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

Percentage Accurate: 99.9% → 99.9%

Time: 4.8s

Alternatives: 10

Speedup: 1.0×

• 24.7% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{x \cdot \left(x \cdot 0.0144\right) - 0.064009}{\frac{0.253}{x} + -0.12} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (/ (- (* x (* x 0.0144)) 0.064009) (+ (/ 0.253 x) -0.12))))

C

double code(double x) {
	return 1.0 + (((x * (x * 0.0144)) - 0.064009) / ((0.253 / x) + -0.12));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (((x * (x * 0.0144d0)) - 0.064009d0) / ((0.253d0 / x) + (-0.12d0)))
end function

Java

public static double code(double x) {
	return 1.0 + (((x * (x * 0.0144)) - 0.064009) / ((0.253 / x) + -0.12));
}

Python

def code(x):
	return 1.0 + (((x * (x * 0.0144)) - 0.064009) / ((0.253 / x) + -0.12))

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(Float64(x * Float64(x * 0.0144)) - 0.064009) / Float64(Float64(0.253 / x) + -0.12)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (((x * (x * 0.0144)) - 0.064009) / ((0.253 / x) + -0.12));
end

Wolfram

code[x_] := N[(1.0 + N[(N[(N[(x * N[(x * 0.0144), $MachinePrecision]), $MachinePrecision] - 0.064009), $MachinePrecision] / N[(N[(0.253 / x), $MachinePrecision] + -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-+ 99.8%

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

   2. associate-*r/ 92.4%

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

   3. metadata-eval 92.4%

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

   4. swap-sqr 92.3%

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

   5. metadata-eval 92.3%

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

   6. *-commutative 92.3%

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

   7. cancel-sign-sub-inv 92.3%

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

   8. metadata-eval 92.3%

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

3. Applied egg-rr 92.3%

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

   1. *-commutative 92.3%

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

   2. associate-/l* 99.5%

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

   3. associate-*l* 99.8%

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

   4. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around 0 99.9%

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

   1. sub-neg 99.9%

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

   2. associate-*r/ 99.9%

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

   3. metadata-eval 99.9%

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

   4. metadata-eval 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.05\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.05))) (* -0.12 (* x x)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.05)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.2d0)) .or. (.not. (x <= 2.05d0))) then
        tmp = (-0.12d0) * (x * x)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.05)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.05):
		tmp = -0.12 * (x * x)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.05))
		tmp = Float64(-0.12 * Float64(x * x));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.2) || ~((x <= 2.05)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 2.05]], $MachinePrecision]], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2.0499999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.8%

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

      1. unpow2 97.8%

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

   4. Simplified 97.8%

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

   5. Taylor expanded in x around inf 97.8%

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

      1. unpow2 97.8%

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

   7. Simplified 97.8%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in x around 0 96.0%

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

4. Final simplification 96.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.05\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2) (* -0.12 (* x x)) (if (<= x 2.05) 1.0 (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = -0.12 * (x * x);
	} else if (x <= 2.05) {
		tmp = 1.0;
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = (-0.12d0) * (x * x)
    else if (x <= 2.05d0) then
        tmp = 1.0d0
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = -0.12 * (x * x);
	} else if (x <= 2.05) {
		tmp = 1.0;
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = -0.12 * (x * x)
	elif x <= 2.05:
		tmp = 1.0
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(-0.12 * Float64(x * x));
	elseif (x <= 2.05)
		tmp = 1.0;
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = -0.12 * (x * x);
	elseif (x <= 2.05)
		tmp = 1.0;
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05], 1.0, N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   7. Simplified 97.0%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 96.0%

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

      1. unpow2 96.0%

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

   4. Simplified 96.0%

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

   5. Taylor expanded in x around 0 96.0%

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

   if 2.0499999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

      2. unpow2 98.3%

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

      3. associate-*r* 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.12\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (* -0.12 (* x x))
   (if (<= x 2.05) (- 1.0 (* x 0.253)) (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = -0.12 * (x * x);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = (-0.12d0) * (x * x)
    else if (x <= 2.05d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = -0.12 * (x * x);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = -0.12 * (x * x)
	elif x <= 2.05:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(-0.12 * Float64(x * x));
	elseif (x <= 2.05)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = -0.12 * (x * x);
	elseif (x <= 2.05)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   7. Simplified 97.0%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   if 2.0499999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

      2. unpow2 98.3%

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

      3. associate-*r* 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.12\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{\frac{-8.333333333333334}{x}}\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.2)
   (/ x (/ -8.333333333333334 x))
   (if (<= x 2.05) (- 1.0 (* x 0.253)) (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x / (-8.333333333333334 / x);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.2d0)) then
        tmp = x / ((-8.333333333333334d0) / x)
    else if (x <= 2.05d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.2) {
		tmp = x / (-8.333333333333334 / x);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.2:
		tmp = x / (-8.333333333333334 / x)
	elif x <= 2.05:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.2)
		tmp = Float64(x / Float64(-8.333333333333334 / x));
	elseif (x <= 2.05)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.2)
		tmp = x / (-8.333333333333334 / x);
	elseif (x <= 2.05)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.2], N[(x / N[(-8.333333333333334 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.20000000000000018

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   4. Simplified 97.0%

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

   5. Taylor expanded in x around inf 97.0%

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

      1. unpow2 97.0%

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

   7. Simplified 97.0%

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

   8. Taylor expanded in x around 0 97.0%

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

      1. *-commutative 97.0%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

      2. unpow2 97.0%

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

      3. associate-*r* 96.9%

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

   10. Simplified 96.9%

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

       1. associate-*r* 97.0%

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

       2. metadata-eval 97.0%

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

       3. div-inv 96.9%

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

       4. associate-/l* 97.0%

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

   12. Applied egg-rr 97.0%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 97.7%

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

      1. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   if 2.0499999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

