Average Error: 28.6 → 0.1
Time: 2.0m
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \left(\frac{x}{\frac{y}{x}} - \frac{\frac{z}{y}}{\frac{1}{z}}\right)}{2}
double f(double x, double y, double z) {
        double r802479 = x;
        double r802480 = r802479 * r802479;
        double r802481 = y;
        double r802482 = r802481 * r802481;
        double r802483 = r802480 + r802482;
        double r802484 = z;
        double r802485 = r802484 * r802484;
        double r802486 = r802483 - r802485;
        double r802487 = 2.0;
        double r802488 = r802481 * r802487;
        double r802489 = r802486 / r802488;
        return r802489;
}
↓
double f(double x, double y, double z) {
        double r802490 = y;
        double r802491 = x;
        double r802492 = r802490 / r802491;
        double r802493 = r802491 / r802492;
        double r802494 = z;
        double r802495 = r802494 / r802490;
        double r802496 = 1.0;
        double r802497 = r802496 / r802494;
        double r802498 = r802495 / r802497;
        double r802499 = r802493 - r802498;
        double r802500 = r802490 + r802499;
        double r802501 = 2.0;
        double r802502 = r802500 / r802501;
        return r802502;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

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Results

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Target

Original28.6
Target0.1
Herbie0.1
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 28.6

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

  2. Simplified0.1

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

  3. Taylor expanded around 0 13.2

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

  4. Simplified0.1

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

  6. Applied div-inv0.1

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

  7. Applied associate-/r*0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 +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)))