Average Error: 0.1 → 0.1
Time: 2.1s
Precision: binary64
\[\left(1 - x\right) + y \cdot \sqrt{x} \]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x \]
\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))
↓
(FPCore (x y) :precision binary64 (- (fma y (sqrt x) 1.0) x))
double code(double x, double y) {
	return (1.0 - x) + (y * sqrt(x));
}
↓
double code(double x, double y) {
	return fma(y, sqrt(x), 1.0) - x;
}

Error

Bits error versus x

Bits error versus y

Derivation

  1. Initial program 0.1

    \[\left(1 - x\right) + y \cdot \sqrt{x} \]

  2. Taylor expanded in y around 0 0.1

    \[\leadsto \color{blue}{\left(y \cdot \sqrt{x} + 1\right) - x} \]

  3. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1\right) - x} \]

  4. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(y, \sqrt{x}, 1\right) - x \]

Reproduce

herbie shell --seed 2021275 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))