Average Error: 29.1 → 0.1
Time: 3.8s
Precision: binary64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[\frac{y + x \cdot \frac{x}{y}}{2} - \frac{z}{y} \cdot \frac{z}{2}\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
\frac{y + x \cdot \frac{x}{y}}{2} - \frac{z}{y} \cdot \frac{z}{2}
(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))
↓
(FPCore (x y z)
 :precision binary64
 (- (/ (+ y (* x (/ x y))) 2.0) (* (/ z y) (/ z 2.0))))
double code(double x, double y, double z) {
	return (((double) (((double) (((double) (x * x)) + ((double) (y * y)))) - ((double) (z * z)))) / ((double) (y * 2.0)));
}
↓
double code(double x, double y, double z) {
	return ((double) ((((double) (y + ((double) (x * (x / y))))) / 2.0) - ((double) ((z / y) * (z / 2.0)))));
}

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

Original29.1
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 Error: 29.1 bits

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

  2. SimplifiedError: 0.2 bits

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

  4. Applied associate-*r/Error: 12.7 bits

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

  5. SimplifiedError: 12.7 bits

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

  7. Applied div-subError: 12.7 bits

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

  8. Applied associate-+r-Error: 12.7 bits

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

  9. Applied div-subError: 12.7 bits

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

  10. SimplifiedError: 7.0 bits

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

  11. SimplifiedError: 0.1 bits

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

  12. Final simplificationError: 0.1 bits

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

Reproduce

herbie shell --seed 2020204 
(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 2.0)))