Average Error: 0.1 → 0.1
Time: 4.5s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\mathsf{fma}\left(\log y, -0.5 - y, y\right) - \left(z - x\right) \]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
\mathsf{fma}\left(\log y, -0.5 - y, y\right) - \left(z - x\right)

(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) (- z x)))
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) - (z - x);
}

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. Applied fma-udef_binary640.1

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

  4. Applied associate--l+_binary640.1

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

  5. Applied associate--r+_binary640.1

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

  6. Simplified0.1

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

  7. Final simplification0.1

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

Reproduce

herbie shell --seed 2022081 
(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))