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

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 6

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))

C

double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((2.30753d0 + (x * 0.27061d0)) / (1.0d0 + (x * (0.99229d0 + (x * 0.04481d0))))) - x
end function

Java

public static double code(double x) {
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
}

Python

def code(x):
	return ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x

Julia

function code(x)
	return Float64(Float64(Float64(2.30753 + Float64(x * 0.27061)) / Float64(1.0 + Float64(x * Float64(0.99229 + Float64(x * 0.04481))))) - x)
end

MATLAB

function tmp = code(x)
	tmp = ((2.30753 + (x * 0.27061)) / (1.0 + (x * (0.99229 + (x * 0.04481))))) - x;
end

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (fma x (+ (* x 0.04481) 0.99229) 1.0)) x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / fma(x, ((x * 0.04481) + 0.99229), 1.0)) - x;
}

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / fma(x, Float64(Float64(x * 0.04481) + 0.99229), 1.0)) - x)
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]
2. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. fma-def 100.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   3. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{x \cdot \left(0.99229 + x \cdot 0.04481\right) + 1}} - x \]

   4. fma-def 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\color{blue}{\mathsf{fma}\left(x, 0.99229 + x \cdot 0.04481, 1\right)}} - x \]

   5. +-commutative 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} - x \]

   6. fma-def 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.04481, 0.99229\right)}, 1\right)} - x \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.04481, 0.99229\right), 1\right)} - x} \]
4. Step-by-step derivation

   1. fma-udef 100.0%

      \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} - x \]

5. Applied egg-rr 100.0%

   \[\leadsto \frac{\mathsf{fma}\left(x, 0.27061, 2.30753\right)}{\mathsf{fma}\left(x, \color{blue}{x \cdot 0.04481 + 0.99229}, 1\right)} - x \]
6. Step-by-step derivation

   1. fma-udef 100.0%

      \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)} - x \]

7. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{x \cdot 0.27061 + 2.30753}}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)} - x \]

8. Final simplification 100.0%

   \[\leadsto \frac{x \cdot 0.27061 + 2.30753}{\mathsf{fma}\left(x, x \cdot 0.04481 + 0.99229, 1\right)} - x \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (+ 1.0 (* x (+ (* x 0.04481) 0.99229)))) x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * ((x * 0.04481) + 0.99229)))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x * 0.27061d0) + 2.30753d0) / (1.0d0 + (x * ((x * 0.04481d0) + 0.99229d0)))) - x
end function

Java

public static double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * ((x * 0.04481) + 0.99229)))) - x;
}

Python

def code(x):
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * ((x * 0.04481) + 0.99229)))) - x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(1.0 + Float64(x * Float64(Float64(x * 0.04481) + 0.99229)))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (((x * 0.27061) + 2.30753) / (1.0 + (x * ((x * 0.04481) + 0.99229)))) - x;
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(x * N[(N[(x * 0.04481), $MachinePrecision] + 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Final simplification 100.0%

   \[\leadsto \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot \left(x \cdot 0.04481 + 0.99229\right)} - x \]

Alternative 3: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot 0.99229} - x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (/ (+ (* x 0.27061) 2.30753) (+ 1.0 (* x 0.99229))) x))

C

double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((x * 0.27061d0) + 2.30753d0) / (1.0d0 + (x * 0.99229d0))) - x
end function

Java

public static double code(double x) {
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
}

Python

def code(x):
	return (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x

Julia

function code(x)
	return Float64(Float64(Float64(Float64(x * 0.27061) + 2.30753) / Float64(1.0 + Float64(x * 0.99229))) - x)
end

MATLAB

function tmp = code(x)
	tmp = (((x * 0.27061) + 2.30753) / (1.0 + (x * 0.99229))) - x;
end

Wolfram

code[x_] := N[(N[(N[(N[(x * 0.27061), $MachinePrecision] + 2.30753), $MachinePrecision] / N[(1.0 + N[(x * 0.99229), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 98.4%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{0.99229 \cdot x}} - x \]
3. Step-by-step derivation

   1. *-commutative 98.4%

      \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

4. Simplified 98.4%

   \[\leadsto \frac{2.30753 + x \cdot 0.27061}{1 + \color{blue}{x \cdot 0.99229}} - x \]

5. Final simplification 98.4%

   \[\leadsto \frac{x \cdot 0.27061 + 2.30753}{1 + x \cdot 0.99229} - x \]

