3frac (problem 3.3.3)

Percentage Accurate: 84.8% → 99.9%

Time: 6.5s

Alternatives: 10

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 2.0 (+ x 1.0)) (* x (+ x -1.0))))

C

double code(double x) {
	return (2.0 / (x + 1.0)) / (x * (x + -1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (2.0d0 / (x + 1.0d0)) / (x * (x + (-1.0d0)))
end function

Java

public static double code(double x) {
	return (2.0 / (x + 1.0)) / (x * (x + -1.0));
}

Python

def code(x):
	return (2.0 / (x + 1.0)) / (x * (x + -1.0))

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(x + 1.0)) / Float64(x * Float64(x + -1.0)))
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (x + 1.0)) / (x * (x + -1.0));
end

Wolfram

code[x_] := N[(N[(2.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
4. Step-by-step derivation

   1. frac-sub 60.1%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]

   2. frac-sub 60.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}} \]

   3. *-un-lft-identity 60.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x + -1\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   4. distribute-rgt-in 60.9%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x + -1 \cdot x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   5. neg-mul-1 60.9%

      \[\leadsto \frac{\left(x \cdot x + \color{blue}{\left(-x\right)}\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   6. sub-neg 60.9%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x - x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   7. *-rgt-identity 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   8. distribute-rgt-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(x \cdot 2 + -1 \cdot 2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   9. metadata-eval 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   10. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{\left(-2\right)}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   11. fma-def 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   12. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   13. distribute-rgt-in 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x + -1 \cdot x\right)}} \]

   14. neg-mul-1 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x + \color{blue}{\left(-x\right)}\right)} \]

   15. sub-neg 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x - x\right)}} \]

5. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}} \]
6. Step-by-step derivation

   1. +-commutative 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(x + 1\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   2. remove-double-neg 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{\left(-\left(-x\right)\right)} + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   3. metadata-eval 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   4. distribute-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\left(\left(-x\right) + -1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   5. neg-mul-1 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   6. *-commutative 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{x \cdot -1} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   7. fma-udef 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\mathsf{fma}\left(x, -1, -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   8. distribute-lft-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   9. distribute-lft-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   10. fma-udef 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\left(x \cdot -1 + -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   11. *-commutative 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   12. neg-mul-1 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{\left(-x\right)} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   13. distribute-neg-in 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(\left(-\left(-x\right)\right) + \left(--1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   14. remove-double-neg 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{x} + \left(--1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   15. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + \color{blue}{1}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   16. +-commutative 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\color{blue}{\left(x + 1\right)} \cdot \left(x \cdot x - x\right)} \]

7. Simplified 60.9%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}} \]

8. Taylor expanded in x around 0 99.3%

   \[\leadsto \frac{\color{blue}{2}}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)} \]
9. Step-by-step derivation

   1. expm1-log1p-u 73.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}\right)\right)} \]

   2. expm1-udef 56.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}\right)} - 1} \]

   3. associate-/r* 56.9%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{x + 1}}{x \cdot x - x}}\right)} - 1 \]

   4. *-un-lft-identity 56.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{x \cdot x - \color{blue}{1 \cdot x}}\right)} - 1 \]

   5. distribute-rgt-out-- 56.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{\color{blue}{x \cdot \left(x - 1\right)}}\right)} - 1 \]

   6. sub-neg 56.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{x \cdot \color{blue}{\left(x + \left(-1\right)\right)}}\right)} - 1 \]

   7. metadata-eval 56.9%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{x \cdot \left(x + \color{blue}{-1}\right)}\right)} - 1 \]

10. Applied egg-rr 56.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)}\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 73.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)}\right)\right)} \]

    2. expm1-log1p 99.9%

       \[\leadsto \color{blue}{\frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)}} \]

12. Simplified 99.9%

    \[\leadsto \color{blue}{\frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)}} \]

13. Final simplification 99.9%

    \[\leadsto \frac{\frac{2}{x + 1}}{x \cdot \left(x + -1\right)} \]

Alternative 2: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 2.0d0 / (x * (x * x))
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = 2.0 / (x * (x * x));
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(2.0 / Float64(x * Float64(x * x)));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(2.0 / N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 66.2%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. unpow3 97.8%

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot x\right) \cdot x}} \]

