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

Percentage Accurate: 100.0% → 100.0%

Time: 8.8s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(x * 0.27061 + 2.30753), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $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. 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-define 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-define 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. fma-undefine 100.0%

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

   6. +-commutative 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-undefine 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. fma-undefine 100.0%

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

   14. fma-undefine 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(x * N[(0.99229 + N[(x * 0.04481), $MachinePrecision]), $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. 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-define 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-define 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. fma-undefine 100.0%

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

   6. +-commutative 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-undefine 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. fma-undefine 100.0%

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

   14. fma-undefine 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 100.0%

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

   1. fma-undefine 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ 6.039053782637804 x) x)
   (if (<= x 1.15)
     (+ 2.30753 (* x (- (* x 1.900161040244073) 3.0191289437)))
     (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437));
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (6.039053782637804d0 / x) - x
    else if (x <= 1.15d0) then
        tmp = 2.30753d0 + (x * ((x * 1.900161040244073d0) - 3.0191289437d0))
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437));
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (6.039053782637804 / x) - x
	elif x <= 1.15:
		tmp = 2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437))
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	elseif (x <= 1.15)
		tmp = Float64(2.30753 + Float64(x * Float64(Float64(x * 1.900161040244073) - 3.0191289437)));
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (6.039053782637804 / x) - x;
	elseif (x <= 1.15)
		tmp = 2.30753 + (x * ((x * 1.900161040244073) - 3.0191289437));
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.15], N[(2.30753 + N[(x * N[(N[(x * 1.900161040244073), $MachinePrecision] - 3.0191289437), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-x)]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \color{blue}{2.30753 + x \cdot \left(1.900161040244073 \cdot x - 3.0191289437\right)} \]

   if 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. 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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753 + x \cdot \left(x \cdot 1.900161040244073 - 3.0191289437\right)\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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]

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. Add Preprocessing
3. Add Preprocessing

Alternative 5: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.05) (not (<= x 1.15)))
   (- x)
   (+ 2.30753 (* x -3.0191289437))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.05) || !(x <= 1.15)) {
		tmp = -x;
	} else {
		tmp = 2.30753 + (x * -3.0191289437);
	}
	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 + (x * (-3.0191289437d0))
    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 + (x * -3.0191289437);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.05) or not (x <= 1.15):
		tmp = -x
	else:
		tmp = 2.30753 + (x * -3.0191289437)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.05) || !(x <= 1.15))
		tmp = Float64(-x);
	else
		tmp = Float64(2.30753 + Float64(x * -3.0191289437));
	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 + (x * -3.0191289437);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.05], N[Not[LessEqual[x, 1.15]], $MachinePrecision]], (-x), N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision]]

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. 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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 99.2%

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

      1. neg-mul-1 99.2%

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

   7. Simplified 99.2%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 2.30753 + \color{blue}{x \cdot -3.0191289437} \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{2.30753 + x \cdot -3.0191289437} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \lor \neg \left(x \leq 1.15\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05:\\ \;\;\;\;\frac{6.039053782637804}{x} - x\\ \mathbf{elif}\;x \leq 1.15:\\ \;\;\;\;2.30753 + x \cdot -3.0191289437\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.05)
   (- (/ 6.039053782637804 x) x)
   (if (<= x 1.15) (+ 2.30753 (* x -3.0191289437)) (- x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * -3.0191289437);
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.05d0)) then
        tmp = (6.039053782637804d0 / x) - x
    else if (x <= 1.15d0) then
        tmp = 2.30753d0 + (x * (-3.0191289437d0))
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.05) {
		tmp = (6.039053782637804 / x) - x;
	} else if (x <= 1.15) {
		tmp = 2.30753 + (x * -3.0191289437);
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.05:
		tmp = (6.039053782637804 / x) - x
	elif x <= 1.15:
		tmp = 2.30753 + (x * -3.0191289437)
	else:
		tmp = -x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.05)
		tmp = Float64(Float64(6.039053782637804 / x) - x);
	elseif (x <= 1.15)
		tmp = Float64(2.30753 + Float64(x * -3.0191289437));
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.05)
		tmp = (6.039053782637804 / x) - x;
	elseif (x <= 1.15)
		tmp = 2.30753 + (x * -3.0191289437);
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.05], N[(N[(6.039053782637804 / x), $MachinePrecision] - x), $MachinePrecision], If[LessEqual[x, 1.15], N[(2.30753 + N[(x * -3.0191289437), $MachinePrecision]), $MachinePrecision], (-x)]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05000000000000004

   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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{6.039053782637804}{x}} - x \]

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{2.30753 + -3.0191289437 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 98.9%

         \[\leadsto 2.30753 + \color{blue}{x \cdot -3.0191289437} \]

   7. Simplified 98.9%

      \[\leadsto \color{blue}{2.30753 + x \cdot -3.0191289437} \]

   if 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. 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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 100.0%

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

      1. neg-mul-1 100.0%

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

   7. Simplified 100.0%

      \[\leadsto \color{blue}{-x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(2.30753 + N[(x * 0.27061), $MachinePrecision]), $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. Add Preprocessing

3. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

5. Simplified 98.6%

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

Alternative 8: 98.4% accurate, 1.4× 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. 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-define 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-define 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-define 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. Add Preprocessing

   5. Taylor expanded in x around inf 99.2%

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

      1. neg-mul-1 99.2%

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

   7. Simplified 99.2%

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

   if -1.05000000000000004 < x < 1.1499999999999999

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. fma-define 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-define 99.9%

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

      5. +-commutative 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. div-inv 99.9%

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

      2. fma-undefine 99.9%

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

      3. +-commutative 99.9%

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

      4. fma-undefine 99.9%

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

      5. fma-undefine 99.9%

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

      6. +-commutative 99.9%

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

      7. +-commutative 99.9%

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

      8. fma-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. fma-undefine 100.0%

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

      11. +-commutative 100.0%

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

      12. +-commutative 100.0%

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

      13. fma-undefine 100.0%

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

      14. fma-undefine 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 97.4%

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

4. Final simplification 98.3%

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

Alternative 9: 98.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(2.30753 / 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. Add Preprocessing

3. Taylor expanded in x around 0 98.6%

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

   1. *-commutative 98.6%

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

5. Simplified 98.6%

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

6. Taylor expanded in x around 0 98.5%

   \[\leadsto \frac{\color{blue}{2.30753}}{1 + x \cdot 0.99229} - x \]
7. Add Preprocessing

Alternative 10: 50.0% 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. 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-define 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-define 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-define 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. Add Preprocessing
5. Step-by-step derivation

   1. div-inv 100.0%

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

   2. fma-undefine 100.0%

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

   3. +-commutative 100.0%

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

   4. fma-undefine 100.0%

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

   5. fma-undefine 100.0%

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

   6. +-commutative 100.0%

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

   7. +-commutative 100.0%

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

   8. fma-neg 100.0%

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

   9. +-commutative 100.0%

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

   10. fma-undefine 100.0%

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

   11. +-commutative 100.0%

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

   12. +-commutative 100.0%

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

   13. fma-undefine 100.0%

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

   14. fma-undefine 100.0%

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0 50.7%

   \[\leadsto \color{blue}{2.30753} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))