Numeric.SpecFunctions:stirlingError from math-functions-0.1.5.2

Average Error: 0.1 → 0.5

Time: 9.1s

Precision: 64

Math

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

↓

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

C

double code(double x, double y, double z) {
	return ((double) (((double) (((double) (x - ((double) (((double) (y + 0.5)) * ((double) log(y)))))) + y)) - z));
}

↓

double code(double x, double y, double z) {
	return ((double) fma(((double) (((double) cbrt(x)) * ((double) cbrt(x)))), ((double) cbrt(x)), ((double) -(((double) (((double) fma(((double) log(y)), ((double) (y + 0.5)), z)) - y))))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	0.1
Target	0.1
Herbie	0.5

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

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

4. Applied add-cube-cbrt 0.5

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

5. Applied fma-neg 0.5

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

6. Final simplification 0.5

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

Reproduce

herbie shell --seed 2020114 +o rules:numerics
(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))