Asymptote A

Percentage Accurate: 77.9% → 99.9%

Time: 6.8s

Alternatives: 7

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: 77.9% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{1 - x\_m}}{-1 - x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ -2.0 (- 1.0 x_m)) (- -1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (1.0 - x_m)) / (-1.0 - x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(-2.0 / Float64(1.0 - x_m)) / Float64(-1.0 - x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. sub-neg 79.9%

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

   2. +-commutative 79.9%

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

   3. distribute-neg-frac2 79.9%

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

   4. neg-sub0 79.9%

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

   5. associate-+l- 79.9%

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

   6. neg-sub0 79.9%

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

   7. remove-double-neg 79.9%

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

   8. distribute-neg-in 79.9%

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

   9. sub-neg 79.9%

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

   10. distribute-neg-frac2 79.9%

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

   11. sub-neg 79.9%

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

   12. +-commutative 79.9%

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

   13. unsub-neg 79.9%

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

   14. sub-neg 79.9%

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

   15. +-commutative 79.9%

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

   16. unsub-neg 79.9%

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

   17. metadata-eval 79.9%

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

3. Simplified 79.9%

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

   1. frac-sub 81.0%

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

   2. *-rgt-identity 81.0%

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

   3. metadata-eval 81.0%

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

   4. div-inv 81.0%

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

   5. associate-/r* 81.0%

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

   6. metadata-eval 81.0%

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

   7. div-inv 81.0%

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

   8. *-un-lft-identity 81.0%

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

   9. associate--l- 83.5%

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

   10. div-inv 83.5%

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

   11. metadata-eval 83.5%

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

   12. *-rgt-identity 83.5%

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

   13. div-inv 83.5%

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

   14. metadata-eval 83.5%

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

   15. *-rgt-identity 83.5%

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

6. Applied egg-rr 83.5%

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

   1. div-sub 83.5%

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

   2. sub-neg 83.5%

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

   3. frac-2neg 83.5%

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

   4. metadata-eval 83.5%

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

   5. flip-- 83.4%

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

   6. metadata-eval 83.4%

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

   7. metadata-eval 83.4%

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

   8. +-commutative 83.4%

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

   9. distribute-neg-frac2 83.4%

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

   10. mul-1-neg 83.4%

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

   11. +-commutative 83.4%

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

   12. distribute-lft-in 83.4%

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

   13. metadata-eval 83.4%

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

   14. neg-mul-1 83.4%

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

   15. sub-neg 83.4%

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

   16. flip-+ 83.5%

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

   17. +-commutative 83.5%

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

8. Applied egg-rr 83.5%

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

   1. sub-neg 83.5%

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

   2. +-commutative 83.5%

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

   3. associate--r- 99.9%

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

   4. +-inverses 99.9%

      \[\leadsto \frac{\frac{1}{x + -1} - \frac{1 - \color{blue}{0}}{1 - x}}{-1 - x} \]

   5. metadata-eval 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. +-commutative 99.9%

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

   8. metadata-eval 99.9%

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

   9. associate--r- 99.9%

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

   10. neg-sub0 99.9%

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

   11. distribute-frac-neg2 99.9%

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

   12. distribute-neg-frac 99.9%

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

   13. metadata-eval 99.9%

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

   14. *-rgt-identity 99.9%

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

   15. metadata-eval 99.9%

       \[\leadsto \frac{\frac{-1}{1 - x} - \frac{\color{blue}{1 - 0}}{1 - x}}{-1 - x} \]

   16. +-inverses 99.9%

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

   17. associate--r- 83.5%

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

   18. +-commutative 83.5%

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

10. Simplified 99.9%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x\_m}}{-x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.0) 2.0 (/ (/ 2.0 x_m) (- x_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], 2.0, N[(N[(2.0 / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.1%

