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

Percentage Accurate: 99.9% → 99.9%

Time: 4.9s

Alternatives: 9

Speedup: 1.0×

• 25.1% 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, 0.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. fma-def 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

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 99.8%

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

   1. +-commutative 99.8%

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

   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: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2.05\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -4.1) (not (<= x 2.05))) (* -0.12 (* x x)) 1.0))

C

double code(double x) {
	double tmp;
	if ((x <= -4.1) || !(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 <= (-4.1d0)) .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 <= -4.1) || !(x <= 2.05)) {
		tmp = -0.12 * (x * x);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -4.1) 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 <= -4.1) || !(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 <= -4.1) || ~((x <= 2.05)))
		tmp = -0.12 * (x * x);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -4.1], 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 < -4.0999999999999996 or 2.0499999999999998 < x

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. +-commutative 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. fma-def 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-neg-in 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. fma-def 99.7%

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

      9. metadata-eval 99.7%

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

      10. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 96.3%

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

      1. unpow2 96.3%

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

   6. Simplified 96.3%

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

   if -4.0999999999999996 < x < 2.0499999999999998

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. fma-def 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.9%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \lor \neg \left(x \leq 2.05\right):\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1) (* -0.12 (* x x)) (if (<= x 2.05) 1.0 (* x (* x -0.12)))))

C

double code(double x) {
	double tmp;
	if (x <= -4.1) {
		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 <= (-4.1d0)) 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 <= -4.1) {
		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 <= -4.1:
		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 <= -4.1)
		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 <= -4.1)
		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, -4.1], 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 < -4.0999999999999996

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. fma-def 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   6. Simplified 97.6%

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

   if -4.0999999999999996 < x < 2.0499999999999998

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. fma-def 100.0%

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

      5. +-commutative 100.0%

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

      6. distribute-neg-in 100.0%

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

      7. distribute-rgt-neg-in 100.0%

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

      8. fma-def 100.0%

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

      9. metadata-eval 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 97.9%

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

   if 2.0499999999999998 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-def 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 95.1%

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

      1. unpow2 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*l* 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -4.1)
   (* -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 <= -4.1) {
		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 <= (-4.1d0)) 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 <= -4.1) {
		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 <= -4.1:
		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 <= -4.1)
		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 <= -4.1)
		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, -4.1], 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 < -4.0999999999999996

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. fma-def 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-neg-in 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. fma-def 99.9%

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

      9. metadata-eval 99.9%

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

      10. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 97.6%

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

      1. unpow2 97.6%

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

   6. Simplified 97.6%

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

   if -4.0999999999999996 < 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 99.2%

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

      1. *-commutative 99.2%

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

   4. Simplified 99.2%

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

   if 2.0499999999999998 < x

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. +-commutative 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

      6. distribute-neg-in 99.6%

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

      7. distribute-rgt-neg-in 99.6%

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

      8. fma-def 99.6%

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

      9. metadata-eval 99.6%

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

      10. metadata-eval 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around inf 95.1%

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

      1. unpow2 95.1%

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

      2. *-commutative 95.1%

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

      3. associate-*l* 95.1%

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

   6. Simplified 95.1%

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

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1:\\ \;\;\;\;-0.12 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 2.05:\\ \;\;\;\;1 - x \cdot 0.253\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -0.12\right)\\ \end{array} \]

Alternative 6: 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 99.8%

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

2. Final simplification 99.8%

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

Alternative 7: 97.7% 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 99.8%

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

2. Taylor expanded in x around inf 97.1%

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

   1. unpow2 97.1%

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

4. Simplified 97.1%

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

5. Final simplification 97.1%

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

Alternative 8: 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 99.8%

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

   1. +-commutative 99.8%

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

   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. Step-by-step derivation

   1. fma-udef 99.8%

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

   2. +-commutative 99.8%

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

   3. flip-+ 99.8%

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

   4. metadata-eval 99.8%

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

   5. *-commutative 99.8%

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

   6. *-commutative 99.8%

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

   7. swap-sqr 99.8%

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

   8. metadata-eval 99.8%

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

   9. *-commutative 99.8%

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

   10. cancel-sign-sub-inv 99.8%

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

   11. metadata-eval 99.8%

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

   12. *-commutative 99.8%

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

5. Applied egg-rr 99.8%

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

6. Taylor expanded in x around inf 97.1%

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

7. Final simplification 97.1%

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

Alternative 9: 49.1% 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 99.8%

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

   1. sub-neg 99.8%

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

   2. +-commutative 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. fma-def 99.9%

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

   5. +-commutative 99.9%

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

   6. distribute-neg-in 99.9%

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

   7. distribute-rgt-neg-in 99.9%

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

   8. fma-def 99.9%

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

   9. metadata-eval 99.9%

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

   10. metadata-eval 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in x around 0 49.8%

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

5. Final simplification 49.8%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023171 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, A"
  :precision binary64
  (- 1.0 (* x (+ 0.253 (* x 0.12)))))