3frac (problem 3.3.3)

Percentage Accurate: 84.8% → 99.7%

Time: 6.8s

Alternatives: 6

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 84.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Simplified 84.5%

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

   1. frac-sub 63.3%

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

   2. frac-sub 63.7%

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

   3. *-un-lft-identity 63.7%

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

   4. distribute-rgt-in 63.7%

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

   5. neg-mul-1 63.7%

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

   6. sub-neg 63.7%

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

   7. *-rgt-identity 63.7%

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

   8. distribute-rgt-in 63.7%

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

   9. metadata-eval 63.7%

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

   10. metadata-eval 63.7%

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

   11. fma-def 63.7%

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

   12. metadata-eval 63.7%

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

   13. distribute-rgt-in 63.7%

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

   14. neg-mul-1 63.7%

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

   15. sub-neg 63.7%

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

5. Applied egg-rr 63.7%

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

   1. +-commutative 63.7%

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

   2. remove-double-neg 63.7%

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

   3. metadata-eval 63.7%

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

   4. distribute-neg-in 63.7%

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

   5. neg-mul-1 63.7%

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

   6. *-commutative 63.7%

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

   7. fma-udef 63.7%

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

   8. distribute-lft-neg-in 63.7%

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

   9. distribute-lft-neg-in 63.7%

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

   10. fma-udef 63.7%

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

   11. *-commutative 63.7%

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

   12. neg-mul-1 63.7%

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

   13. distribute-neg-in 63.7%

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

   14. remove-double-neg 63.7%

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

   15. metadata-eval 63.7%

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

   16. +-commutative 63.7%

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

7. Simplified 63.7%

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

8. Taylor expanded in x around 0 99.9%

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

   1. frac-2neg 99.9%

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

   2. div-inv 99.9%

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

   3. metadata-eval 99.9%

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

   4. +-commutative 99.9%

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

   5. *-commutative 99.9%

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

   6. distribute-rgt-neg-in 99.9%

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

   7. fma-neg 99.9%

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

   8. distribute-neg-in 99.9%

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

   9. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. metadata-eval 99.9%

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

    3. fma-def 99.9%

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

    4. neg-mul-1 99.9%

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

    5. distribute-rgt-in 99.9%

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

    6. *-commutative 99.9%

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

    7. associate-*l* 99.9%

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

    8. unsub-neg 99.9%

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

12. Simplified 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.85) (not (<= x 1.0)))
   (/ 2.0 (* (+ x 1.0) (* x x)))
   (- (* -2.0 x) (/ 2.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -0.85) || !(x <= 1.0)) {
		tmp = 2.0 / ((x + 1.0) * (x * x));
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.85d0)) .or. (.not. (x <= 1.0d0))) then
        tmp = 2.0d0 / ((x + 1.0d0) * (x * x))
    else
        tmp = ((-2.0d0) * x) - (2.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.85) || !(x <= 1.0)) {
		tmp = 2.0 / ((x + 1.0) * (x * x));
	} else {
		tmp = (-2.0 * x) - (2.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.85) or not (x <= 1.0):
		tmp = 2.0 / ((x + 1.0) * (x * x))
	else:
		tmp = (-2.0 * x) - (2.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.85) || !(x <= 1.0))
		tmp = Float64(2.0 / Float64(Float64(x + 1.0) * Float64(x * x)));
	else
		tmp = Float64(Float64(-2.0 * x) - Float64(2.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.85) || ~((x <= 1.0)))
		tmp = 2.0 / ((x + 1.0) * (x * x));
	else
		tmp = (-2.0 * x) - (2.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.85], N[Not[LessEqual[x, 1.0]], $MachinePrecision]], N[(2.0 / N[(N[(x + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.849999999999999978 or 1 < x

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

      1. frac-sub 18.3%

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

      2. frac-sub 19.2%

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

      3. *-un-lft-identity 19.2%

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

      4. distribute-rgt-in 19.3%

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

      5. neg-mul-1 19.3%

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

      6. sub-neg 19.3%

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

      7. *-rgt-identity 19.3%

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

      8. distribute-rgt-in 19.3%

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

      9. metadata-eval 19.3%

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

      10. metadata-eval 19.3%

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

      11. fma-def 19.3%

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

      12. metadata-eval 19.3%

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

      13. distribute-rgt-in 19.3%

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

      14. neg-mul-1 19.3%

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

      15. sub-neg 19.3%

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

   5. Applied egg-rr 19.3%

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

      1. +-commutative 19.3%

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

      2. remove-double-neg 19.3%

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

      3. metadata-eval 19.3%

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

      4. distribute-neg-in 19.3%

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

      5. neg-mul-1 19.3%

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

      6. *-commutative 19.3%

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

      7. fma-udef 19.3%

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

      8. distribute-lft-neg-in 19.3%

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

      9. distribute-lft-neg-in 19.3%

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

      10. fma-udef 19.3%

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

      11. *-commutative 19.3%

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

      12. neg-mul-1 19.3%

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

      13. distribute-neg-in 19.3%

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

      14. remove-double-neg 19.3%

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

      15. metadata-eval 19.3%

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

      16. +-commutative 19.3%

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

   7. Simplified 19.3%

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

   8. Taylor expanded in x around 0 99.7%

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

   9. Taylor expanded in x around inf 97.3%

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

       1. unpow2 97.3%

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

   11. Simplified 97.3%

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

   if -0.849999999999999978 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.7%

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

      1. associate-*r/ 99.7%

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

      2. metadata-eval 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.85 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{2}{\left(x + 1\right) \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot x - \frac{2}{x}\\ \end{array} \]

Alternative 3: 76.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		tmp = -1.0 / (x * x);
	} 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)) .or. (.not. (x <= 1.0d0))) then
        tmp = (-1.0d0) / (x * x)
    else
        tmp = (-2.0d0) / x
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in x around inf 63.9%

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

   5. Taylor expanded in x around inf 50.1%

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

      1. unpow2 50.1%

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

   7. Simplified 50.1%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 99.1%

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

4. Final simplification 77.1%

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

Alternative 4: 83.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in x around 0 56.2%

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

5. Taylor expanded in x around 0 83.1%

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

6. Final simplification 83.1%

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

Alternative 5: 51.9% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -2.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in x around 0 56.7%

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

5. Final simplification 56.7%

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

Alternative 6: 3.3% accurate, 15.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 84.5%

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

3. Simplified 84.5%

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

4. Taylor expanded in x around 0 56.2%

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

5. Taylor expanded in x around inf 11.9%

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

6. Taylor expanded in x around inf 3.4%

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

7. Final simplification 3.4%

   \[\leadsto 1 \]

Developer target: 99.7% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023273 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

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

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