Alternative 4: 98.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15))) (- x) 2.30753))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.05d0)) .or. (.not. (x <= 1.15d0))) then
        tmp = -x
    else
        tmp = 2.30753d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = -x
	else:
		tmp = 2.30753
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = 2.30753;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.05) || ~((x <= 1.15)))
		tmp = -x;
	else
		tmp = 2.30753;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), 2.30753]

Derivation

1. Split input into 2 regimes

2. if x < -1.05000000000000004 or 1.1499999999999999 < x

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0 98.0%

      \[\leadsto \color{blue}{2.30753} - x \]

   3. Taylor expanded in x around inf 98.9%

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

      1. neg-mul-1 98.9%

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

   5. Simplified 98.9%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 100.0%

      \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

   2. Taylor expanded in x around 0 98.3%

      \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]
   3. Step-by-step derivation

      1. associate--l+ 98.3%

         \[\leadsto \color{blue}{2.30753 + \left(-2.0191289437 \cdot x - x\right)} \]

      2. flip-+ 98.3%

         \[\leadsto \color{blue}{\frac{2.30753 \cdot 2.30753 - \left(-2.0191289437 \cdot x - x\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)}} \]

      3. metadata-eval 96.8%

         \[\leadsto \frac{\color{blue}{5.3246947009} - \left(-2.0191289437 \cdot x - x\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      4. *-un-lft-identity 96.8%

         \[\leadsto \frac{5.3246947009 - \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      5. distribute-rgt-out-- 96.8%

         \[\leadsto \frac{5.3246947009 - \color{blue}{\left(x \cdot \left(-2.0191289437 - 1\right)\right)} \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      6. metadata-eval 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot \color{blue}{-3.0191289437}\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      7. *-un-lft-identity 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      8. distribute-rgt-out-- 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(x \cdot \left(-2.0191289437 - 1\right)\right)}}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      9. metadata-eval 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{-3.0191289437}\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

      10. *-un-lft-identity 96.8%

          \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right)} \]

      11. distribute-rgt-out-- 96.8%

          \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - \color{blue}{x \cdot \left(-2.0191289437 - 1\right)}} \]

      12. metadata-eval 96.8%

          \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - x \cdot \color{blue}{-3.0191289437}} \]

   4. Applied egg-rr 96.8%

      \[\leadsto \color{blue}{\frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - x \cdot -3.0191289437}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 47.0%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(\sqrt{x \cdot -3.0191289437} \cdot \sqrt{x \cdot -3.0191289437}\right)}}{2.30753 - x \cdot -3.0191289437} \]

      2. sqrt-prod 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\sqrt{\left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}}}{2.30753 - x \cdot -3.0191289437} \]

      3. swap-sqr 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(-3.0191289437 \cdot -3.0191289437\right)}}}{2.30753 - x \cdot -3.0191289437} \]

      4. sqrt-prod 96.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(\sqrt{x \cdot x} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}}{2.30753 - x \cdot -3.0191289437} \]

      5. sqrt-unprod 49.8%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

      6. add-sqr-sqrt 96.7%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(\color{blue}{x} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

      7. metadata-eval 96.7%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \sqrt{\color{blue}{9.115139578687078}}\right)}{2.30753 - x \cdot -3.0191289437} \]

      8. metadata-eval 96.7%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

      9. metadata-eval 96.7%

         \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{\left(--3.0191289437\right)}\right)}{2.30753 - x \cdot -3.0191289437} \]

      10. distribute-rgt-neg-in 96.7%

          \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(-x \cdot -3.0191289437\right)}}{2.30753 - x \cdot -3.0191289437} \]

   6. Applied egg-rr 96.7%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(-x \cdot -3.0191289437\right)}}{2.30753 - x \cdot -3.0191289437} \]

   7. Taylor expanded in x around 0 97.6%

      \[\leadsto \color{blue}{2.30753} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753\\ \end{array} \]

Alternative 5: 98.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ 2.30753 - x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- 2.30753 x))

C

double code(double x) {
	return 2.30753 - x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0 - x
end function

Java

public static double code(double x) {
	return 2.30753 - x;
}

Python

def code(x):
	return 2.30753 - x

Julia

function code(x)
	return Float64(2.30753 - x)
end

MATLAB

function tmp = code(x)
	tmp = 2.30753 - x;
end

Wolfram

code[x_] := N[(2.30753 - x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 97.8%

   \[\leadsto \color{blue}{2.30753} - x \]

3. Final simplification 97.8%

   \[\leadsto 2.30753 - x \]

Alternative 6: 50.6% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 2.30753 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.30753)

C

double code(double x) {
	return 2.30753;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.30753d0
end function

