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

Percentage Accurate: 99.9% → 99.9%

Time: 7.0s

Alternatives: 7

Speedup: 1.0×

• 24.4% 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. distribute-rgt-in 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-rgt-in 99.8%

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

   4. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto 1 - x \cdot \mathsf{fma}\left(x, 0.12, 0.253\right) \]

Alternative 2: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 3: 97.8% 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 99.8%

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

2. Taylor expanded in x around inf 97.6%

   \[\leadsto 1 - \color{blue}{0.12 \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 97.6%

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

4. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 4: 97.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around inf 97.6%

   \[\leadsto 1 - \color{blue}{0.12 \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 97.6%

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

4. Simplified 97.6%

   \[\leadsto 1 - \color{blue}{0.12 \cdot \left(x \cdot x\right)} \]
5. Step-by-step derivation

   1. *-commutative 97.6%

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

   2. associate-*l* 97.6%

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

   3. /-rgt-identity 97.6%

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

   4. associate-*l/ 97.6%

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

   5. associate-*l* 97.6%

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

   6. *-commutative 97.6%

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

   7. frac-2neg 97.6%

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

   8. frac-2neg 97.6%

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

   9. *-commutative 97.6%

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

   10. associate-*l* 97.6%

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

   11. distribute-rgt-neg-in 97.6%

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

   12. distribute-rgt-neg-in 97.6%

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

   13. metadata-eval 97.6%

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

   14. metadata-eval 97.6%

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

   15. metadata-eval 97.6%

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

6. Applied egg-rr 97.6%

   \[\leadsto 1 - \color{blue}{\frac{-x \cdot \left(x \cdot -0.12\right)}{1}} \]
7. Step-by-step derivation

   1. /-rgt-identity 97.6%

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

   2. distribute-rgt-neg-in 97.6%

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

   3. distribute-rgt-neg-in 97.6%

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

   4. metadata-eval 97.6%

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

   5. *-commutative 97.6%

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

8. Simplified 97.6%

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

9. Final simplification 97.6%

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

Alternative 5: 50.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 1.0 (* x -0.253)))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = x * (-0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 1.0
	else:
		tmp = x * -0.253
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = Float64(x * -0.253);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 1.0;
	else
		tmp = x * -0.253;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 1.0, N[(x * -0.253), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 64.4%

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

      1. *-commutative 64.4%

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

   4. Simplified 64.4%

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

   5. Taylor expanded in x around 0 64.1%

      \[\leadsto \color{blue}{1} \]

   if 2 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 7.5%

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

      1. *-commutative 7.5%

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

   4. Simplified 7.5%

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

   5. Taylor expanded in x around inf 7.5%

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

      1. *-commutative 7.5%

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

   7. Simplified 7.5%

      \[\leadsto \color{blue}{x \cdot -0.253} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \]

Alternative 6: 51.4% 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 99.8%

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

2. Taylor expanded in x around 0 48.2%

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

   1. *-commutative 48.2%

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

4. Simplified 48.2%

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

5. Final simplification 48.2%

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

Alternative 7: 49.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0 48.2%

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

   1. *-commutative 48.2%

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

4. Simplified 48.2%

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

5. Taylor expanded in x around 0 46.1%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 46.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023297 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))