied *-un-lft-identity6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{{z}^{2}}{\color{blue}{1 \cdot y}}\right)\]

  9. Applied add-sqr-sqrt6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \frac{\color{blue}{\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}}}{1 \cdot y}\right)\]

  10. Applied times-frac6.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{x}{\frac{y}{x}}\right) - \color{blue}{\frac{\sqrt{{z}^{2}}}{1} \cdot \frac{\sqrt{{z}^{2}}}{y}}\right)\]

  11. Simplified6.9

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

  12. Simplified0.2

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

  14. Applied associate-/r/0.2

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

  15. Final simplification0.2

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

Reproduce

herbie shell --seed 2019354 
(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)))