   6. Applied egg-rr 97.8%

      \[\leadsto \frac{2}{\color{blue}{\left(x \cdot x\right) \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto -2 \cdot x - \color{blue}{\frac{2 \cdot 1}{x}} \]

      2. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 98.3%

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

Alternative 3: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (2.0 / x) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = (2.0d0 / x) / (x * x)
    else
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = (2.0 / x) / (x * x);
	} else {
		tmp = (x * -2.0) - (2.0 / x);
	}
	return tmp;
}

Python

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

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.0))
		tmp = Float64(Float64(2.0 / x) / Float64(x * x));
	else
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(N[(2.0 / x), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 66.2%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 97.8%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. clear-num 97.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

      2. associate-/r/ 97.8%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 99.2%

         \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

      4. metadata-eval 99.2%

         \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

   6. Applied egg-rr 99.2%

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

      1. metadata-eval 99.2%

         \[\leadsto {x}^{\color{blue}{\left(-3\right)}} \cdot 2 \]

      2. pow-flip 97.8%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}}} \cdot 2 \]

      3. pow3 97.8%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot x}} \cdot 2 \]

      4. associate-/r/ 97.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot x}{2}}} \]

      5. clear-num 97.8%

         \[\leadsto \color{blue}{\frac{2}{\left(x \cdot x\right) \cdot x}} \]

      6. associate-*l* 97.8%

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      7. associate-/r* 99.0%

         \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x}} \]

   8. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto -2 \cdot x - \color{blue}{\frac{2 \cdot 1}{x}} \]

      2. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

4. Final simplification 98.9%

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

Alternative 4: 99.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (/ (/ 2.0 x) (* x x))
   (if (<= x 1.0) (- (* x -2.0) (/ 2.0 x)) (/ (/ 2.0 (* x x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) / (x * x);
	} else if (x <= 1.0) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (2.0 / (x * x)) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (2.0d0 / x) / (x * x)
    else if (x <= 1.0d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = (2.0d0 / (x * x)) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) / (x * x);
	} else if (x <= 1.0) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (2.0 / (x * x)) / x;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 61.5%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 61.5%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. clear-num 98.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

      2. associate-/r/ 98.7%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 99.6%

         \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

      4. metadata-eval 99.6%

         \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

   6. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

         \[\leadsto {x}^{\color{blue}{\left(-3\right)}} \cdot 2 \]

      2. pow-flip 98.7%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}}} \cdot 2 \]

      3. pow3 98.6%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot x}} \cdot 2 \]

      4. associate-/r/ 98.6%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot x}{2}}} \]

      5. clear-num 98.6%

         \[\leadsto \color{blue}{\frac{2}{\left(x \cdot x\right) \cdot x}} \]

      6. associate-*l* 98.6%

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(x \cdot x\right)}} \]

      7. associate-/r* 99.4%

         \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x}} \]

   8. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\frac{2}{x}}{x \cdot x}} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 98.8%

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

      1. associate-*r/ 98.8%

         \[\leadsto -2 \cdot x - \color{blue}{\frac{2 \cdot 1}{x}} \]

      2. metadata-eval 98.8%

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

   6. Simplified 98.8%

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

   if 1 < x

   1. Initial program 71.5%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 71.5%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around inf 96.9%

      \[\leadsto \color{blue}{\frac{2}{{x}^{3}}} \]
   5. Step-by-step derivation

      1. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{{x}^{3}}{2}}} \]

      2. associate-/r/ 96.9%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}} \cdot 2} \]

      3. pow-flip 98.7%

         \[\leadsto \color{blue}{{x}^{\left(-3\right)}} \cdot 2 \]

      4. metadata-eval 98.7%

         \[\leadsto {x}^{\color{blue}{-3}} \cdot 2 \]

   6. Applied egg-rr 98.7%

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

      1. metadata-eval 98.7%

         \[\leadsto {x}^{\color{blue}{\left(-3\right)}} \cdot 2 \]

      2. pow-flip 96.9%

         \[\leadsto \color{blue}{\frac{1}{{x}^{3}}} \cdot 2 \]

      3. pow3 96.9%

         \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right) \cdot x}} \cdot 2 \]

      4. associate-/r/ 96.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(x \cdot x\right) \cdot x}{2}}} \]

      5. clear-num 96.9%

         \[\leadsto \color{blue}{\frac{2}{\left(x \cdot x\right) \cdot x}} \]

