3frac (problem 3.3.3)

Percentage Accurate: 85.1% → 99.9%

Time: 6.1s

Alternatives: 9

Speedup: 1.0×

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: 85.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. frac-sub 62.5%

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

   2. frac-2neg 62.5%

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

   3. metadata-eval 62.5%

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

   4. frac-add 63.3%

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

3. Applied egg-rr 62.5%

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

   1. *-commutative 62.5%

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

   2. neg-mul-1 62.5%

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

   3. fma-neg 63.3%

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

   4. *-commutative 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in x around 0 99.4%

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

   1. clear-num 99.4%

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

   2. associate-/r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-/r* 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{x + -1}\\ t_1 := \left(\frac{1}{1 + x} - \frac{2}{x}\right) + t_0\\ \mathbf{if}\;t_1 \leq -5000:\\ \;\;\;\;t_0 + \left(x + 2\right) \cdot \frac{-1}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ x -1.0))) (t_1 (+ (- (/ 1.0 (+ 1.0 x)) (/ 2.0 x)) t_0)))
   (if (<= t_1 -5000.0)
     (+ t_0 (* (+ x 2.0) (/ -1.0 (* x (+ 1.0 x)))))
     (if (<= t_1 0.0) (/ 2.0 (pow x 3.0)) t_1))))

C

double code(double x) {
	double t_0 = 1.0 / (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 + ((x + 2.0) * (-1.0 / (x * (1.0 + x))));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / pow(x, 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / (x + (-1.0d0))
    t_1 = ((1.0d0 / (1.0d0 + x)) - (2.0d0 / x)) + t_0
    if (t_1 <= (-5000.0d0)) then
        tmp = t_0 + ((x + 2.0d0) * ((-1.0d0) / (x * (1.0d0 + x))))
    else if (t_1 <= 0.0d0) then
        tmp = 2.0d0 / (x ** 3.0d0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (x + -1.0);
	double t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0;
	double tmp;
	if (t_1 <= -5000.0) {
		tmp = t_0 + ((x + 2.0) * (-1.0 / (x * (1.0 + x))));
	} else if (t_1 <= 0.0) {
		tmp = 2.0 / Math.pow(x, 3.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x):
	t_0 = 1.0 / (x + -1.0)
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0
	tmp = 0
	if t_1 <= -5000.0:
		tmp = t_0 + ((x + 2.0) * (-1.0 / (x * (1.0 + x))))
	elif t_1 <= 0.0:
		tmp = 2.0 / math.pow(x, 3.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(x + -1.0))
	t_1 = Float64(Float64(Float64(1.0 / Float64(1.0 + x)) - Float64(2.0 / x)) + t_0)
	tmp = 0.0
	if (t_1 <= -5000.0)
		tmp = Float64(t_0 + Float64(Float64(x + 2.0) * Float64(-1.0 / Float64(x * Float64(1.0 + x)))));
	elseif (t_1 <= 0.0)
		tmp = Float64(2.0 / (x ^ 3.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 1.0 / (x + -1.0);
	t_1 = ((1.0 / (1.0 + x)) - (2.0 / x)) + t_0;
	tmp = 0.0;
	if (t_1 <= -5000.0)
		tmp = t_0 + ((x + 2.0) * (-1.0 / (x * (1.0 + x))));
	elseif (t_1 <= 0.0)
		tmp = 2.0 / (x ^ 3.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(x + -1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[t$95$1, -5000.0], N[(t$95$0 + N[(N[(x + 2.0), $MachinePrecision] * N[(-1.0 / N[(x * N[(1.0 + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(2.0 / N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < -5e3

   1. Initial program 100.0%

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

      1. frac-sub 99.9%

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

      2. div-inv 100.0%

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

      3. /-rgt-identity 100.0%

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

      4. *-un-lft-identity 100.0%

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

      5. /-rgt-identity 100.0%

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

      6. +-commutative 100.0%

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

      7. *-commutative 100.0%

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

      8. +-commutative 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in x around 0 100.0%

