Average Error: 28.4 → 0.2
Time: 3.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - z \cdot \frac{z}{y}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \left|x\right| \cdot \frac{\left|x\right|}{y}\right) - z \cdot \frac{z}{y}\right)
double f(double x, double y, double z) {
        double r911240 = x;
        double r911241 = r911240 * r911240;
        double r911242 = y;
        double r911243 = r911242 * r911242;
        double r911244 = r911241 + r911243;
        double r911245 = z;
        double r911246 = r911245 * r911245;
        double r911247 = r911244 - r911246;
        double r911248 = 2.0;
        double r911249 = r911242 * r911248;
        double r911250 = r911247 / r911249;
        return r911250;
}
↓
double f(double x, double y, double z) {
        double r911251 = 0.5;
        double r911252 = y;
        double r911253 = x;
        double r911254 = fabs(r911253);
        double r911255 = r911254 / r911252;
        double r911256 = r911254 * r911255;
        double r911257 = r911252 + r911256;
        double r911258 = z;
        double r911259 = r911258 / r911252;
        double r911260 = r911258 * r911259;
        double r911261 = r911257 - r911260;
        double r911262 = r911251 * r911261;
        return r911262;
}

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 12.8

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

  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 *-un-lft-identity12.8

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

  6. Applied add-sqr-sqrt38.3

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

  7. Applied unpow-prod-down38.3

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

  8. Applied times-frac35.5

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

  9. Simplified35.5

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

  10. Simplified6.9

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

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

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

  13. Applied add-sqr-sqrt6.9

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

  14. Applied times-frac6.9

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

  15. Simplified6.9

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

  16. Simplified0.2

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

  17. Final simplification0.2

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

Reproduce

herbie shell --seed 2020057 
(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)))