Average Error: 28.5 → 0.1
Time: 18.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y}{2} - \frac{\frac{x + z}{2}}{\frac{y}{z - x}}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y}{2} - \frac{\frac{x + z}{2}}{\frac{y}{z - x}}
double f(double x, double y, double z) {
        double r43069362 = x;
        double r43069363 = r43069362 * r43069362;
        double r43069364 = y;
        double r43069365 = r43069364 * r43069364;
        double r43069366 = r43069363 + r43069365;
        double r43069367 = z;
        double r43069368 = r43069367 * r43069367;
        double r43069369 = r43069366 - r43069368;
        double r43069370 = 2.0;
        double r43069371 = r43069364 * r43069370;
        double r43069372 = r43069369 / r43069371;
        return r43069372;
}
↓
double f(double x, double y, double z) {
        double r43069373 = y;
        double r43069374 = 2.0;
        double r43069375 = r43069373 / r43069374;
        double r43069376 = x;
        double r43069377 = z;
        double r43069378 = r43069376 + r43069377;
        double r43069379 = r43069378 / r43069374;
        double r43069380 = r43069377 - r43069376;
        double r43069381 = r43069373 / r43069380;
        double r43069382 = r43069379 / r43069381;
        double r43069383 = r43069375 - r43069382;
        return r43069383;
}

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

Derivation

  1. Initial program 28.5

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

  2. Simplified12.7

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

  4. Applied difference-of-squares12.7

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

  5. Applied associate-/l*0.1

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

  7. Applied div-inv0.2

    \[\leadsto \frac{y - \frac{z + x}{\color{blue}{y \cdot \frac{1}{z - x}}}}{2}\]
  8. Using strategy rm

  9. Applied div-sub0.2

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

  10. Simplified0.1

    \[\leadsto \frac{y}{2} - \color{blue}{\frac{\frac{x + z}{2}}{\frac{y}{z - x}}}\]

  11. Final simplification0.1

    \[\leadsto \frac{y}{2} - \frac{\frac{x + z}{2}}{\frac{y}{z - x}}\]

Reproduce

herbie shell --seed 2019174 
(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)))