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

Percentage Accurate: 99.9% → 99.9%

Time: 4.7s

Alternatives: 3

Speedup: 1.0×

• 25.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \left(0.253 + x \cdot 0.12\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 97.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - 0.12 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Step-by-step derivation

   1. flip-+ 99.9%

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

   2. associate-*r/ 91.3%

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

   3. metadata-eval 91.3%

      \[\leadsto 1 - \frac{x \cdot \left(\color{blue}{0.064009} - \left(x \cdot 0.12\right) \cdot \left(x \cdot 0.12\right)\right)}{0.253 - x \cdot 0.12} \]

   4. swap-sqr 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - \color{blue}{\left(x \cdot x\right) \cdot \left(0.12 \cdot 0.12\right)}\right)}{0.253 - x \cdot 0.12} \]

   5. pow2 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - \color{blue}{{x}^{2}} \cdot \left(0.12 \cdot 0.12\right)\right)}{0.253 - x \cdot 0.12} \]

   6. metadata-eval 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - {x}^{2} \cdot \color{blue}{0.0144}\right)}{0.253 - x \cdot 0.12} \]

   7. *-commutative 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{0.253 - \color{blue}{0.12 \cdot x}} \]

   8. cancel-sign-sub-inv 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{\color{blue}{0.253 + \left(-0.12\right) \cdot x}} \]

   9. metadata-eval 91.2%

      \[\leadsto 1 - \frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{0.253 + \color{blue}{-0.12} \cdot x} \]

3. Applied egg-rr 91.2%

   \[\leadsto 1 - \color{blue}{\frac{x \cdot \left(0.064009 - {x}^{2} \cdot 0.0144\right)}{0.253 + -0.12 \cdot x}} \]
4. Step-by-step derivation

   1. associate-/l* 99.8%

      \[\leadsto 1 - \color{blue}{\frac{x}{\frac{0.253 + -0.12 \cdot x}{0.064009 - {x}^{2} \cdot 0.0144}}} \]

   2. *-commutative 99.8%

      \[\leadsto 1 - \frac{x}{\frac{0.253 + \color{blue}{x \cdot -0.12}}{0.064009 - {x}^{2} \cdot 0.0144}} \]

5. Simplified 99.8%

   \[\leadsto 1 - \color{blue}{\frac{x}{\frac{0.253 + x \cdot -0.12}{0.064009 - {x}^{2} \cdot 0.0144}}} \]

6. Taylor expanded in x around inf 97.9%

   \[\leadsto 1 - \frac{x}{\color{blue}{\frac{8.333333333333334}{x}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 97.9%

      \[\leadsto 1 - \frac{\color{blue}{1 \cdot x}}{\frac{8.333333333333334}{x}} \]

   2. div-inv 97.9%

      \[\leadsto 1 - \frac{1 \cdot x}{\color{blue}{8.333333333333334 \cdot \frac{1}{x}}} \]

   3. times-frac 98.0%

      \[\leadsto 1 - \color{blue}{\frac{1}{8.333333333333334} \cdot \frac{x}{\frac{1}{x}}} \]

   4. metadata-eval 98.0%

      \[\leadsto 1 - \color{blue}{0.12} \cdot \frac{x}{\frac{1}{x}} \]

8. Applied egg-rr 98.0%

   \[\leadsto 1 - \color{blue}{0.12 \cdot \frac{x}{\frac{1}{x}}} \]
9. Step-by-step derivation

   1. div-inv 98.0%

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

   2. remove-double-div 98.0%

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

10. Applied egg-rr 98.0%

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

11. Final simplification 98.0%

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

Alternative 3: 49.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot 0.253 \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in x around 0 49.6%

   \[\leadsto 1 - \color{blue}{0.253 \cdot x} \]
3. Step-by-step derivation

   1. *-commutative 49.6%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

4. Simplified 49.6%

   \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]

5. Final simplification 49.6%

   \[\leadsto 1 - x \cdot 0.253 \]

Reproduce

herbie shell --seed 2023311 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))