Average Error: 0.0 → 0.0
Time: 1.4s
Precision: binary64
\[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]
↓
\[\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right) \]
\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673
↓
\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right)

(FPCore (x y)
 :precision binary64
 (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))
↓
(FPCore (x y) :precision binary64 (fma y (+ x -0.5) (- 0.918938533204673 x)))
double code(double x, double y) {
	return ((x * (y - 1.0)) - (y * 0.5)) + 0.918938533204673;
}
↓
double code(double x, double y) {
	return fma(y, (x + -0.5), (0.918938533204673 - x));
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.0

    \[\left(x \cdot \left(y - 1\right) - y \cdot 0.5\right) + 0.918938533204673 \]

  2. Simplified0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -0.5, 0.918938533204673\right) - x} \]
  3. Using strategy rm

  4. Applied fma-udef_binary640.0

    \[\leadsto \color{blue}{\left(y \cdot \left(x + -0.5\right) + 0.918938533204673\right)} - x \]

  5. Applied associate--l+_binary640.0

    \[\leadsto \color{blue}{y \cdot \left(x + -0.5\right) + \left(0.918938533204673 - x\right)} \]

  6. Applied fma-def_binary640.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right)} \]

  7. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(y, x + -0.5, 0.918938533204673 - x\right) \]

Reproduce

herbie shell --seed 2021211 
(FPCore (x y)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (+ (- (* x (- y 1.0)) (* y 0.5)) 0.918938533204673))