Average Error: 0.1 → 0.1
Time: 5.0s
Precision: binary64
Cost: 13120
\[\left(1 - x\right) + y \cdot \sqrt{x} \]
↓
\[\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right) \]
(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));
}

function code(x, y)
	return Float64(Float64(1.0 - x) + Float64(y * sqrt(x)))
end
↓
function code(x, y)
	return fma(y, sqrt(x), Float64(1.0 - x))
end

code[x_, y_] := N[(N[(1.0 - x), $MachinePrecision] + N[(y * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_] := N[(y * N[Sqrt[x], $MachinePrecision] + N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

\left(1 - x\right) + y \cdot \sqrt{x}
↓
\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)

Error

Derivation

  1. Initial program 0.1

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

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)} \]
    Proof
    (fma.f64 y (sqrt.f64 x) (-.f64 1 x)): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y (sqrt.f64 x)) (-.f64 1 x))): 2 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (-.f64 1 x) (*.f64 y (sqrt.f64 x)))): 0 points increase in error, 0 points decrease in error

  3. Final simplification0.1

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

Alternatives

Alternative 1
Error5.0
Cost6856
\[\begin{array}{l} t_0 := y \cdot \sqrt{x}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+93}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+55}:\\ \;\;\;\;1 - x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 2
Error0.1
Cost6848
\[\left(1 - x\right) + y \cdot \sqrt{x} \]
Alternative 3
Error21.8
Cost260
\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
Alternative 4
Error20.9
Cost192
\[1 - x \]
Alternative 5
Error42.5
Cost64
\[1 \]

Error

Reproduce

herbie shell --seed 2022330 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))