Asymptote A

Percentage Accurate: 77.0% → 99.9%

Time: 7.3s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{1 + x}}{1 - x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac2 78.4%

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

   4. neg-sub0 78.4%

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

   5. associate-+l- 78.4%

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

   6. neg-sub0 78.4%

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

   7. remove-double-neg 78.4%

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

   8. distribute-neg-in 78.4%

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

   9. sub-neg 78.4%

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

   10. distribute-neg-frac2 78.4%

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

   11. sub-neg 78.4%

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

   12. +-commutative 78.4%

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

   13. unsub-neg 78.4%

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

   14. sub-neg 78.4%

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

   15. +-commutative 78.4%

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

   16. unsub-neg 78.4%

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

   17. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. distribute-neg-frac 78.4%

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

   3. metadata-eval 78.4%

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

6. Applied egg-rr 78.4%

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

8. Simplified 99.0%

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

   1. associate-/r* 99.9%

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

   2. metadata-eval 99.9%

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

   3. sub-neg 99.9%

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

   4. flip-- 99.0%

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

   5. metadata-eval 99.0%

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

   6. difference-of-sqr-1 99.0%

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

   7. sub-neg 99.0%

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

   8. metadata-eval 99.0%

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

   9. associate-/r/ 95.3%

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

   10. +-commutative 95.3%

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

   11. metadata-eval 95.3%

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

   12. sub-neg 95.3%

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

   13. difference-of-sqr-1 95.3%

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

   14. fmm-def 95.3%

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

   15. metadata-eval 95.3%

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

   16. +-commutative 95.3%

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

10. Applied egg-rr 95.3%

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

    1. associate-/r/ 99.0%

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

    2. metadata-eval 99.0%

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

    3. fmm-def 99.0%

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

    4. metadata-eval 99.0%

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

    5. +-commutative 99.0%

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

    6. metadata-eval 99.0%

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

    7. sub-neg 99.0%

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

    8. flip-+ 99.9%

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

    9. frac-2neg 99.9%

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

    10. distribute-neg-frac 99.9%

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

    11. metadata-eval 99.9%

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

12. Applied egg-rr 99.9%

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

13. Taylor expanded in x around 0 99.9%

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

    1. neg-mul-1 99.9%

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

    2. sub-neg 99.9%

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

15. Simplified 99.9%

    \[\leadsto \frac{\frac{2}{1 + x}}{\color{blue}{1 - x}} \]
16. Add Preprocessing

Alternative 2: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{1 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.75) 2.0 (/ (/ 2.0 x) (- 1.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = (2.0d0 / x) / (1.0d0 - x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = (2.0 / x) / (1.0 - x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.75:
		tmp = 2.0
	else:
		tmp = (2.0 / x) / (1.0 - x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(Float64(2.0 / x) / Float64(1.0 - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = (2.0 / x) / (1.0 - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.75], 2.0, N[(N[(2.0 / x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 88.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 88.7%

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

      2. +-commutative 88.7%

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

      3. distribute-neg-frac2 88.7%

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

      4. neg-sub0 88.7%

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

      5. associate-+l- 88.7%

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

      6. neg-sub0 88.7%

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

      7. remove-double-neg 88.7%

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

      8. distribute-neg-in 88.7%

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

      9. sub-neg 88.7%

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

      10. distribute-neg-frac2 88.7%

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

      11. sub-neg 88.7%

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

      12. +-commutative 88.7%

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

      13. unsub-neg 88.7%

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

      14. sub-neg 88.7%

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

      15. +-commutative 88.7%

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

      16. unsub-neg 88.7%

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

      17. metadata-eval 88.7%

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

   3. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{2} \]