      2. unpow2 98.3%

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

      3. associate-*r* 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.12\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2:\\ \;\;\;\;\frac{x}{\frac{-8.333333333333334}{x}}\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 6: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;1.5334083333333333 + \frac{x \cdot x}{-8.333333333333334}\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (+ 1.5334083333333333 (/ (* x x) -8.333333333333334))
   (if (<= x 2.05) (- 1.0 (* x 0.253)) (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = 1.5334083333333333 + ((x * x) / -8.333333333333334);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.3d0)) then
        tmp = 1.5334083333333333d0 + ((x * x) / (-8.333333333333334d0))
    else if (x <= 2.05d0) then
        tmp = 1.0d0 - (x * 0.253d0)
    else
        tmp = x * (x * (-0.12d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = 1.5334083333333333 + ((x * x) / -8.333333333333334);
	} else if (x <= 2.05) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = x * (x * -0.12);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.3:
		tmp = 1.5334083333333333 + ((x * x) / -8.333333333333334)
	elif x <= 2.05:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = x * (x * -0.12)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(1.5334083333333333 + Float64(Float64(x * x) / -8.333333333333334));
	elseif (x <= 2.05)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(x * Float64(x * -0.12));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.3)
		tmp = 1.5334083333333333 + ((x * x) / -8.333333333333334);
	elseif (x <= 2.05)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = x * (x * -0.12);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(1.5334083333333333 + N[(N[(x * x), $MachinePrecision] / -8.333333333333334), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.05], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 99.7%

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

      1. flip-+ 99.6%

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

      2. associate-*r/ 84.2%

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

      3. metadata-eval 84.2%

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

      4. swap-sqr 84.2%

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

      5. metadata-eval 84.2%

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

      6. *-commutative 84.2%

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

      7. cancel-sign-sub-inv 84.2%

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

      8. metadata-eval 84.2%

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

   3. Applied egg-rr 84.2%

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

      1. *-commutative 84.2%

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

      2. associate-/l* 99.6%

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

      3. associate-*l* 99.6%

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

      4. *-commutative 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around inf 95.4%

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

      1. div-sub 95.4%

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

      2. associate--r- 95.4%

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

      3. metadata-eval 95.4%

         \[\leadsto \left(1 - \color{blue}{-0.5334083333333334}\right) + \frac{x \cdot \left(x \cdot 0.0144\right)}{-0.12} \]

      4. metadata-eval 95.4%

         \[\leadsto \color{blue}{1.5334083333333333} + \frac{x \cdot \left(x \cdot 0.0144\right)}{-0.12} \]

      5. associate-*r* 95.4%

         \[\leadsto 1.5334083333333333 + \frac{\color{blue}{\left(x \cdot x\right) \cdot 0.0144}}{-0.12} \]

      6. associate-/l* 95.4%

         \[\leadsto 1.5334083333333333 + \color{blue}{\frac{x \cdot x}{\frac{-0.12}{0.0144}}} \]

      7. metadata-eval 95.4%

         \[\leadsto 1.5334083333333333 + \frac{x \cdot x}{\color{blue}{-8.333333333333334}} \]

   8. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{1.5334083333333333 + \frac{x \cdot x}{-8.333333333333334}} \]

   if -1.30000000000000004 < x < 2.0499999999999998

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

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

   4. Simplified 98.3%

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

   if 2.0499999999999998 < x

   1. Initial program 99.7%

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

   2. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   4. Simplified 98.3%

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

   5. Taylor expanded in x around inf 98.3%

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

      1. unpow2 98.3%

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

   7. Simplified 98.3%

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

   8. Taylor expanded in x around 0 98.3%

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

      1. *-commutative 98.3%

         \[\leadsto \color{blue}{{x}^{2} \cdot -0.12} \]

      2. unpow2 98.3%

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

      3. associate-*r* 99.3%

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

   10. Simplified 99.3%

       \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.12\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;1.5334083333333333 + \frac{x \cdot x}{-8.333333333333334}\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 7: 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 8: 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 99.9%

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

2. Taylor expanded in x around inf 96.8%

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

   1. unpow2 96.8%

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

4. Simplified 96.8%

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

5. Final simplification 96.8%

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

Alternative 9: 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-+ 99.8%

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

   2. associate-*r/ 92.4%

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

   3. metadata-eval 92.4%

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

   4. swap-sqr 92.3%

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

   5. metadata-eval 92.3%

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

   6. *-commutative 92.3%

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

   7. cancel-sign-sub-inv 92.3%

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

   8. metadata-eval 92.3%

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

3. Applied egg-rr 92.3%

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

   1. *-commutative 92.3%

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

   2. associate-/l* 99.5%

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

   3. associate-*l* 99.8%

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

   4. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Taylor expanded in x around inf 96.8%

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

   1. *-commutative 96.8%

      \[\leadsto 1 - \color{blue}{{x}^{2} \cdot 0.12} \]

   2. unpow2 96.8%

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

   3. associate-*r* 97.1%

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

8. Simplified 97.1%

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

9. Final simplification 97.1%

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

Alternative 10: 49.8% 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 inf 96.8%

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

   1. unpow2 96.8%

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

4. Simplified 96.8%

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

5. Taylor expanded in x around 0 51.8%

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

6. Final simplification 51.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))