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

Percentage Accurate: 99.9% → 99.9%

Time: 7.6s

Alternatives: 8

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, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. *-commutative 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. swap-sqr 99.8%

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

   8. pow2 99.8%

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

   9. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 99.9%

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

   1. associate-*r/ 99.9%

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

   2. metadata-eval 99.9%

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

   3. +-commutative 99.9%

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

7. Simplified 99.9%

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

Alternative 2: 98.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.05)))
   (* x (- (* x -0.12) 0.253))
   (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.05)) {
		tmp = x * ((x * -0.12) - 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 <= (-4.2d0)) .or. (.not. (x <= 2.05d0))) then
        tmp = x * ((x * (-0.12d0)) - 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 <= -4.2) || !(x <= 2.05)) {
		tmp = x * ((x * -0.12) - 0.253);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.2) or not (x <= 2.05):
		tmp = x * ((x * -0.12) - 0.253)
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.2) || !(x <= 2.05))
		tmp = Float64(x * Float64(Float64(x * -0.12) - 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 <= -4.2) || ~((x <= 2.05)))
		tmp = x * ((x * -0.12) - 0.253);
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.2], N[Not[LessEqual[x, 2.05]], $MachinePrecision]], N[(x * N[(N[(x * -0.12), $MachinePrecision] - 0.253), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2.0499999999999998 < 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 inf 99.5%

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

      1. mul-1-neg 99.5%

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

      2. distribute-rgt-neg-in 99.5%

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

      3. distribute-neg-in 99.5%

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

      4. metadata-eval 99.5%

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

      5. associate-*r/ 99.5%

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

      6. metadata-eval 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in x around 0 99.6%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   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 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 98.9%

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

Alternative 3: 98.2% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.2) (not (<= x 2.05))) (* -0.12 (* x x)) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.2) || !(x <= 2.05)) {
		tmp = -0.12 * (x * x);
	} 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)) .or. (.not. (x <= 2.05d0))) then
        tmp = (-0.12d0) * (x * x)
    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) || !(x <= 2.05)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	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 - (x * 0.253)
	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 = Float64(1.0 - Float64(x * 0.253));
	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 - (x * 0.253);
	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], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.20000000000000018 or 2.0499999999999998 < 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 inf 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. unpow2 97.7%

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

   7. Applied egg-rr 97.7%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   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 98.0%

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

      1. *-commutative 98.0%

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

   5. Simplified 98.0%

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

4. Final simplification 97.8%

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

Alternative 4: 97.7% accurate, 0.6× 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 99.8%

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

   3. Taylor expanded in x around inf 97.7%

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

      1. *-commutative 97.7%

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

   5. Simplified 97.7%

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

      1. unpow2 97.7%

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

   7. Applied egg-rr 97.7%

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

   if -4.20000000000000018 < x < 2.0499999999999998

   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 97.0%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \lor \neg \left(x \leq 2.05\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
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: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + 0.12 \cdot \frac{x}{\frac{-1}{x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(1.0 + N[(0.12 * N[(x / N[(-1.0 / x), $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. 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. clear-num 99.8%

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

   3. *-commutative 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. swap-sqr 99.8%

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

   8. pow2 99.8%

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

   9. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 97.4%

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

   1. un-div-inv 97.4%

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

7. Applied egg-rr 97.4%

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

   1. *-un-lft-identity 97.4%

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

   2. div-inv 97.4%

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

   3. times-frac 97.4%

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

   4. metadata-eval 97.4%

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

9. Applied egg-rr 97.4%

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

10. Final simplification 97.4%

    \[\leadsto 1 + 0.12 \cdot \frac{x}{\frac{-1}{x}} \]
11. Add Preprocessing

Alternative 7: 97.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

   \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
2. Add Preprocessing
3. 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. clear-num 99.8%

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

   3. *-commutative 99.8%

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

   4. cancel-sign-sub-inv 99.8%

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

   5. metadata-eval 99.8%

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

   6. metadata-eval 99.8%

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

   7. swap-sqr 99.8%

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

   8. pow2 99.8%

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

   9. metadata-eval 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in x around inf 97.4%

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

   1. un-div-inv 97.4%

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

7. Applied egg-rr 97.4%

   \[\leadsto 1 - \color{blue}{\frac{x}{\frac{8.333333333333334}{x}}} \]
8. Add Preprocessing

Alternative 8: 49.2% 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 45.9%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))