Average Error: 28.7 → 0.1
Time: 16.1s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + \left(x + z\right) \cdot \frac{x - z}{y}}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + \left(x + z\right) \cdot \frac{x - z}{y}}{2}
double f(double x, double y, double z) {
        double r434496 = x;
        double r434497 = r434496 * r434496;
        double r434498 = y;
        double r434499 = r434498 * r434498;
        double r434500 = r434497 + r434499;
        double r434501 = z;
        double r434502 = r434501 * r434501;
        double r434503 = r434500 - r434502;
        double r434504 = 2.0;
        double r434505 = r434498 * r434504;
        double r434506 = r434503 / r434505;
        return r434506;
}
↓
double f(double x, double y, double z) {
        double r434507 = y;
        double r434508 = x;
        double r434509 = z;
        double r434510 = r434508 + r434509;
        double r434511 = r434508 - r434509;
        double r434512 = r434511 / r434507;
        double r434513 = r434510 * r434512;
        double r434514 = r434507 + r434513;
        double r434515 = 2.0;
        double r434516 = r434514 / r434515;
        return r434516;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original28.7
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 28.7

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

  2. Simplified12.6

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

  4. Applied *-un-lft-identity12.6

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

  5. Applied difference-of-squares12.6

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

  6. Applied times-frac0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2019235 +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)))