Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A

Average Error: 0.1 → 0.2

Time: 1.2min

Precision: binary64

Cost: 6976

Math

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

↓

\[1 - \left(\mathsf{fma}\left(x, 0.4743083003952569, 1\right) \cdot x\right) \cdot 0.253 \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (+ 0.253 (* x 0.12)))))

↓

(FPCore (x)
 :precision binary64
 (- 1.0 (* (* (fma x 0.4743083003952569 1.0) x) 0.253)))

C

double code(double x) {
	return 1.0 - (x * (0.253 + (x * 0.12)));
}

↓

double code(double x) {
	return 1.0 - ((fma(x, 0.4743083003952569, 1.0) * x) * 0.253);
}

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(0.253 + Float64(x * 0.12))))
end

↓

function code(x)
	return Float64(1.0 - Float64(Float64(fma(x, 0.4743083003952569, 1.0) * x) * 0.253))
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(0.253 + N[(x * 0.12), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(1.0 - N[(N[(N[(x * 0.4743083003952569 + 1.0), $MachinePrecision] * x), $MachinePrecision] * 0.253), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.1

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

2. Applied egg-rr 0.2

   \[\leadsto 1 - \color{blue}{\left(\mathsf{fma}\left(x, 0.4743083003952569, 1\right) \cdot x\right) \cdot 0.253} \]

Alternatives

Alternative 1
Error	0.1
Cost	704

\[1 - x \cdot \left(\frac{x \cdot 0.4743083003952569}{3.952569169960474} + 0.253\right) \]

Alternative 2
Error	0.1
Cost	576

\[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]

Alternative 3
Error	1.9
Cost	448

\[1 - \left(x \cdot 0.12\right) \cdot x \]

Alternative 4
Error	20.7
Cost	320

\[1 - x \cdot 0.253 \]

Alternative 5
Error	21.8
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023033 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))