Average Error: 0.1 → 0.1
Time: 5.1s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\mathsf{fma}\left(\log y, -0.5 - y, y + x\right) - z \]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\mathsf{fma}\left(\log y, -0.5 - y, y + x\right) - z

(FPCore (x y z) :precision binary64 (- (+ (- x (* (+ y 0.5) (log y))) y) z))
↓
(FPCore (x y z) :precision binary64 (- (fma (log y) (- -0.5 y) (+ y x)) 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 fma(log(y), (-0.5 - y), (y + x)) - 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. Taylor expanded in x around 0 0.1

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

  3. Simplified0.1

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

  4. Applied fma-udef_binary640.1

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

  5. Applied associate-+l+_binary640.1

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

  6. Taylor expanded in y around 0 0.1

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

  7. Simplified0.1

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

  8. Final simplification0.1

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

Reproduce

herbie shell --seed 2022077 
(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))