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

Percentage Accurate: 99.9% → 99.9%

Time: 5.1s

Alternatives: 6

Speedup: 1.0×

• 25.8% 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-+ 99.9%

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

   2. associate-*r/ 95.9%

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

   3. metadata-eval 95.9%

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

   4. swap-sqr 95.9%

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

   5. metadata-eval 95.9%

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

   6. *-commutative 95.9%

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

   7. cancel-sign-sub-inv 95.9%

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

   8. metadata-eval 95.9%

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

3. Applied egg-rr 95.9%

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

   1. *-commutative 95.9%

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

   2. associate-/l* 99.8%

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

   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: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.0) (not (<= x 2.0))) (* -0.12 (* x x)) 1.5334083333333333))

C

double code(double x) {
	double tmp;
	if ((x <= -4.0) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.5334083333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.0d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = (-0.12d0) * (x * x)
    else
        tmp = 1.5334083333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.0) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.5334083333333333;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.0) or not (x <= 2.0):
		tmp = -0.12 * (x * x)
	else:
		tmp = 1.5334083333333333
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.0) || !(x <= 2.0))
		tmp = Float64(-0.12 * Float64(x * x));
	else
		tmp = 1.5334083333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.0) || ~((x <= 2.0)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.5334083333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.0], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(-0.12 * N[(x * x), $MachinePrecision]), $MachinePrecision], 1.5334083333333333]

Derivation

1. Split input into 2 regimes

2. if x < -4 or 2 < x

   1. Initial program 99.7%

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

      1. flip-+ 99.7%

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

      2. associate-*r/ 91.2%

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

      3. metadata-eval 91.2%

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

      4. swap-sqr 91.3%

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

      5. metadata-eval 91.3%

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

      6. *-commutative 91.3%

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

      7. cancel-sign-sub-inv 91.3%

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

      8. metadata-eval 91.3%

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

   3. Applied egg-rr 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-/l* 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 97.9%

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

   7. Taylor expanded in x around inf 98.0%

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

      1. unpow2 98.0%

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

   9. Simplified 98.0%

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

   if -4 < x < 2

   1. Initial program 100.0%

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

      1. flip-+ 100.0%

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

      2. associate-*r/ 100.0%

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

      3. metadata-eval 100.0%

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

      4. swap-sqr 100.0%

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

      5. metadata-eval 100.0%

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

      6. *-commutative 100.0%

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

      7. cancel-sign-sub-inv 100.0%

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

      8. metadata-eval 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 20.2%

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

   7. Taylor expanded in x around 0 20.2%

      \[\leadsto \color{blue}{1.5334083333333333} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.0) (not (<= x 2.0))) (* -0.12 (* x x)) (- 1.0 (* x 0.253))))

C

double code(double x) {
	double tmp;
	if ((x <= -4.0) || !(x <= 2.0)) {
		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 <= (-4.0d0)) .or. (.not. (x <= 2.0d0))) 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 <= -4.0) || !(x <= 2.0)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0 - (x * 0.253);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.0) or not (x <= 2.0):
		tmp = -0.12 * (x * x)
	else:
		tmp = 1.0 - (x * 0.253)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.0) || !(x <= 2.0))
		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 <= -4.0) || ~((x <= 2.0)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0 - (x * 0.253);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.0], N[Not[LessEqual[x, 2.0]], $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 or 2 < x

   1. Initial program 99.7%

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

      1. flip-+ 99.7%

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

      2. associate-*r/ 91.2%

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

      3. metadata-eval 91.2%

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

      4. swap-sqr 91.3%

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

      5. metadata-eval 91.3%

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

      6. *-commutative 91.3%

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

      7. cancel-sign-sub-inv 91.3%

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

      8. metadata-eval 91.3%

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

   3. Applied egg-rr 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-/l* 99.7%

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

      3. associate-*l* 99.7%

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

      4. *-commutative 99.7%

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf 97.9%

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

   7. Taylor expanded in x around inf 98.0%

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

      1. unpow2 98.0%

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

   9. Simplified 98.0%

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

   if -4 < x < 2

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.7%

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

      1. *-commutative 99.7%

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

   4. Simplified 99.7%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1 - x \cdot 0.253\\ \end{array} \]

Alternative 4: 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 5: 97.9% 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 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. Final simplification 98.3%

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

Alternative 6: 10.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 1.5334083333333333 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.5334083333333333)

C

double code(double x) {
	return 1.5334083333333333;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.5334083333333333d0
end function

Java

public static double code(double x) {
	return 1.5334083333333333;
}

Python

def code(x):
	return 1.5334083333333333

Julia

function code(x)
	return 1.5334083333333333
end

MATLAB

function tmp = code(x)
	tmp = 1.5334083333333333;
end

Wolfram

code[x_] := 1.5334083333333333

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/ 95.9%

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

   3. metadata-eval 95.9%

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

   4. swap-sqr 95.9%

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

   5. metadata-eval 95.9%

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

   6. *-commutative 95.9%

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

   7. cancel-sign-sub-inv 95.9%

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

   8. metadata-eval 95.9%

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

3. Applied egg-rr 95.9%

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

   1. *-commutative 95.9%

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

   2. associate-/l* 99.8%

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

   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 56.9%

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

7. Taylor expanded in x around 0 11.0%

   \[\leadsto \color{blue}{1.5334083333333333} \]

8. Final simplification 11.0%

   \[\leadsto 1.5334083333333333 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))