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

      1. sub-neg 89.1%

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

      2. +-commutative 89.1%

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

      3. distribute-neg-frac2 89.1%

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

      4. neg-sub0 89.1%

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

      5. associate-+l- 89.1%

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

      6. neg-sub0 89.1%

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

      7. remove-double-neg 89.1%

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

      8. distribute-neg-in 89.1%

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

      9. sub-neg 89.1%

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

      10. distribute-neg-frac2 89.1%

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

      11. sub-neg 89.1%

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

      12. +-commutative 89.1%

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

      13. unsub-neg 89.1%

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

      14. sub-neg 89.1%

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

      15. +-commutative 89.1%

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

      16. unsub-neg 89.1%

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

      17. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 72.7%

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

   if 1 < x

   1. Initial program 53.0%

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

      1. sub-neg 53.0%

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

      2. +-commutative 53.0%

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

      3. distribute-neg-frac2 53.0%

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

      4. neg-sub0 53.0%

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

      5. associate-+l- 53.0%

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

      6. neg-sub0 53.0%

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

      7. remove-double-neg 53.0%

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

      8. distribute-neg-in 53.0%

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

      9. sub-neg 53.0%

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

      10. distribute-neg-frac2 53.0%

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

      11. sub-neg 53.0%

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

      12. +-commutative 53.0%

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

      13. unsub-neg 53.0%

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

      14. sub-neg 53.0%

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

      15. +-commutative 53.0%

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

      16. unsub-neg 53.0%

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

      17. metadata-eval 53.0%

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

   3. Simplified 53.0%

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

      1. sub-neg 53.0%

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

      2. distribute-neg-frac 53.0%

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

      3. metadata-eval 53.0%

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

   6. Applied egg-rr 53.0%

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

   8. Simplified 97.0%

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

   9. Taylor expanded in x around inf 93.3%

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

       1. div-inv 93.3%

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

       2. *-commutative 93.3%

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

       3. associate-/r* 96.0%

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

   11. Applied egg-rr 96.0%

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

       1. associate-/l/ 93.3%

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

       2. associate-*r/ 93.3%

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

       3. metadata-eval 93.3%

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

       4. metadata-eval 93.3%

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

       5. distribute-neg-frac 93.3%

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

       6. associate-/r* 96.0%

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

       7. distribute-neg-frac2 96.0%

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

       8. +-commutative 96.0%

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

       9. distribute-neg-in 96.0%

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

       10. metadata-eval 96.0%

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

       11. sub-neg 96.0%

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

   13. Simplified 96.0%

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

   14. Taylor expanded in x around inf 96.9%

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

       1. neg-mul-1 96.9%

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

   16. Simplified 96.9%

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

Alternative 3: 53.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{1 + x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.0) 2.0 (/ -2.0 (+ 1.0 x_m))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], 2.0, N[(-2.0 / N[(1.0 + x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.1%

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

      1. sub-neg 89.1%

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

      2. +-commutative 89.1%

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

      3. distribute-neg-frac2 89.1%

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

      4. neg-sub0 89.1%

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

      5. associate-+l- 89.1%

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

      6. neg-sub0 89.1%

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

      7. remove-double-neg 89.1%

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

      8. distribute-neg-in 89.1%

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

      9. sub-neg 89.1%

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

      10. distribute-neg-frac2 89.1%

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

      11. sub-neg 89.1%

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

      12. +-commutative 89.1%

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

      13. unsub-neg 89.1%

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

      14. sub-neg 89.1%

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

      15. +-commutative 89.1%

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

      16. unsub-neg 89.1%

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

      17. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 72.7%

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

   if 1 < x

   1. Initial program 53.0%

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

      1. sub-neg 53.0%

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

      2. +-commutative 53.0%

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

      3. distribute-neg-frac2 53.0%

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

      4. neg-sub0 53.0%

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

      5. associate-+l- 53.0%

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

      6. neg-sub0 53.0%

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

      7. remove-double-neg 53.0%

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

      8. distribute-neg-in 53.0%

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

      9. sub-neg 53.0%

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

      10. distribute-neg-frac2 53.0%

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

      11. sub-neg 53.0%

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

      12. +-commutative 53.0%

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

      13. unsub-neg 53.0%

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

      14. sub-neg 53.0%

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

      15. +-commutative 53.0%

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

      16. unsub-neg 53.0%

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

      17. metadata-eval 53.0%

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

   3. Simplified 53.0%

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

      1. frac-sub 55.2%

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

      2. *-rgt-identity 55.2%

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

      3. metadata-eval 55.2%

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

      4. div-inv 55.2%

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

      5. associate-/r* 55.3%

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

      6. metadata-eval 55.3%

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

      7. div-inv 55.3%

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

      8. *-un-lft-identity 55.3%

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

      9. associate--l- 61.1%

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

      10. div-inv 61.1%

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

      11. metadata-eval 61.1%

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

      12. *-rgt-identity 61.1%

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

      13. div-inv 61.1%

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

      14. metadata-eval 61.1%

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

      15. *-rgt-identity 61.1%

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

   6. Applied egg-rr 61.1%

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

   7. Taylor expanded in x around 0 4.6%

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

      1. add-sqr-sqrt 4.6%

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

      2. sqrt-unprod 41.8%

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

      3. frac-times 46.4%

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

      4. metadata-eval 46.4%

         \[\leadsto \sqrt{\frac{\color{blue}{4}}{\left(-1 - x\right) \cdot \left(-1 - x\right)}} \]

