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

Percentage Accurate: 99.9% → 99.9%

Time: 6.6s

Alternatives: 8

Speedup: 1.0×

• 25.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 - N[(N[(x * 0.12), $MachinePrecision] * x + N[(x * 0.253), $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. +-commutative 99.9%

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

   2. distribute-rgt-in 99.9%

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

   3. fma-def 99.9%

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

   4. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

Alternative 3: 51.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{3.952569169960474}\\ \mathbf{elif}\;x \leq 2.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (/ x 3.952569169960474) (if (<= x 2.1) 1.0 (* x -0.253))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x / 3.952569169960474;
	} else if (x <= 2.1) {
		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 <= (-4.2d0)) then
        tmp = x / 3.952569169960474d0
    else if (x <= 2.1d0) 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 <= -4.2) {
		tmp = x / 3.952569169960474;
	} else if (x <= 2.1) {
		tmp = 1.0;
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x / 3.952569169960474
	elif x <= 2.1:
		tmp = 1.0
	else:
		tmp = x * -0.253
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x / 3.952569169960474);
	elseif (x <= 2.1)
		tmp = 1.0;
	else
		tmp = Float64(x * -0.253);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x / 3.952569169960474;
	elseif (x <= 2.1)
		tmp = 1.0;
	else
		tmp = x * -0.253;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x / 3.952569169960474), $MachinePrecision], If[LessEqual[x, 2.1], 1.0, N[(x * -0.253), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   4. Simplified 0.5%

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

   5. Taylor expanded in x around inf 0.5%

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

      1. *-commutative 0.5%

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

   7. Simplified 0.5%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 0.5%

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

      3. swap-sqr 0.5%

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

      4. metadata-eval 0.5%

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

      5. metadata-eval 0.5%

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

      6. swap-sqr 0.5%

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

      7. metadata-eval 0.5%

         \[\leadsto \sqrt{\left(x \cdot \color{blue}{\frac{1}{3.952569169960474}}\right) \cdot \left(x \cdot 0.253\right)} \]

      8. div-inv 0.5%

         \[\leadsto \sqrt{\color{blue}{\frac{x}{3.952569169960474}} \cdot \left(x \cdot 0.253\right)} \]

      9. metadata-eval 0.5%

         \[\leadsto \sqrt{\frac{x}{3.952569169960474} \cdot \left(x \cdot \color{blue}{\frac{1}{3.952569169960474}}\right)} \]

      10. div-inv 0.5%

          \[\leadsto \sqrt{\frac{x}{3.952569169960474} \cdot \color{blue}{\frac{x}{3.952569169960474}}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{3.952569169960474}} \cdot \sqrt{\frac{x}{3.952569169960474}}} \]

      12. add-sqr-sqrt 7.0%

          \[\leadsto \color{blue}{\frac{x}{3.952569169960474}} \]

   9. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\frac{x}{3.952569169960474}} \]

   if -4.20000000000000018 < x < 2.10000000000000009

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.4%

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

      1. *-commutative 99.4%

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

   4. Simplified 99.4%

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

   5. Taylor expanded in x around 0 98.0%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 7.0%

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

      1. *-commutative 7.0%

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

   4. Simplified 7.0%

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

   5. Taylor expanded in x around inf 7.0%

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

      1. *-commutative 7.0%

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

   7. Simplified 7.0%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{3.952569169960474}\\ \mathbf{elif}\;x \leq 2.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \]

Alternative 4: 52.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{3.952569169960474}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (/ x 3.952569169960474) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x / 3.952569169960474;
	} 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 <= (-4.2d0)) then
        tmp = x / 3.952569169960474d0
    else
        tmp = 1.0d0 - (x * 0.253d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x / 3.952569169960474;
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x / 3.952569169960474
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x / 3.952569169960474);
	else
		tmp = Float64(1.0 - Float64(x * 0.253));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x / 3.952569169960474;
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x / 3.952569169960474), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 0.5%

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

      1. *-commutative 0.5%

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

   4. Simplified 0.5%

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

   5. Taylor expanded in x around inf 0.5%

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

      1. *-commutative 0.5%

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

   7. Simplified 0.5%

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

      1. add-sqr-sqrt 0.5%

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

      2. sqrt-unprod 0.5%

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

      3. swap-sqr 0.5%

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

      4. metadata-eval 0.5%

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

      5. metadata-eval 0.5%

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

      6. swap-sqr 0.5%

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

      7. metadata-eval 0.5%

         \[\leadsto \sqrt{\left(x \cdot \color{blue}{\frac{1}{3.952569169960474}}\right) \cdot \left(x \cdot 0.253\right)} \]

      8. div-inv 0.5%

         \[\leadsto \sqrt{\color{blue}{\frac{x}{3.952569169960474}} \cdot \left(x \cdot 0.253\right)} \]

      9. metadata-eval 0.5%

         \[\leadsto \sqrt{\frac{x}{3.952569169960474} \cdot \left(x \cdot \color{blue}{\frac{1}{3.952569169960474}}\right)} \]

      10. div-inv 0.5%

          \[\leadsto \sqrt{\frac{x}{3.952569169960474} \cdot \color{blue}{\frac{x}{3.952569169960474}}} \]

