Average Error: 28.4 → 0.2
Time: 18.9s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{\frac{x + z}{y} \cdot \left(x - z\right) + y}{2}
double f(double x, double y, double z) {
        double r497349 = x;
        double r497350 = r497349 * r497349;
        double r497351 = y;
        double r497352 = r497351 * r497351;
        double r497353 = r497350 + r497352;
        double r497354 = z;
        double r497355 = r497354 * r497354;
        double r497356 = r497353 - r497355;
        double r497357 = 2.0;
        double r497358 = r497351 * r497357;
        double r497359 = r497356 / r497358;
        return r497359;
}
↓
double f(double x, double y, double z) {
        double r497360 = x;
        double r497361 = z;
        double r497362 = r497360 + r497361;
        double r497363 = y;
        double r497364 = r497362 / r497363;
        double r497365 = r497360 - r497361;
        double r497366 = r497364 * r497365;
        double r497367 = r497366 + r497363;
        double r497368 = 2.0;
        double r497369 = r497367 / r497368;
        return r497369;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

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. Simplified0.2

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

  4. Applied fma-udef0.2

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

  5. Final simplification0.2

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

Reproduce

herbie shell --seed 2019303 +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)))