   if 0.75 < x

   1. Initial program 44.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. +-commutative 44.7%

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

      3. distribute-neg-frac2 44.7%

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

      4. neg-sub0 44.7%

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

      5. associate-+l- 44.7%

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

      6. neg-sub0 44.7%

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

      7. remove-double-neg 44.7%

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

      8. distribute-neg-in 44.7%

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

      9. sub-neg 44.7%

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

      10. distribute-neg-frac2 44.7%

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

      11. sub-neg 44.7%

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

      12. +-commutative 44.7%

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

      13. unsub-neg 44.7%

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

      14. sub-neg 44.7%

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

      15. +-commutative 44.7%

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

      16. unsub-neg 44.7%

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

      17. metadata-eval 44.7%

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

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. distribute-neg-frac 44.7%

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

      3. metadata-eval 44.7%

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

   6. Applied egg-rr 44.7%

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

   8. Simplified 97.4%

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

      1. add-sqr-sqrt 97.0%

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

      2. pow2 97.0%

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

      3. +-commutative 97.0%

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

   10. Applied egg-rr 97.0%

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

   11. Taylor expanded in x around inf 96.4%

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

       1. frac-2neg 96.4%

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

       2. metadata-eval 96.4%

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

       3. div-inv 96.4%

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

       4. unpow2 96.4%

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

       5. add-sqr-sqrt 96.7%

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

       6. distribute-rgt-neg-in 96.7%

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

       7. add-sqr-sqrt 96.4%

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

       8. add-sqr-sqrt 0.0%

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

       9. sqrt-unprod 42.9%

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

       10. sqr-neg 42.9%

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

       11. sqrt-unprod 42.9%

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

       12. add-sqr-sqrt 42.9%

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

       13. unpow2 42.9%

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

       14. frac-2neg 42.9%

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

       15. metadata-eval 42.9%

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

       16. unpow2 42.9%

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

       17. add-sqr-sqrt 42.9%

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

       18. distribute-rgt-neg-in 42.9%

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

   13. Applied egg-rr 96.7%

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

       1. associate-*r/ 96.7%

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

       2. metadata-eval 96.7%

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

       3. sub-neg 96.7%

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

       4. unpow2 96.7%

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

       5. sqr-neg 96.7%

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

       6. distribute-lft1-in 96.7%

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

       7. metadata-eval 96.7%

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

       8. distribute-neg-in 96.7%

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

       9. associate-/l/ 99.0%

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

       10. distribute-neg-frac2 99.0%

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

       11. distribute-neg-frac 99.0%

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

       12. metadata-eval 99.0%

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

       13. sub0-neg 99.0%

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

       14. +-commutative 99.0%

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

       15. associate--r+ 99.0%

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

       16. metadata-eval 99.0%

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

   15. Simplified 99.0%

       \[\leadsto \color{blue}{\frac{\frac{2}{x}}{1 - x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 73.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = (-2.0 / (1.0 + 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
    else
        tmp = ((-2.0d0) / (1.0d0 + x)) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 88.7%

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

      2. +-commutative 88.7%

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

      3. distribute-neg-frac2 88.7%

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

      4. neg-sub0 88.7%

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

      5. associate-+l- 88.7%

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

      6. neg-sub0 88.7%

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

      7. remove-double-neg 88.7%

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

      8. distribute-neg-in 88.7%

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

      9. sub-neg 88.7%

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

      10. distribute-neg-frac2 88.7%

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

      11. sub-neg 88.7%

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

      12. +-commutative 88.7%

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

      13. unsub-neg 88.7%

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

      14. sub-neg 88.7%

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

      15. +-commutative 88.7%

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

      16. unsub-neg 88.7%

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

      17. metadata-eval 88.7%

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

   3. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 44.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. +-commutative 44.7%

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

      3. distribute-neg-frac2 44.7%

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

      4. neg-sub0 44.7%

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

      5. associate-+l- 44.7%

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

      6. neg-sub0 44.7%

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

      7. remove-double-neg 44.7%

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

      8. distribute-neg-in 44.7%

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

      9. sub-neg 44.7%

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

      10. distribute-neg-frac2 44.7%

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

      11. sub-neg 44.7%

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

      12. +-commutative 44.7%

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

      13. unsub-neg 44.7%

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

      14. sub-neg 44.7%

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

      15. +-commutative 44.7%

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

      16. unsub-neg 44.7%

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

      17. metadata-eval 44.7%

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

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-sub 45.4%

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

      2. *-rgt-identity 45.4%

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

      3. metadata-eval 45.4%

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

      4. div-inv 45.4%

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

      5. associate-/r* 45.3%

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

      6. metadata-eval 45.3%

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

      7. div-inv 45.3%

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

      8. *-un-lft-identity 45.3%

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

      9. associate--l- 53.0%

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

      10. div-inv 53.0%

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

      11. metadata-eval 53.0%

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

      12. *-rgt-identity 53.0%

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

      13. div-inv 53.0%

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

      14. metadata-eval 53.0%

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

      15. *-rgt-identity 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in x around inf 99.0%

