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

Percentage Accurate: 99.9% → 99.9%

Time: 6.1s

Alternatives: 9

Speedup: 1.0×

• 24.8% 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, 0.1× 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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(1.0 - N[(x * N[(x * 0.12 + 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. fma-def 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 58.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.1) (not (<= x 2.0))) (* x (* x -0.12)) 1.5334083333333333))

C

double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = x * (x * -0.12);
	} else {
		tmp = 1.5334083333333333;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-4.1d0)) .or. (.not. (x <= 2.0d0))) then
        tmp = x * (x * (-0.12d0))
    else
        tmp = 1.5334083333333333d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(x <= 2.0)) {
		tmp = x * (x * -0.12);
	} else {
		tmp = 1.5334083333333333;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.1) or not (x <= 2.0):
		tmp = x * (x * -0.12)
	else:
		tmp = 1.5334083333333333
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -4.1) || !(x <= 2.0))
		tmp = Float64(x * Float64(x * -0.12));
	else
		tmp = 1.5334083333333333;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -4.1) || ~((x <= 2.0)))
		tmp = x * (x * -0.12);
	else
		tmp = 1.5334083333333333;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.1], N[Not[LessEqual[x, 2.0]], $MachinePrecision]], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], 1.5334083333333333]

Derivation

1. Split input into 2 regimes

2. if x < -4.0999999999999996 or 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   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/ 84.5%

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

      3. metadata-eval 84.5%

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

      4. swap-sqr 84.4%

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

      5. metadata-eval 84.4%

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

      6. *-commutative 84.4%

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

      7. cancel-sign-sub-inv 84.4%

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

      8. metadata-eval 84.4%

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

   3. Applied egg-rr 84.4%

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

      1. *-commutative 84.4%

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

      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 98.1%

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

   7. Taylor expanded in x around inf 98.2%

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

      1. *-commutative 98.2%

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

      2. unpow2 98.2%

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

      3. associate-*l* 98.3%

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

   9. Simplified 98.3%

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

   if -4.0999999999999996 < 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 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2\right):\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{else}:\\ \;\;\;\;1.5334083333333333\\ \end{array} \]

Alternative 4: 58.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1.5334083333333333\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1)
   (* x (* x -0.12))
   (if (<= x 2.0) 1.5334083333333333 (* (* x x) -0.12))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.5334083333333333;
	} else {
		tmp = (x * x) * -0.12;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-4.1d0)) then
        tmp = x * (x * (-0.12d0))
    else if (x <= 2.0d0) then
        tmp = 1.5334083333333333d0
    else
        tmp = (x * x) * (-0.12d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.5334083333333333;
	} else {
		tmp = (x * x) * -0.12;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1:
		tmp = x * (x * -0.12)
	elif x <= 2.0:
		tmp = 1.5334083333333333
	else:
		tmp = (x * x) * -0.12
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1)
		tmp = Float64(x * Float64(x * -0.12));
	elseif (x <= 2.0)
		tmp = 1.5334083333333333;
	else
		tmp = Float64(Float64(x * x) * -0.12);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.1)
		tmp = x * (x * -0.12);
	elseif (x <= 2.0)
		tmp = 1.5334083333333333;
	else
		tmp = (x * x) * -0.12;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], 1.5334083333333333, N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999996

   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/ 82.9%

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

      3. metadata-eval 82.9%

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

      4. swap-sqr 82.8%

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

      5. metadata-eval 82.8%

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

      6. *-commutative 82.8%

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

      7. cancel-sign-sub-inv 82.8%

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

      8. metadata-eval 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. *-commutative 82.8%

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

      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 98.8%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

      3. associate-*l* 99.0%

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

   9. Simplified 99.0%

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

   if -4.0999999999999996 < 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%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   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/ 86.1%

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

      3. metadata-eval 86.1%

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

      4. swap-sqr 86.0%

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

      5. metadata-eval 86.0%

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

      6. *-commutative 86.0%

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

      7. cancel-sign-sub-inv 86.0%

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

      8. metadata-eval 86.0%

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

   3. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-/l* 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* 99.7%