      11. sqrt-unprod 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{3.952569169960474}} \cdot \sqrt{\frac{x}{3.952569169960474}}} \]

      12. add-sqr-sqrt 7.0%

          \[\leadsto \color{blue}{\frac{x}{3.952569169960474}} \]

   9. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{\frac{x}{3.952569169960474}} \]

   if -4.20000000000000018 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 64.6%

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

      1. *-commutative 64.6%

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

   4. Simplified 64.6%

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

4. Final simplification 47.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{3.952569169960474}\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]

Alternative 5: 97.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 1 - x \cdot \frac{x}{8.333333333333334} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 1.0 (* x (/ x 8.333333333333334))))

C

double code(double x) {
	return 1.0 - (x * (x / 8.333333333333334));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (x * (x / 8.333333333333334d0))
end function

Java

public static double code(double x) {
	return 1.0 - (x * (x / 8.333333333333334));
}

Python

def code(x):
	return 1.0 - (x * (x / 8.333333333333334))

Julia

function code(x)
	return Float64(1.0 - Float64(x * Float64(x / 8.333333333333334)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (x * (x / 8.333333333333334));
end

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x / 8.333333333333334), $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/ 88.3%

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

   3. metadata-eval 88.3%

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

   4. pow2 88.3%

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

   5. *-commutative 88.3%

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

   6. cancel-sign-sub-inv 88.3%

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

   7. metadata-eval 88.3%

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

3. Applied egg-rr 88.3%

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

   1. associate-/l* 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around inf 97.7%

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

   1. clear-num 97.7%

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

   2. inv-pow 97.7%

      \[\leadsto 1 - \color{blue}{{\left(\frac{\frac{8.333333333333334}{x}}{x}\right)}^{-1}} \]

   3. associate-/l/ 97.8%

      \[\leadsto 1 - {\color{blue}{\left(\frac{8.333333333333334}{x \cdot x}\right)}}^{-1} \]

   4. pow2 97.8%

      \[\leadsto 1 - {\left(\frac{8.333333333333334}{\color{blue}{{x}^{2}}}\right)}^{-1} \]

8. Applied egg-rr 97.8%

   \[\leadsto 1 - \color{blue}{{\left(\frac{8.333333333333334}{{x}^{2}}\right)}^{-1}} \]
9. Step-by-step derivation

   1. unpow-1 97.8%

      \[\leadsto 1 - \color{blue}{\frac{1}{\frac{8.333333333333334}{{x}^{2}}}} \]

   2. clear-num 97.7%

      \[\leadsto 1 - \color{blue}{\frac{{x}^{2}}{8.333333333333334}} \]

   3. unpow2 97.7%

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

   4. associate-/l* 97.7%

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

10. Applied egg-rr 97.7%

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

    1. associate-/r/ 97.7%

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

12. Simplified 97.7%

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

13. Final simplification 97.7%

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

Alternative 6: 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 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
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/ 88.3%

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

   3. metadata-eval 88.3%

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

   4. pow2 88.3%

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

   5. *-commutative 88.3%

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

   6. cancel-sign-sub-inv 88.3%

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

   7. metadata-eval 88.3%

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

3. Applied egg-rr 88.3%

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

   1. associate-/l* 99.8%

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

   2. *-commutative 99.8%

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

   3. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around inf 97.7%

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

   1. frac-2neg 97.7%

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

   2. associate-/r/ 97.7%

      \[\leadsto 1 - \color{blue}{\frac{x}{-8.333333333333334} \cdot \left(-x\right)} \]

   3. metadata-eval 97.7%

      \[\leadsto 1 - \frac{x}{\color{blue}{-8.333333333333334}} \cdot \left(-x\right) \]

8. Applied egg-rr 97.7%

   \[\leadsto 1 - \color{blue}{\frac{x}{-8.333333333333334} \cdot \left(-x\right)} \]

9. Taylor expanded in x around 0 97.8%

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

    1. *-commutative 97.8%

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

11. Simplified 97.8%

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

12. Final simplification 97.8%

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

Alternative 7: 50.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.1) 1.0 (* x -0.253)))

C

double code(double x) {
	double tmp;
	if (x <= 2.1) {
		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 <= 2.1d0) 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 <= 2.1) {
		tmp = 1.0;
	} else {
		tmp = x * -0.253;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.1:
		tmp = 1.0
	else:
		tmp = x * -0.253
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1)
		tmp = 1.0;
	else
		tmp = Float64(x * -0.253);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.1)
		tmp = 1.0;
	else
		tmp = x * -0.253;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.1], 1.0, N[(x * -0.253), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 58.6%

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

      1. *-commutative 58.6%

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

   4. Simplified 58.6%

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

   5. Taylor expanded in x around 0 57.9%

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

   if 2.10000000000000009 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0 7.0%

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

      1. *-commutative 7.0%

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

   4. Simplified 7.0%

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

   5. Taylor expanded in x around inf 7.0%

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

      1. *-commutative 7.0%

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

   7. Simplified 7.0%

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

4. Final simplification 44.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.253\\ \end{array} \]

Alternative 8: 48.8% 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. Taylor expanded in x around 0 45.1%

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

   1. *-commutative 45.1%

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

4. Simplified 45.1%

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

5. Taylor expanded in x around 0 43.0%

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

6. Final simplification 43.0%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))