      6. associate-/r* 98.6%

         \[\leadsto \color{blue}{\frac{\frac{2}{x \cdot x}}{x}} \]

   8. Applied egg-rr 98.6%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{\frac{2}{x}}{x \cdot x}\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x \cdot x}}{x}\\ \end{array} \]

Alternative 5: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* (+ x 1.0) (- (* x x) x))))

C

double code(double x) {
	return 2.0 / ((x + 1.0) * ((x * x) - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / ((x + 1.0d0) * ((x * x) - x))
end function

Java

public static double code(double x) {
	return 2.0 / ((x + 1.0) * ((x * x) - x));
}

Python

def code(x):
	return 2.0 / ((x + 1.0) * ((x * x) - x))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(x + 1.0) * Float64(Float64(x * x) - x)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / ((x + 1.0) * ((x * x) - x));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
4. Step-by-step derivation

   1. frac-sub 60.1%

      \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{2 \cdot \left(x + -1\right) - x \cdot 1}{x \cdot \left(x + -1\right)}} \]

   2. frac-sub 60.9%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(x \cdot \left(x + -1\right)\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)}} \]

   3. *-un-lft-identity 60.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(x + -1\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   4. distribute-rgt-in 60.9%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x + -1 \cdot x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   5. neg-mul-1 60.9%

      \[\leadsto \frac{\left(x \cdot x + \color{blue}{\left(-x\right)}\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   6. sub-neg 60.9%

      \[\leadsto \frac{\color{blue}{\left(x \cdot x - x\right)} - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - x \cdot 1\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   7. *-rgt-identity 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(2 \cdot \left(x + -1\right) - \color{blue}{x}\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   8. distribute-rgt-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\left(x \cdot 2 + -1 \cdot 2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   9. metadata-eval 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   10. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\left(x \cdot 2 + \color{blue}{\left(-2\right)}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   11. fma-def 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)} - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   12. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, \color{blue}{-2}\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot \left(x + -1\right)\right)} \]

   13. distribute-rgt-in 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x + -1 \cdot x\right)}} \]

   14. neg-mul-1 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x + \color{blue}{\left(-x\right)}\right)} \]

   15. sub-neg 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \color{blue}{\left(x \cdot x - x\right)}} \]

5. Applied egg-rr 60.9%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(1 + x\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)}} \]
6. Step-by-step derivation

   1. +-commutative 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(x + 1\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   2. remove-double-neg 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{\left(-\left(-x\right)\right)} + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   3. metadata-eval 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\left(-\left(-x\right)\right) + \color{blue}{\left(--1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   4. distribute-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\left(\left(-x\right) + -1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   5. neg-mul-1 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   6. *-commutative 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{x \cdot -1} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   7. fma-udef 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\mathsf{fma}\left(x, -1, -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   8. distribute-lft-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   9. distribute-lft-neg-in 60.9%

      \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(-\mathsf{fma}\left(x, -1, -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   10. fma-udef 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\color{blue}{\left(x \cdot -1 + -1\right)}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   11. *-commutative 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{-1 \cdot x} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   12. neg-mul-1 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(-\left(\color{blue}{\left(-x\right)} + -1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   13. distribute-neg-in 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \color{blue}{\left(\left(-\left(-x\right)\right) + \left(--1\right)\right)} \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   14. remove-double-neg 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(\color{blue}{x} + \left(--1\right)\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   15. metadata-eval 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + \color{blue}{1}\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(1 + x\right) \cdot \left(x \cdot x - x\right)} \]

   16. +-commutative 60.9%

       \[\leadsto \frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\color{blue}{\left(x + 1\right)} \cdot \left(x \cdot x - x\right)} \]

7. Simplified 60.9%

   \[\leadsto \color{blue}{\frac{\left(x \cdot x - x\right) - \left(x + 1\right) \cdot \left(\mathsf{fma}\left(x, 2, -2\right) - x\right)}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)}} \]

8. Taylor expanded in x around 0 99.3%

   \[\leadsto \frac{\color{blue}{2}}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)} \]

9. Final simplification 99.3%

   \[\leadsto \frac{2}{\left(x + 1\right) \cdot \left(x \cdot x - x\right)} \]

Alternative 6: 83.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (- (/ -2.0 x) x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = (-2.0 / x) - x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = ((-2.0d0) / x) - x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = (-2.0 / x) - x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = (-2.0 / x) - x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = Float64(Float64(-2.0 / x) - x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = (-2.0 / x) - x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(N[(-2.0 / x), $MachinePrecision] - x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 66.2%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