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

      1. div-inv 98.7%

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

      2. div-inv 98.7%

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

      3. associate-*l* 98.7%

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

      4. frac-2neg 98.7%

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

      5. metadata-eval 98.7%

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

      6. flip-- 96.3%

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

      7. metadata-eval 96.3%

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

      8. metadata-eval 96.3%

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

      9. +-commutative 96.3%

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

      10. distribute-neg-frac2 96.3%

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

      11. +-commutative 96.3%

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

      12. distribute-neg-in 96.3%

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

      13. metadata-eval 96.3%

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

      14. sub-neg 96.3%

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

      15. flip-+ 98.7%

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

      16. +-commutative 98.7%

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

   9. Applied egg-rr 98.7%

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

       1. associate-*r* 98.7%

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

       2. associate-*r/ 98.7%

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

       3. metadata-eval 98.7%

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

       4. associate-*l/ 99.0%

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

       5. associate-*r/ 99.0%

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

       6. metadata-eval 99.0%

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

       7. +-commutative 99.0%

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

   11. Simplified 99.0%

       \[\leadsto \color{blue}{\frac{\frac{-2}{1 + x}}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.75:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x \cdot \left(x + -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 0.75) 2.0 (/ -2.0 (* x (+ x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.75d0) then
        tmp = 2.0d0
    else
        tmp = (-2.0d0) / (x * (x + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.75) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / (x * (x + -1.0));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.75:
		tmp = 2.0
	else:
		tmp = -2.0 / (x * (x + -1.0))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = Float64(-2.0 / Float64(x * Float64(x + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.75)
		tmp = 2.0;
	else
		tmp = -2.0 / (x * (x + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.75], 2.0, N[(-2.0 / N[(x * N[(x + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.75

   1. Initial program 88.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 88.7%

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

      2. +-commutative 88.7%

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

      3. distribute-neg-frac2 88.7%

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

      4. neg-sub0 88.7%

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

      5. associate-+l- 88.7%

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

      6. neg-sub0 88.7%

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

      7. remove-double-neg 88.7%

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

      8. distribute-neg-in 88.7%

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

      9. sub-neg 88.7%

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

      10. distribute-neg-frac2 88.7%

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

      11. sub-neg 88.7%

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

      12. +-commutative 88.7%

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

      13. unsub-neg 88.7%

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

      14. sub-neg 88.7%

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

      15. +-commutative 88.7%

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

      16. unsub-neg 88.7%

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

      17. metadata-eval 88.7%

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

   3. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{2} \]

   if 0.75 < x

   1. Initial program 44.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. +-commutative 44.7%

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

      3. distribute-neg-frac2 44.7%

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

      4. neg-sub0 44.7%

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

      5. associate-+l- 44.7%

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

      6. neg-sub0 44.7%

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

      7. remove-double-neg 44.7%

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

      8. distribute-neg-in 44.7%

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

      9. sub-neg 44.7%

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

      10. distribute-neg-frac2 44.7%

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

      11. sub-neg 44.7%

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

      12. +-commutative 44.7%

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

      13. unsub-neg 44.7%

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

      14. sub-neg 44.7%

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

      15. +-commutative 44.7%

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

      16. unsub-neg 44.7%

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

      17. metadata-eval 44.7%

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

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. distribute-neg-frac 44.7%

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

      3. metadata-eval 44.7%

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

   6. Applied egg-rr 44.7%

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

   8. Simplified 97.4%

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

   9. Taylor expanded in x around inf 96.7%

      \[\leadsto \frac{-2}{\color{blue}{x} \cdot \left(x + -1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac2 78.4%

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

   4. neg-sub0 78.4%

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

   5. associate-+l- 78.4%

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

   6. neg-sub0 78.4%

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

   7. remove-double-neg 78.4%

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

   8. distribute-neg-in 78.4%

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

   9. sub-neg 78.4%

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

   10. distribute-neg-frac2 78.4%

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

   11. sub-neg 78.4%

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

   12. +-commutative 78.4%

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

   13. unsub-neg 78.4%

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

   14. sub-neg 78.4%

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

   15. +-commutative 78.4%

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

   16. unsub-neg 78.4%

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

   17. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. distribute-neg-frac 78.4%

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

   3. metadata-eval 78.4%

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

6. Applied egg-rr 78.4%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

Alternative 6: 51.2% accurate, 1.4× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = 2.0;
	} else {
		tmp = -2.0 / 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
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 88.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 88.7%

