Average Error: 0.1 → 0.1
Time: 5.9s
Precision: binary64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z \]
↓
\[\mathsf{fma}\left(\log y, -0.5 - y, y\right) + \left(x - z\right) \]
\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(x - z\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) (- 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 add-cube-cbrt_binary640.5

    \[\leadsto \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x\right) - \color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}} \]

  5. Applied add-cube-cbrt_binary641.3

    \[\leadsto \color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x}\right) \cdot \sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x}} - \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z} \]

  6. Applied prod-diff_binary641.3

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x} \cdot \sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x}, \sqrt[3]{\mathsf{fma}\left(\log y, -0.5 - y, y\right) + x}, -\sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right)} \]

  7. Simplified0.1

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) + \left(x - z\right)\right)} + \mathsf{fma}\left(-\sqrt[3]{z}, \sqrt[3]{z} \cdot \sqrt[3]{z}, \sqrt[3]{z} \cdot \left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right)\right) \]

  8. Simplified0.1

    \[\leadsto \left(\mathsf{fma}\left(\log y, -0.5 - y, y\right) + \left(x - z\right)\right) + \color{blue}{0} \]

  9. Final simplification0.1

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

Reproduce

herbie shell --seed 2022004 
(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))