      5. metadata-eval 46.4%

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

      6. sub-neg 46.4%

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

      7. metadata-eval 46.4%

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

      8. distribute-neg-in 46.4%

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

      9. +-commutative 46.4%

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

      10. sub-neg 46.4%

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

      11. metadata-eval 46.4%

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

      12. distribute-neg-in 46.4%

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

      13. +-commutative 46.4%

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

      14. frac-times 41.8%

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

      15. metadata-eval 41.8%

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

      16. frac-2neg 41.8%

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

      17. metadata-eval 41.8%

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

      18. frac-2neg 41.8%

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

      19. sqrt-unprod 0.0%

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

      20. add-sqr-sqrt 7.0%

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

      21. div-inv 7.0%

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

   9. Applied egg-rr 7.0%

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

       1. exp-neg 7.0%

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

       2. log1p-undefine 7.0%

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

       3. rem-exp-log 7.0%

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

       4. associate-*r/ 7.0%

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

       5. metadata-eval 7.0%

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

       6. +-commutative 7.0%

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

   11. Simplified 7.0%

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

4. Final simplification 56.0%

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

Alternative 4: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{\frac{-2}{-1 - x\_m}}{1 - x\_m} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ (/ -2.0 (- -1.0 x_m)) (- 1.0 x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	return (-2.0 / (-1.0 - x_m)) / (1.0 - x_m);
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (-2.0 / (-1.0 - x_m)) / (1.0 - x_m);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (-2.0 / (-1.0 - x_m)) / (1.0 - x_m)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(-2.0 / Float64(-1.0 - x_m)) / Float64(1.0 - x_m))
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(-2.0 / N[(-1.0 - x$95$m), $MachinePrecision]), $MachinePrecision] / N[(1.0 - x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. sub-neg 79.9%

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

   2. +-commutative 79.9%

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

   3. distribute-neg-frac2 79.9%

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

   4. neg-sub0 79.9%

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

   5. associate-+l- 79.9%

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

   6. neg-sub0 79.9%

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

   7. remove-double-neg 79.9%

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

   8. distribute-neg-in 79.9%

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

   9. sub-neg 79.9%

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

   10. distribute-neg-frac2 79.9%

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

   11. sub-neg 79.9%

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

   12. +-commutative 79.9%

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

   13. unsub-neg 79.9%

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

   14. sub-neg 79.9%

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

   15. +-commutative 79.9%

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

   16. unsub-neg 79.9%

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

   17. metadata-eval 79.9%

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

3. Simplified 79.9%

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

   1. sub-neg 79.9%

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

   2. distribute-neg-frac 79.9%

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

   3. metadata-eval 79.9%

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

6. Applied egg-rr 79.9%

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

8. Simplified 98.9%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.9%

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

10. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. frac-2neg 99.9%

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

    3. *-rgt-identity 99.9%

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

    4. frac-2neg 99.9%

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

    5. metadata-eval 99.9%

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

    6. +-commutative 99.9%

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

    7. distribute-neg-in 99.9%

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

    8. metadata-eval 99.9%

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

    9. sub-neg 99.9%

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

    10. distribute-frac-neg2 99.9%

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

    11. metadata-eval 99.9%

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

    12. frac-2neg 99.9%

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

    13. +-commutative 99.9%

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

    14. distribute-neg-in 99.9%

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

    15. metadata-eval 99.9%

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

    16. sub-neg 99.9%

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

12. Applied egg-rr 99.9%

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

Alternative 5: 99.4% accurate, 1.2× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (/ -2.0 (* (+ 1.0 x_m) (+ x_m -1.0))))

C

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

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return -2.0 / ((1.0 + x_m) * (x_m + -1.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return -2.0 / ((1.0 + x_m) * (x_m + -1.0))

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(-2.0 / N[(N[(1.0 + x$95$m), $MachinePrecision] * N[(x$95$m + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.9%