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

      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 97.4%

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

   7. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. unpow2 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1.5334083333333333\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \]

Alternative 5: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1)
   (* x (* x -0.12))
   (if (<= x 2.0) (- 1.0 (* x 0.253)) (* (* x x) -0.12))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		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 <= (-4.1d0)) then
        tmp = x * (x * (-0.12d0))
    else if (x <= 2.0d0) 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 <= -4.1) {
		tmp = x * (x * -0.12);
	} else if (x <= 2.0) {
		tmp = 1.0 - (x * 0.253);
	} else {
		tmp = (x * x) * -0.12;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -4.1:
		tmp = x * (x * -0.12)
	elif x <= 2.0:
		tmp = 1.0 - (x * 0.253)
	else:
		tmp = (x * x) * -0.12
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -4.1)
		tmp = Float64(x * Float64(x * -0.12));
	elseif (x <= 2.0)
		tmp = Float64(1.0 - Float64(x * 0.253));
	else
		tmp = Float64(Float64(x * x) * -0.12);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -4.1)
		tmp = x * (x * -0.12);
	elseif (x <= 2.0)
		tmp = 1.0 - (x * 0.253);
	else
		tmp = (x * x) * -0.12;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -4.1], N[(x * N[(x * -0.12), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.0], N[(1.0 - N[(x * 0.253), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.12), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999996

   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/ 82.9%

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

      3. metadata-eval 82.9%

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

      4. swap-sqr 82.8%

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

      5. metadata-eval 82.8%

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

      6. *-commutative 82.8%

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

      7. cancel-sign-sub-inv 82.8%

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

      8. metadata-eval 82.8%

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

   3. Applied egg-rr 82.8%

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

      1. *-commutative 82.8%

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

      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 98.8%

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

   7. Taylor expanded in x around inf 98.9%

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

      1. *-commutative 98.9%

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

      2. unpow2 98.9%

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

      3. associate-*l* 99.0%

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

   9. Simplified 99.0%

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

   if -4.0999999999999996 < 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 98.1%

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

      1. *-commutative 98.1%

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

   4. Simplified 98.1%

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

   if 2 < x

   1. Initial program 99.8%

      \[1 - x \cdot \left(0.253 + x \cdot 0.12\right) \]
   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/ 86.1%

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

      3. metadata-eval 86.1%

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

      4. swap-sqr 86.0%

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

      5. metadata-eval 86.0%

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

      6. *-commutative 86.0%

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

      7. cancel-sign-sub-inv 86.0%

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

      8. metadata-eval 86.0%

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

   3. Applied egg-rr 86.0%

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

      1. *-commutative 86.0%

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

      2. associate-/l* 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* 99.7%

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

      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 97.4%

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

   7. Taylor expanded in x around inf 97.6%

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

      1. *-commutative 97.6%

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

      2. unpow2 97.6%

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

   9. Simplified 97.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \mathbf{elif}\;x \leq 2:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot x\right) \cdot -0.12\\ \end{array} \]

Alternative 6: 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 7: 97.6% 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 97.6%

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

   1. unpow2 97.6%

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

4. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 8: 97.7% 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. +-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. Taylor expanded in x around inf 97.6%

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

   1. unpow2 97.6%

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

   2. associate-*r* 97.6%

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

   3. *-commutative 97.6%

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

6. Simplified 97.6%

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

7. Final simplification 97.6%

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

Alternative 9: 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/ 92.0%

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

   3. metadata-eval 92.0%

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

   4. swap-sqr 92.0%

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

   5. metadata-eval 92.0%

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

   6. *-commutative 92.0%

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

   7. cancel-sign-sub-inv 92.0%

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

   8. metadata-eval 92.0%

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

3. Applied egg-rr 92.0%

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

   1. *-commutative 92.0%

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

   2. associate-/l* 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 60.4%

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

7. Taylor expanded in x around 0 10.2%

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

8. Final simplification 10.2%

   \[\leadsto 1.5334083333333333 \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))