Java

public static double code(double x) {
	return 2.30753;
}

Python

def code(x):
	return 2.30753

Julia

function code(x)
	return 2.30753
end

MATLAB

function tmp = code(x)
	tmp = 2.30753;
end

Wolfram

code[x_] := 2.30753

Derivation

1. Initial program 100.0%

   \[\frac{2.30753 + x \cdot 0.27061}{1 + x \cdot \left(0.99229 + x \cdot 0.04481\right)} - x \]

2. Taylor expanded in x around 0 58.9%

   \[\leadsto \color{blue}{\left(2.30753 + -2.0191289437 \cdot x\right)} - x \]
3. Step-by-step derivation

   1. associate--l+ 58.9%

      \[\leadsto \color{blue}{2.30753 + \left(-2.0191289437 \cdot x - x\right)} \]

   2. flip-+ 56.2%

      \[\leadsto \color{blue}{\frac{2.30753 \cdot 2.30753 - \left(-2.0191289437 \cdot x - x\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)}} \]

   3. metadata-eval 55.5%

      \[\leadsto \frac{\color{blue}{5.3246947009} - \left(-2.0191289437 \cdot x - x\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   4. *-un-lft-identity 55.5%

      \[\leadsto \frac{5.3246947009 - \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   5. distribute-rgt-out-- 55.5%

      \[\leadsto \frac{5.3246947009 - \color{blue}{\left(x \cdot \left(-2.0191289437 - 1\right)\right)} \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   6. metadata-eval 55.5%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot \color{blue}{-3.0191289437}\right) \cdot \left(-2.0191289437 \cdot x - x\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   7. *-un-lft-identity 55.5%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   8. distribute-rgt-out-- 55.5%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(x \cdot \left(-2.0191289437 - 1\right)\right)}}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   9. metadata-eval 55.5%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{-3.0191289437}\right)}{2.30753 - \left(-2.0191289437 \cdot x - x\right)} \]

   10. *-un-lft-identity 55.5%

       \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - \left(-2.0191289437 \cdot x - \color{blue}{1 \cdot x}\right)} \]

   11. distribute-rgt-out-- 55.5%

       \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - \color{blue}{x \cdot \left(-2.0191289437 - 1\right)}} \]

   12. metadata-eval 55.5%

       \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - x \cdot \color{blue}{-3.0191289437}} \]

4. Applied egg-rr 55.5%

   \[\leadsto \color{blue}{\frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}{2.30753 - x \cdot -3.0191289437}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 27.2%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(\sqrt{x \cdot -3.0191289437} \cdot \sqrt{x \cdot -3.0191289437}\right)}}{2.30753 - x \cdot -3.0191289437} \]

   2. sqrt-prod 52.8%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\sqrt{\left(x \cdot -3.0191289437\right) \cdot \left(x \cdot -3.0191289437\right)}}}{2.30753 - x \cdot -3.0191289437} \]

   3. swap-sqr 52.8%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(-3.0191289437 \cdot -3.0191289437\right)}}}{2.30753 - x \cdot -3.0191289437} \]

   4. sqrt-prod 52.8%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(\sqrt{x \cdot x} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}}{2.30753 - x \cdot -3.0191289437} \]

   5. sqrt-unprod 25.6%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

   6. add-sqr-sqrt 49.9%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(\color{blue}{x} \cdot \sqrt{-3.0191289437 \cdot -3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

   7. metadata-eval 49.9%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \sqrt{\color{blue}{9.115139578687078}}\right)}{2.30753 - x \cdot -3.0191289437} \]

   8. metadata-eval 49.9%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{3.0191289437}\right)}{2.30753 - x \cdot -3.0191289437} \]

   9. metadata-eval 49.9%

      \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \left(x \cdot \color{blue}{\left(--3.0191289437\right)}\right)}{2.30753 - x \cdot -3.0191289437} \]

   10. distribute-rgt-neg-in 49.9%

       \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(-x \cdot -3.0191289437\right)}}{2.30753 - x \cdot -3.0191289437} \]

6. Applied egg-rr 49.9%

   \[\leadsto \frac{5.3246947009 - \left(x \cdot -3.0191289437\right) \cdot \color{blue}{\left(-x \cdot -3.0191289437\right)}}{2.30753 - x \cdot -3.0191289437} \]

7. Taylor expanded in x around 0 51.5%

   \[\leadsto \color{blue}{2.30753} \]

8. Final simplification 51.5%

   \[\leadsto 2.30753 \]

Reproduce

herbie shell --seed 2023335 
(FPCore (x)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, C"
  :precision binary64
  (- (/ (+ 2.30753 (* x 0.27061)) (+ 1.0 (* x (+ 0.99229 (* x 0.04481))))) x))