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

   1. sub-neg 79.9%

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

   2. +-commutative 79.9%

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

   3. distribute-neg-frac2 79.9%

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

   4. neg-sub0 79.9%

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

   5. associate-+l- 79.9%

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

   6. neg-sub0 79.9%

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

   7. remove-double-neg 79.9%

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

   8. distribute-neg-in 79.9%

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

   9. sub-neg 79.9%

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

   10. distribute-neg-frac2 79.9%

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

   11. sub-neg 79.9%

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

   12. +-commutative 79.9%

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

   13. unsub-neg 79.9%

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

   14. sub-neg 79.9%

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

   15. +-commutative 79.9%

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

   16. unsub-neg 79.9%

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

   17. metadata-eval 79.9%

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

3. Simplified 79.9%

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

   1. sub-neg 79.9%

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

   2. distribute-neg-frac 79.9%

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

   3. metadata-eval 79.9%

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

6. Applied egg-rr 79.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 53.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1:\\ \;\;\;\;2\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.0) 2.0 (/ -2.0 x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.0], 2.0, N[(-2.0 / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 89.1%

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

      1. sub-neg 89.1%

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

      2. +-commutative 89.1%

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

      3. distribute-neg-frac2 89.1%

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

      4. neg-sub0 89.1%

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

      5. associate-+l- 89.1%

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

      6. neg-sub0 89.1%

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

      7. remove-double-neg 89.1%

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

      8. distribute-neg-in 89.1%

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

      9. sub-neg 89.1%

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

      10. distribute-neg-frac2 89.1%

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

      11. sub-neg 89.1%

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

      12. +-commutative 89.1%

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

      13. unsub-neg 89.1%

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

      14. sub-neg 89.1%

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

      15. +-commutative 89.1%

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

      16. unsub-neg 89.1%

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

      17. metadata-eval 89.1%

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

   3. Simplified 89.1%

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

   5. Taylor expanded in x around 0 72.7%

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

   if 1 < x

   1. Initial program 53.0%

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

      1. sub-neg 53.0%

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

      2. +-commutative 53.0%

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

      3. distribute-neg-frac2 53.0%

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

      4. neg-sub0 53.0%

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

      5. associate-+l- 53.0%

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

      6. neg-sub0 53.0%

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

      7. remove-double-neg 53.0%

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

      8. distribute-neg-in 53.0%

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

      9. sub-neg 53.0%

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

      10. distribute-neg-frac2 53.0%

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

      11. sub-neg 53.0%

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

      12. +-commutative 53.0%

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

      13. unsub-neg 53.0%

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

      14. sub-neg 53.0%

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

      15. +-commutative 53.0%

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

      16. unsub-neg 53.0%

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

      17. metadata-eval 53.0%

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

   3. Simplified 53.0%

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

      1. frac-sub 55.2%

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

      2. *-rgt-identity 55.2%

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

      3. metadata-eval 55.2%

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

      4. div-inv 55.2%

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

      5. associate-/r* 55.3%

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

      6. metadata-eval 55.3%

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

      7. div-inv 55.3%

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

      8. *-un-lft-identity 55.3%

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

      9. associate--l- 61.1%

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

      10. div-inv 61.1%

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

      11. metadata-eval 61.1%

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

      12. *-rgt-identity 61.1%

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

      13. div-inv 61.1%

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

      14. metadata-eval 61.1%

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

      15. *-rgt-identity 61.1%

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

   6. Applied egg-rr 61.1%

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

   7. Taylor expanded in x around inf 96.0%

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

   8. Taylor expanded in x around 0 7.0%

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

Alternative 7: 51.1% accurate, 11.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

x_m = abs(x)
function code(x_m)
	return 2.0
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 79.9%

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

   1. sub-neg 79.9%

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

   2. +-commutative 79.9%

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

   3. distribute-neg-frac2 79.9%

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

   4. neg-sub0 79.9%

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

   5. associate-+l- 79.9%

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

   6. neg-sub0 79.9%

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

   7. remove-double-neg 79.9%

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

   8. distribute-neg-in 79.9%

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

   9. sub-neg 79.9%

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

   10. distribute-neg-frac2 79.9%

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

   11. sub-neg 79.9%

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

   12. +-commutative 79.9%

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

   13. unsub-neg 79.9%

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

   14. sub-neg 79.9%

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

   15. +-commutative 79.9%

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

   16. unsub-neg 79.9%

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

   17. metadata-eval 79.9%

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

3. Simplified 79.9%

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

5. Taylor expanded in x around 0 54.9%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (x)
  :name "Asymptote A"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 (- x 1.0))))