Average Error: 9.2 → 0.0
Time: 2.2s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
↓
\[\frac{x}{y + \frac{y}{x}} + \frac{x}{x + 1} \]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\frac{x}{y + \frac{y}{x}} + \frac{x}{x + 1}

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
↓
(FPCore (x y) :precision binary64 (+ (/ x (+ y (/ y x))) (/ x (+ x 1.0))))
double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}
↓
double code(double x, double y) {
	return (x / (y + (y / x))) + (x / (x + 1.0));
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.2
Target0.1
Herbie0.0
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \]

Derivation

  1. Initial program 9.2

    \[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]

  2. Simplified9.2

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, \frac{x}{y}, x\right)}{x + 1}} \]

  3. Taylor expanded in y around 0 13.7

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

  4. Applied unpow2_binary6413.7

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

  5. Applied associate-/l*_binary644.7

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

  6. Simplified4.7

    \[\leadsto \frac{x}{\color{blue}{\frac{\mathsf{fma}\left(x, y, y\right)}{x}}} + \frac{x}{1 + x} \]

  7. Taylor expanded in x around 0 0.0

    \[\leadsto \frac{x}{\color{blue}{\frac{y}{x} + y}} + \frac{x}{1 + x} \]

  8. Simplified0.0

    \[\leadsto \frac{x}{\color{blue}{y + \frac{y}{x}}} + \frac{x}{1 + x} \]

  9. Final simplification0.0

    \[\leadsto \frac{x}{y + \frac{y}{x}} + \frac{x}{x + 1} \]

Reproduce

herbie shell --seed 2021280 
(FPCore (x y)
  :name "Codec.Picture.Types:toneMapping from JuicyPixels-3.2.6.1"
  :precision binary64

  :herbie-target
  (* (/ x 1.0) (/ (+ (/ x y) 1.0) (+ x 1.0)))

  (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))