Average Error: 0.1 → 0.1
Time: 2.0s
Precision: binary64
\[\frac{x \cdot x - 3}{6} \]
↓
\[\mathsf{fma}\left(x, \frac{x}{6}, -0.5\right) \]
\frac{x \cdot x - 3}{6}
↓
\mathsf{fma}\left(x, \frac{x}{6}, -0.5\right)

(FPCore (x) :precision binary64 (/ (- (* x x) 3.0) 6.0))
↓
(FPCore (x) :precision binary64 (fma x (/ x 6.0) -0.5))
double code(double x) {
	return ((x * x) - 3.0) / 6.0;
}
↓
double code(double x) {
	return fma(x, (x / 6.0), -0.5);
}

Error

Bits error versus x

Derivation

  1. Initial program 0.1

    \[\frac{x \cdot x - 3}{6} \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, -3\right)}{6}} \]

  3. Applied add-sqr-sqrt_binary640.8

    \[\leadsto \frac{\mathsf{fma}\left(x, x, -3\right)}{\color{blue}{\sqrt{6} \cdot \sqrt{6}}} \]

  4. Applied associate-/r*_binary640.8

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(x, x, -3\right)}{\sqrt{6}}}{\sqrt{6}}} \]

  5. Taylor expanded in x around 0 0.3

    \[\leadsto \color{blue}{\frac{{x}^{2}}{{\left(\sqrt{6}\right)}^{2}} - 3 \cdot \frac{1}{{\left(\sqrt{6}\right)}^{2}}} \]

  6. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{x}{6}, -0.5\right)} \]

  7. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(x, \frac{x}{6}, -0.5\right) \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, H"
  :precision binary64
  (/ (- (* x x) 3.0) 6.0))