Average Error: 28.5 → 0.2
Time: 26.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\left(y + \frac{\frac{x}{y}}{\frac{1}{x}}\right) - \frac{z}{\frac{y}{z}}}{2}
double f(double x, double y, double z) {
        double r28497458 = x;
        double r28497459 = r28497458 * r28497458;
        double r28497460 = y;
        double r28497461 = r28497460 * r28497460;
        double r28497462 = r28497459 + r28497461;
        double r28497463 = z;
        double r28497464 = r28497463 * r28497463;
        double r28497465 = r28497462 - r28497464;
        double r28497466 = 2.0;
        double r28497467 = r28497460 * r28497466;
        double r28497468 = r28497465 / r28497467;
        return r28497468;
}
↓
double f(double x, double y, double z) {
        double r28497469 = y;
        double r28497470 = x;
        double r28497471 = r28497470 / r28497469;
        double r28497472 = 1.0;
        double r28497473 = r28497472 / r28497470;
        double r28497474 = r28497471 / r28497473;
        double r28497475 = r28497469 + r28497474;
        double r28497476 = z;
        double r28497477 = r28497469 / r28497476;
        double r28497478 = r28497476 / r28497477;
        double r28497479 = r28497475 - r28497478;
        double r28497480 = 2.0;
        double r28497481 = r28497479 / r28497480;
        return r28497481;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.5
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 28.5

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

  2. Simplified0.1

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

  4. Applied *-un-lft-identity0.1

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

  5. Applied associate-/r*0.1

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

  6. Simplified0.1

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x - z, \frac{z + x}{y}, y\right)}}{2}\]

  7. Taylor expanded around 0 12.7

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

  8. Simplified0.1

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

  10. Applied div-inv0.2

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

  11. Applied associate-/r*0.2

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

  12. Final simplification0.2

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

Reproduce

herbie shell --seed 2019174 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))