   5. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - -1\right) \]

   6. Taylor expanded in x around inf 65.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 98.2%

      \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

   5. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{-1 \cdot x - 2 \cdot \frac{1}{x}} \]
   6. Step-by-step derivation

      1. neg-mul-1 98.2%

         \[\leadsto \color{blue}{\left(-x\right)} - 2 \cdot \frac{1}{x} \]

      2. associate-*r/ 98.2%

         \[\leadsto \left(-x\right) - \color{blue}{\frac{2 \cdot 1}{x}} \]

      3. metadata-eval 98.2%

         \[\leadsto \left(-x\right) - \frac{\color{blue}{2}}{x} \]

   7. Simplified 98.2%

      \[\leadsto \color{blue}{\left(-x\right) - \frac{2}{x}} \]

   8. Taylor expanded in x around 0 98.2%

      \[\leadsto \color{blue}{-1 \cdot x - 2 \cdot \frac{1}{x}} \]
   9. Step-by-step derivation

      1. mul-1-neg 98.2%

         \[\leadsto \color{blue}{\left(-x\right)} - 2 \cdot \frac{1}{x} \]

      2. neg-sub0 98.2%

         \[\leadsto \color{blue}{\left(0 - x\right)} - 2 \cdot \frac{1}{x} \]

      3. associate-*r/ 98.2%

         \[\leadsto \left(0 - x\right) - \color{blue}{\frac{2 \cdot 1}{x}} \]

      4. metadata-eval 98.2%

         \[\leadsto \left(0 - x\right) - \frac{\color{blue}{2}}{x} \]

      5. associate--r+ 98.2%

         \[\leadsto \color{blue}{0 - \left(x + \frac{2}{x}\right)} \]

      6. +-commutative 98.2%

         \[\leadsto 0 - \color{blue}{\left(\frac{2}{x} + x\right)} \]

      7. associate--r+ 98.2%

         \[\leadsto \color{blue}{\left(0 - \frac{2}{x}\right) - x} \]

      8. neg-sub0 98.2%

         \[\leadsto \color{blue}{\left(-\frac{2}{x}\right)} - x \]

      9. distribute-neg-frac 98.2%

         \[\leadsto \color{blue}{\frac{-2}{x}} - x \]

      10. metadata-eval 98.2%

          \[\leadsto \frac{\color{blue}{-2}}{x} - x \]

   10. Simplified 98.2%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x} - x\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 7: 83.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (/ -2.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 1.0d0) then
        tmp = (-2.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = -2.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = -2.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = Float64(-2.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 1.0)
		tmp = -2.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(-2.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 66.2%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 3.4%

      \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

   5. Taylor expanded in x around 0 65.0%

      \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - -1\right) \]

   6. Taylor expanded in x around inf 65.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

   4. Taylor expanded in x around 0 98.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 8: 83.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(-1 - \frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- -1.0 (/ 2.0 x))))

C

double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-1.0d0) - (2.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Python

def code(x):
	return 1.0 + (-1.0 - (2.0 / x))

Julia

function code(x)
	return Float64(1.0 + Float64(-1.0 - Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (-1.0 - (2.0 / x));
end

Wolfram

code[x_] := N[(1.0 + N[(-1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 51.5%

   \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

5. Taylor expanded in x around 0 81.8%

   \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - -1\right) \]

6. Final simplification 81.8%

   \[\leadsto 1 + \left(-1 - \frac{2}{x}\right) \]

Alternative 9: 3.3% accurate, 15.0× 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 83.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 51.5%

   \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

5. Taylor expanded in x around inf 3.4%

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

6. Final simplification 3.4%

   \[\leadsto -1 \]

Alternative 10: 35.1% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 83.4%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.4%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]

4. Taylor expanded in x around 0 51.5%

   \[\leadsto \frac{1}{1 + x} - \left(\frac{2}{x} - \color{blue}{-1}\right) \]

5. Taylor expanded in x around 0 81.8%

   \[\leadsto \color{blue}{1} - \left(\frac{2}{x} - -1\right) \]

6. Taylor expanded in x around inf 33.2%

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

7. Final simplification 33.2%

   \[\leadsto 0 \]

Developer target: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023264 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))