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

Percentage Accurate: 99.9% → 99.9%

Time: 6.9s

Alternatives: 8

Speedup: 1.0×

• 25.0% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. +-commutative 99.9%

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

   3. distribute-rgt-neg-in 99.9%

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

   4. fma-define 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. metadata-eval 99.9%

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

   9. metadata-eval 99.9%

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

   10. fma-define 99.9%

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

   11. metadata-eval 99.9%

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

   12. metadata-eval 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, -0.12, -0.253\right), 1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 52.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1) (* x 0.253) (if (<= x 2.0) 1.0 (* x -0.253))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * 0.253;
	} else 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 <= (-4.1d0)) then
        tmp = x * 0.253d0
    else 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 <= -4.1) {
		tmp = x * 0.253;
	} else if (x <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1:
		tmp = x * 0.253
	elif x <= 2.0:
		tmp = 1.0
	else:
		tmp = x * -0.253
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1)
		tmp = Float64(x * 0.253);
	elseif (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 <= -4.1)
		tmp = x * 0.253;
	elseif (x <= 2.0)
		tmp = 1.0;
	else
		tmp = x * -0.253;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1], N[(x * 0.253), $MachinePrecision], If[LessEqual[x, 2.0], 1.0, N[(x * -0.253), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999996

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   5. Simplified 0.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.9%

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

      3. swap-sqr 50.1%

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

      4. metadata-eval 50.1%

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

      5. metadata-eval 50.1%

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

      6. swap-sqr 49.9%

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

      7. *-commutative 49.9%

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

      8. *-commutative 49.9%

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

      9. sqrt-unprod 6.7%

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

      10. add-sqr-sqrt 6.7%

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

      11. *-commutative 6.7%

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

      12. cancel-sign-sub-inv 6.7%

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

   7. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{1 + \left(-x\right) \cdot -0.253} \]
   8. Step-by-step derivation

      1. *-commutative 6.7%

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

      2. distribute-rgt-neg-in 6.7%

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

      3. distribute-lft-neg-in 6.7%

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

      4. metadata-eval 6.7%

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

      5. *-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around inf 6.7%

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

   if -4.0999999999999996 < x < 2

   1. Initial program 100.0%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 96.8%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.8%

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

      1. *-commutative 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 6.8%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1) (* x 0.253) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * 0.253;
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.1d0)) then
        tmp = x * 0.253d0
    else
        tmp = 1.0d0 - (x * 0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * 0.253;
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1:
		tmp = x * 0.253
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1)
		tmp = Float64(x * 0.253);
	else
		tmp = Float64(1.0 - Float64(x * 0.253));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.1)
		tmp = x * 0.253;
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1], N[(x * 0.253), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.0999999999999996

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   5. Simplified 0.5%

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

      1. add-sqr-sqrt 0.0%

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

      2. sqrt-unprod 49.9%

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

      3. swap-sqr 50.1%

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

      4. metadata-eval 50.1%

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

      5. metadata-eval 50.1%

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

      6. swap-sqr 49.9%

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

      7. *-commutative 49.9%

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

      8. *-commutative 49.9%

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

      9. sqrt-unprod 6.7%

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

      10. add-sqr-sqrt 6.7%

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

      11. *-commutative 6.7%

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

      12. cancel-sign-sub-inv 6.7%

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

   7. Applied egg-rr 6.7%

      \[\leadsto \color{blue}{1 + \left(-x\right) \cdot -0.253} \]
   8. Step-by-step derivation

      1. *-commutative 6.7%

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

      2. distribute-rgt-neg-in 6.7%

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

      3. distribute-lft-neg-in 6.7%

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

      4. metadata-eval 6.7%

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

      5. *-commutative 6.7%

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

   9. Simplified 6.7%

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

   10. Taylor expanded in x around inf 6.7%

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

   if -4.0999999999999996 < x

   1. Initial program 99.9%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.9%

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

      1. *-commutative 64.9%

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

   5. Simplified 64.9%

      \[\leadsto 1 - \color{blue}{x \cdot 0.253} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 1.0 + (x * 0.253);
	} 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 + (x * 0.253d0)
    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 + (x * 0.253);
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = Float64(1.0 + Float64(x * 0.253));
	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 + (x * 0.253);
	else
		tmp = x * -0.253;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], N[(1.0 + N[(x * 0.253), $MachinePrecision]), $MachinePrecision], 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. Add Preprocessing

   3. Taylor expanded in x around 0 62.6%

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

      1. *-commutative 62.6%

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

   5. Simplified 62.6%

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

      1. add-sqr-sqrt 28.4%

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

      2. sqrt-unprod 80.0%

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

      3. swap-sqr 80.1%

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

      4. metadata-eval 80.1%

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

      5. metadata-eval 80.1%

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

      6. swap-sqr 80.0%

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

      7. *-commutative 80.0%

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

      8. *-commutative 80.0%

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

      9. sqrt-unprod 35.6%

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

      10. add-sqr-sqrt 63.4%

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

      11. *-commutative 63.4%

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

      12. cancel-sign-sub-inv 63.4%

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

   7. Applied egg-rr 63.4%

      \[\leadsto \color{blue}{1 + \left(-x\right) \cdot -0.253} \]
   8. Step-by-step derivation

      1. *-commutative 63.4%

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

      2. distribute-rgt-neg-in 63.4%

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

      3. distribute-lft-neg-in 63.4%

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

      4. metadata-eval 63.4%

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

      5. *-commutative 63.4%

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

   9. Simplified 63.4%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.8%

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

      1. *-commutative 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 6.8%

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

4. Final simplification 48.2%

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

Alternative 5: 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. Add Preprocessing
3. Add Preprocessing

Alternative 6: 51.1% accurate, 1.1× 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. Add Preprocessing

   3. Taylor expanded in x around 0 61.4%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 6.8%

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

      1. *-commutative 6.8%

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

   5. Simplified 6.8%

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

   6. Taylor expanded in x around inf 6.8%

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

4. Final simplification 46.7%

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

Alternative 7: 97.6% 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 99.9%

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

4. Taylor expanded in x around inf 97.4%

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

Alternative 8: 49.6% 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0 45.1%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))