3frac (problem 3.3.3)

Percentage Accurate: 84.5% → 99.9%

Time: 7.7s

Alternatives: 7

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: 84.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. frac-sub 59.0%

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

   2. frac-add 59.6%

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

   3. fma-def 59.0%

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

   4. /-rgt-identity 59.0%

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

   5. *-un-lft-identity 59.0%

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

   6. /-rgt-identity 59.0%

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

   7. +-commutative 59.0%

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

   8. sub-neg 59.0%

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

   9. metadata-eval 59.0%

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

   10. *-commutative 59.0%

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

   11. +-commutative 59.0%

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

   12. *-commutative 59.0%

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

   13. +-commutative 59.0%

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

   14. sub-neg 59.0%

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

   15. metadata-eval 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in x around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-/r* 99.9%

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

   4. div-inv 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. pow2 99.9%

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

   8. sub-neg 99.9%

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

   9. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-*r/ 99.9%

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

   2. *-rgt-identity 99.9%

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

   3. +-commutative 99.9%

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

   4. unpow2 99.9%

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

   5. fma-def 99.9%

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

8. Simplified 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 x) (/ 2.0 x))
   (if (<= x 0.65)
     (- (* x -2.0) (/ 2.0 x))
     (+ (/ 1.0 (+ x -1.0)) (/ -1.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = (2.0d0 / x) - (2.0d0 / x)
    else if (x <= 0.65d0) then
        tmp = (x * (-2.0d0)) - (2.0d0 / x)
    else
        tmp = (1.0d0 / (x + (-1.0d0))) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / x) - (2.0 / x);
	} else if (x <= 0.65) {
		tmp = (x * -2.0) - (2.0 / x);
	} else {
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = (2.0 / x) - (2.0 / x)
	elif x <= 0.65:
		tmp = (x * -2.0) - (2.0 / x)
	else:
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / x) - Float64(2.0 / x));
	elseif (x <= 0.65)
		tmp = Float64(Float64(x * -2.0) - Float64(2.0 / x));
	else
		tmp = Float64(Float64(1.0 / Float64(x + -1.0)) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = (2.0 / x) - (2.0 / x);
	elseif (x <= 0.65)
		tmp = (x * -2.0) - (2.0 / x);
	else
		tmp = (1.0 / (x + -1.0)) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / x), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.65], N[(N[(x * -2.0), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 63.0%

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

      1. +-commutative 63.0%

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

      2. associate-+r- 62.8%

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

      3. sub-neg 62.8%

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

      4. metadata-eval 62.8%

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

      5. +-commutative 62.8%

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

   3. Applied egg-rr 62.8%

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

   4. Taylor expanded in x around inf 60.5%

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

   if -1 < x < 0.650000000000000022

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   4. Simplified 99.1%

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

   if 0.650000000000000022 < x

   1. Initial program 68.5%

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

   2. Taylor expanded in x around inf 65.9%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{x} - \frac{2}{x}\\ \mathbf{elif}\;x \leq 0.65:\\ \;\;\;\;x \cdot -2 - \frac{2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x + -1} + \frac{-1}{x}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. frac-sub 59.0%

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

   2. frac-add 59.6%

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

   3. fma-def 59.0%

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

   4. /-rgt-identity 59.0%

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

   5. *-un-lft-identity 59.0%

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

   6. /-rgt-identity 59.0%

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

   7. +-commutative 59.0%

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

   8. sub-neg 59.0%

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

   9. metadata-eval 59.0%

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

   10. *-commutative 59.0%

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

   11. +-commutative 59.0%

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

   12. *-commutative 59.0%

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

   13. +-commutative 59.0%

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

   14. sub-neg 59.0%

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

   15. metadata-eval 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in x around 0 99.6%

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

   1. metadata-eval 99.6%

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

   2. sub-neg 99.6%

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

   3. associate-/r* 99.9%

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

   4. div-inv 99.8%

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

   5. distribute-lft-in 99.9%

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

   6. *-rgt-identity 99.9%

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

   7. pow2 99.9%

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

   8. sub-neg 99.9%

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

   9. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. associate-*l/ 99.9%

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

8. Simplified 99.9%

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

   1. unpow2 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 4: 83.5% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 65.9%

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

      1. +-commutative 65.9%

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

      2. associate-+r- 65.8%

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

      3. sub-neg 65.8%

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

      4. metadata-eval 65.8%

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

      5. +-commutative 65.8%

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

   3. Applied egg-rr 65.8%

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

   4. Taylor expanded in x around inf 63.0%

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

   if -1 < x < 1

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. associate-*r/ 99.1%

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

      2. metadata-eval 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 81.2%

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

Alternative 5: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.1%

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

   1. frac-sub 59.0%

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

   2. frac-add 59.6%

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

   3. fma-def 59.0%

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

   4. /-rgt-identity 59.0%

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

   5. *-un-lft-identity 59.0%

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

   6. /-rgt-identity 59.0%

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

   7. +-commutative 59.0%

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

   8. sub-neg 59.0%

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

   9. metadata-eval 59.0%

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

   10. *-commutative 59.0%

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

   11. +-commutative 59.0%

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

   12. *-commutative 59.0%

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

   13. +-commutative 59.0%

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

   14. sub-neg 59.0%

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

   15. metadata-eval 59.0%

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

3. Applied egg-rr 59.0%

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

4. Taylor expanded in x around 0 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 52.8% 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 83.1%

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

2. Taylor expanded in x around 0 52.0%

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

3. Final simplification 52.0%

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

Alternative 7: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 83.1%

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

2. Taylor expanded in x around 0 51.2%

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

3. Taylor expanded in x around inf 11.1%

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

4. Taylor expanded in x around inf 3.3%

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

5. Final simplification 3.3%

   \[\leadsto -1 \]

Developer target: 99.6% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023307 
(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))))