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

Average Error: 0.18% → 0.18%

Time: 7.7s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (0.253d0 + (x * 0.12d0)))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 + (x * (x * (-0.12d0)))) + (x * (-0.253d0))
end function

Java

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

↓

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

Python

def code(x):
	return 1.0 - (x * (0.253 + (x * 0.12)))

↓

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

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (0.253 + (x * 0.12)));
end

↓

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

Derivation

1. Initial program 0.18

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

2. Applied egg-rr 0.18

   \[\leadsto \color{blue}{\left(1 + x \cdot \left(x \cdot -0.12\right)\right) + x \cdot -0.253} \]

3. Final simplification 0.18

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

Alternatives

Alternative 1
Error	2.78%
Cost	585

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	2.81%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 3
Error	2.15%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -4:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 + x \cdot -0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 4
Error	0.18%
Cost	576

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

Alternative 5
Error	2.85%
Cost	448

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

Alternative 6
Error	34.41%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))