Average Error: 0.1 → 0.1
Time: 8.7s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\left(y + x\right) - \mathsf{fma}\left(\log y, y + 0.5, z\right) \]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\left(y + x\right) - \mathsf{fma}\left(\log y, y + 0.5, z\right)

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

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original0.1
Target0.1
Herbie0.1
\[\left(\left(y + x\right) - z\right) - \left(y + 0.5\right) \cdot \log y \]

Derivation

  1. Initial program 0.1

    \[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]

  2. Simplified0.1

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

  3. Taylor expanded in y around 0 0.1

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

  4. Simplified0.1

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

  5. Final simplification0.1

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

Reproduce

herbie shell --seed 2021332 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (- (- (+ y x) z) (* (+ y 0.5) (log y)))

  (- (+ (- x (* (+ y 0.5) (log y))) y) z))