Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E

Average Error: 0.1 → 0.1

Time: 4.3s

Precision: binary64

Cost: 13120

Math

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

↓

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

FPCore

(FPCore (x y) :precision binary64 (+ (- 1.0 x) (* y (sqrt x))))

↓

(FPCore (x y) :precision binary64 (fma y (sqrt x) (- 1.0 x)))

C

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));
}

Julia

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

Wolfram

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]

Derivation

1. Initial program 0.1

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

2. Simplified 0.1

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

   [Start] 0.1	\[ \left(1 - x\right) + y \cdot \sqrt{x} \]
   +-commutative [=>] 0.1	\[ \color{blue}{y \cdot \sqrt{x} + \left(1 - x\right)} \]
   fma-def [=>] 0.1	\[ \color{blue}{\mathsf{fma}\left(y, \sqrt{x}, 1 - x\right)} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	4.9
Cost	6857

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+89} \lor \neg \left(y \leq 2.4 \cdot 10^{+49}\right):\\ \;\;\;\;y \cdot \sqrt{x}\\ \mathbf{else}:\\ \;\;\;\;1 - x\\ \end{array} \]

Alternative 2
Error	0.1
Cost	6848

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

Alternative 3
Error	21.7
Cost	260

\[\begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]

Alternative 4
Error	21.1
Cost	192

\[1 - x \]

Alternative 5
Error	42.6
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023031 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, E"
  :precision binary64
  (+ (- 1.0 x) (* y (sqrt x))))