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

      2. +-commutative 88.7%

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

      3. distribute-neg-frac2 88.7%

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

      4. neg-sub0 88.7%

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

      5. associate-+l- 88.7%

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

      6. neg-sub0 88.7%

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

      7. remove-double-neg 88.7%

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

      8. distribute-neg-in 88.7%

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

      9. sub-neg 88.7%

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

      10. distribute-neg-frac2 88.7%

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

      11. sub-neg 88.7%

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

      12. +-commutative 88.7%

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

      13. unsub-neg 88.7%

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

      14. sub-neg 88.7%

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

      15. +-commutative 88.7%

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

      16. unsub-neg 88.7%

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

      17. metadata-eval 88.7%

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

   3. Simplified 88.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 67.2%

      \[\leadsto \color{blue}{2} \]

   if 1 < x

   1. Initial program 44.7%

      \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
   2. Step-by-step derivation

      1. sub-neg 44.7%

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

      2. +-commutative 44.7%

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

      3. distribute-neg-frac2 44.7%

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

      4. neg-sub0 44.7%

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

      5. associate-+l- 44.7%

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

      6. neg-sub0 44.7%

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

      7. remove-double-neg 44.7%

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

      8. distribute-neg-in 44.7%

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

      9. sub-neg 44.7%

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

      10. distribute-neg-frac2 44.7%

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

      11. sub-neg 44.7%

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

      12. +-commutative 44.7%

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

      13. unsub-neg 44.7%

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

      14. sub-neg 44.7%

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

      15. +-commutative 44.7%

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

      16. unsub-neg 44.7%

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

      17. metadata-eval 44.7%

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

   3. Simplified 44.7%

      \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. frac-sub 45.4%

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

      2. *-rgt-identity 45.4%

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

      3. metadata-eval 45.4%

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

      4. div-inv 45.4%

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

      5. associate-/r* 45.3%

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

      6. metadata-eval 45.3%

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

      7. div-inv 45.3%

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

      8. *-un-lft-identity 45.3%

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

      9. associate--l- 53.0%

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

      10. div-inv 53.0%

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

      11. metadata-eval 53.0%

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

      12. *-rgt-identity 53.0%

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

      13. div-inv 53.0%

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

      14. metadata-eval 53.0%

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

      15. *-rgt-identity 53.0%

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

   6. Applied egg-rr 53.0%

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

   7. Taylor expanded in x around inf 99.0%

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

   8. Taylor expanded in x around 0 6.8%

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

Alternative 7: 50.2% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0
end function

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

function tmp = code(x)
	tmp = 2.0;
end

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 78.4%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac2 78.4%

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

   4. neg-sub0 78.4%

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

   5. associate-+l- 78.4%

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

   6. neg-sub0 78.4%

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

   7. remove-double-neg 78.4%

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

   8. distribute-neg-in 78.4%

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

   9. sub-neg 78.4%

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

   10. distribute-neg-frac2 78.4%

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

   11. sub-neg 78.4%

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

   12. +-commutative 78.4%

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

   13. unsub-neg 78.4%

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

   14. sub-neg 78.4%

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

   15. +-commutative 78.4%

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

   16. unsub-neg 78.4%

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

   17. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 52.1%

   \[\leadsto \color{blue}{2} \]
6. Add Preprocessing

Alternative 8: 10.6% accurate, 11.0× 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 78.4%

   \[\frac{1}{x + 1} - \frac{1}{x - 1} \]
2. Step-by-step derivation

   1. sub-neg 78.4%

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

   2. +-commutative 78.4%

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

   3. distribute-neg-frac2 78.4%

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

   4. neg-sub0 78.4%

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

   5. associate-+l- 78.4%

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

   6. neg-sub0 78.4%

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

   7. remove-double-neg 78.4%

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

   8. distribute-neg-in 78.4%

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

   9. sub-neg 78.4%

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

   10. distribute-neg-frac2 78.4%

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

   11. sub-neg 78.4%

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

   12. +-commutative 78.4%

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

   13. unsub-neg 78.4%

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

   14. sub-neg 78.4%

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

   15. +-commutative 78.4%

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

   16. unsub-neg 78.4%

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

   17. metadata-eval 78.4%

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

3. Simplified 78.4%

   \[\leadsto \color{blue}{\frac{1}{1 - x} - \frac{1}{-1 - x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 51.5%

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

6. Taylor expanded in x around inf 11.0%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))