Average Error: 27.9 → 0.2
Time: 4.4s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \frac{z}{\frac{y}{z}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + x \cdot \frac{x}{y}\right) - \frac{z}{\frac{y}{z}}\right)
double f(double x, double y, double z) {
        double r807281 = x;
        double r807282 = r807281 * r807281;
        double r807283 = y;
        double r807284 = r807283 * r807283;
        double r807285 = r807282 + r807284;
        double r807286 = z;
        double r807287 = r807286 * r807286;
        double r807288 = r807285 - r807287;
        double r807289 = 2.0;
        double r807290 = r807283 * r807289;
        double r807291 = r807288 / r807290;
        return r807291;
}
↓
double f(double x, double y, double z) {
        double r807292 = 0.5;
        double r807293 = y;
        double r807294 = x;
        double r807295 = r807294 / r807293;
        double r807296 = r807294 * r807295;
        double r807297 = r807293 + r807296;
        double r807298 = z;
        double r807299 = r807293 / r807298;
        double r807300 = r807298 / r807299;
        double r807301 = r807297 - r807300;
        double r807302 = r807292 * r807301;
        return r807302;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original27.9
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 27.9

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

  2. Taylor expanded around 0 12.0

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

  3. Simplified12.0

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

  5. Applied unpow212.0

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

  6. Applied associate-/l*6.8

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

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

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

  9. Applied add-sqr-sqrt35.0

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

  10. Applied unpow-prod-down35.0

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

  11. Applied times-frac31.6

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

  12. Simplified31.5

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

  13. Simplified0.2

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

  14. Final simplification0.2

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

Reproduce

herbie shell --seed 2019362 +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)))