Average Error: 28.6 → 0.1
Time: 15.2s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y - \left(\left(z + x\right) \cdot \frac{z}{y} - \frac{x}{\frac{y}{z + x}}\right)}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y - \left(\left(z + x\right) \cdot \frac{z}{y} - \frac{x}{\frac{y}{z + x}}\right)}{2}
double f(double x, double y, double z) {
        double r569465 = x;
        double r569466 = r569465 * r569465;
        double r569467 = y;
        double r569468 = r569467 * r569467;
        double r569469 = r569466 + r569468;
        double r569470 = z;
        double r569471 = r569470 * r569470;
        double r569472 = r569469 - r569471;
        double r569473 = 2.0;
        double r569474 = r569467 * r569473;
        double r569475 = r569472 / r569474;
        return r569475;
}
↓
double f(double x, double y, double z) {
        double r569476 = y;
        double r569477 = z;
        double r569478 = x;
        double r569479 = r569477 + r569478;
        double r569480 = r569477 / r569476;
        double r569481 = r569479 * r569480;
        double r569482 = r569476 / r569479;
        double r569483 = r569478 / r569482;
        double r569484 = r569481 - r569483;
        double r569485 = r569476 - r569484;
        double r569486 = 2.0;
        double r569487 = r569485 / r569486;
        return r569487;
}

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

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

  4. Applied div-sub0.1

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

  5. Simplified0.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2019196 
(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)))