Average Error: 9.5 → 0.0
Time: 3.4s
Precision: binary64
\[\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1} \]
↓
\[\begin{array}{l} t_0 := \frac{x}{x + 1}\\ \mathsf{fma}\left(t_0, \frac{x}{y}, t_0\right) \end{array} \]
\frac{x \cdot \left(\frac{x}{y} + 1\right)}{x + 1}
↓
\begin{array}{l}
t_0 := \frac{x}{x + 1}\\
\mathsf{fma}\left(t_0, \frac{x}{y}, t_0\right)
\end{array}

(FPCore (x y) :precision binary64 (/ (* x (+ (/ x y) 1.0)) (+ x 1.0)))
↓
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ x (+ x 1.0)))) (fma t_0 (/ x y) t_0)))
double code(double x, double y) {
	return (x * ((x / y) + 1.0)) / (x + 1.0);
}
↓
double code(double x, double y) {
	double t_0 = x / (x + 1.0);
	return fma(t_0, (x / y), t_0);
}

Error

Bits error versus x

Bits error versus y

Target

Original9.5
Target0.1
Herbie0.0
\[\frac{x}{1} \cdot \frac{\frac{x}{y} + 1}{x + 1} \]

Derivation

  1. Initial program 9.5

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

  2. Simplified9.5

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

  3. Taylor expanded in y around 0 13.8

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

  4. Applied sqr-pow_binary6413.8

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

  5. Applied times-frac_binary640.0

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

  6. Applied fma-def_binary640.0

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

  7. Final simplification0.0

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

Reproduce

herbie shell --seed 2022077 
(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)))