Average Error: 0.1 → 0.1
Time: 4.4s
Precision: 64
\[\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z\]
↓
\[x - \left(\mathsf{fma}\left(\log y, y + 0.5, z - y\right) + \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\right)\]
\left(\left(x - \left(y + 0.5\right) \cdot \log y\right) + y\right) - z
↓
x - \left(\mathsf{fma}\left(\log y, y + 0.5, z - y\right) + \mathsf{fma}\left(-\sqrt{y}, \sqrt{y}, \sqrt{y} \cdot \sqrt{y}\right)\right)
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 (x - (fma(log(y), (y + 0.5), (z - y)) + fma(-sqrt(y), sqrt(y), (sqrt(y) * sqrt(y)))));
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Applied fma-udef0.1

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

  5. Applied associate--l+0.1

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

  7. Applied add-sqr-sqrt0.1

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

  8. Applied add-sqr-sqrt31.8

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

  9. Applied prod-diff31.8

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

  10. Applied associate-+r+31.8

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

  11. Simplified0.1

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

  12. Final simplification0.1

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

Reproduce

herbie shell --seed 2020060 +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))