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

   if -5e3 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1))) < 0.0

   1. Initial program 63.0%

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

   2. Taylor expanded in x around inf 98.3%

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

   if 0.0 < (+.f64 (-.f64 (/.f64 1 (+.f64 x 1)) (/.f64 2 x)) (/.f64 1 (-.f64 x 1)))

   1. Initial program 100.0%

      \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq -5000:\\ \;\;\;\;\frac{1}{x + -1} + \left(x + 2\right) \cdot \frac{-1}{x \cdot \left(1 + x\right)}\\ \mathbf{elif}\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1} \leq 0:\\ \;\;\;\;\frac{2}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{1 + x} - \frac{2}{x}\right) + \frac{1}{x + -1}\\ \end{array} \]

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

   1. frac-sub 62.5%

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

   2. frac-2neg 62.5%

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

   3. metadata-eval 62.5%

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

   4. frac-add 63.3%

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

3. Applied egg-rr 62.5%

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

   1. *-commutative 62.5%

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

   2. neg-mul-1 62.5%

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

   3. fma-neg 63.3%

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

   4. *-commutative 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.9%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 4: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x - {x}^{3}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 (- x (pow x 3.0))))

C

double code(double x) {
	return -2.0 / (x - pow(x, 3.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / (x - (x ** 3.0d0))
end function

Java

public static double code(double x) {
	return -2.0 / (x - Math.pow(x, 3.0));
}

Python

def code(x):
	return -2.0 / (x - math.pow(x, 3.0))

Julia

function code(x)
	return Float64(-2.0 / Float64(x - (x ^ 3.0)))
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / (x - (x ^ 3.0));
end

Wolfram

code[x_] := N[(-2.0 / N[(x - N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

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

   1. frac-sub 62.5%

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

   2. frac-2neg 62.5%

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

   3. metadata-eval 62.5%

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

   4. frac-add 63.3%

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

3. Applied egg-rr 62.5%

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

   1. *-commutative 62.5%

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

   2. neg-mul-1 62.5%

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

   3. fma-neg 63.3%

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

   4. *-commutative 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in x around 0 99.4%

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

7. Taylor expanded in x around 0 99.5%

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

   1. +-commutative 99.5%

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

   2. mul-1-neg 99.5%

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

   3. unsub-neg 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

    \[\leadsto \frac{-2}{x - {x}^{3}} \]

Alternative 5: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.2%

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

2. Final simplification 83.2%

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

Alternative 6: 76.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.38) (not (<= x 1.0))) (/ 1.0 (* x x)) (/ -2.0 x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.38) || !(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 <= (-0.38d0)) .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 <= -0.38) || !(x <= 1.0)) {
		tmp = 1.0 / (x * x);
	} else {
		tmp = -2.0 / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.38) 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 <= -0.38) || !(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 <= -0.38) || ~((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, -0.38], 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 < -0.38 or 1 < x

   1. Initial program 63.7%

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

   2. Taylor expanded in x around inf 61.3%

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

   3. Taylor expanded in x around inf 48.2%

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

      1. unpow2 48.2%

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

   5. Simplified 48.2%

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

   if -0.38 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.38 \lor \neg \left(x \leq 1\right):\\ \;\;\;\;\frac{1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{x}\\ \end{array} \]

Alternative 7: 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 83.2%

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

2. Taylor expanded in x around 0 55.3%

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

3. Taylor expanded in x around 0 81.7%

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

4. Final simplification 81.7%

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

Alternative 8: 52.4% 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 83.2%

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

2. Taylor expanded in x around 0 55.9%

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

3. Final simplification 55.9%

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

Alternative 9: 3.3% accurate, 15.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 83.2%

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

2. Taylor expanded in x around 0 55.3%

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

3. Taylor expanded in x around inf 3.4%

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

4. Final simplification 3.4%

   \[\leadsto -1 \]

Developer target: 99.5% 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 2023228 
(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))))