Average Error: 9.0 → 0.1
Time: 2.8s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
↓
\[\left(1 + \frac{x}{y}\right) \cdot \frac{x}{1 + x} \]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\left(1 + \frac{x}{y}\right) \cdot \frac{x}{1 + x}

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

Error

Bits error versus x

Bits error versus y

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Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

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

Derivation

  1. Initial program 9.0

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

  2. Simplified9.0

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

  3. Applied *-un-lft-identity_binary649.0

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

  4. Applied *-un-lft-identity_binary649.0

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

  5. Applied times-frac_binary649.0

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

  6. Simplified9.0

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2021307 
(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)))