2frac (problem 3.3.1)

Percentage Accurate: 78.0% → 99.9%

Time: 6.4s

Alternatives: 6

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 78.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. frac-sub 78.4%

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

   2. *-commutative 78.4%

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

   3. associate-/r* 78.4%

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

   4. *-un-lft-identity 78.4%

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

   5. *-rgt-identity 78.4%

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

   6. sub-neg 78.4%

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

   7. neg-sub0 78.4%

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

   8. +-commutative 78.4%

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

   9. associate--r+ 78.4%

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

   10. metadata-eval 78.4%

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

   11. +-commutative 78.4%

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

3. Applied egg-rr 78.4%

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

4. Taylor expanded in x around 0 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 76.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 3.9e+61) (+ (- 1.0 x) (/ -1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 3.9d+61) then
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 3.9e+61) {
		tmp = (1.0 - x) + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 3.9e+61:
		tmp = (1.0 - x) + (-1.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 3.9e+61)
		tmp = (1.0 - x) + (-1.0 / x);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 3.9e+61], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.89999999999999987e61 < x

   1. Initial program 55.8%

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

      1. frac-sub 56.4%

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

      2. div-inv 56.4%

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

      3. metadata-eval 56.4%

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

      4. frac-times 56.4%

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

      5. *-commutative 56.4%

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

      6. un-div-inv 56.4%

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

      7. associate-/l/ 56.4%

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

      8. +-commutative 56.4%

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

      9. *-un-lft-identity 56.4%

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

      10. *-rgt-identity 56.4%

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

      11. sub-neg 56.4%

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

      12. neg-sub0 56.4%

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

      13. +-commutative 56.4%

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

      14. associate--r+ 56.4%

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

      15. metadata-eval 56.4%

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

   3. Applied egg-rr 56.4%

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

      1. associate-*l/ 56.4%

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

      2. *-lft-identity 56.4%

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

      3. associate-+r- 56.4%

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

      4. +-commutative 56.4%

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

      5. associate--l+ 97.5%

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

      6. +-commutative 97.5%

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

   5. Simplified 97.5%

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

   7. Applied egg-rr 52.0%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 3.89999999999999987e61

   1. Initial program 93.9%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 91.2%

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

      1. mul-1-neg 91.2%

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

      2. sub-neg 91.2%

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

   4. Simplified 91.2%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{+61}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3: 76.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 1.0) (+ 1.0 (/ -1.0 x)) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

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

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 1.0) {
		tmp = 1.0 + (-1.0 / x);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 1.0:
		tmp = 1.0 + (-1.0 / x)
	else:
		tmp = 0.0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 1.0], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 52.8%

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

      1. frac-sub 54.3%

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

      2. div-inv 54.3%

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

      3. metadata-eval 54.3%

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

      4. frac-times 54.3%

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

      5. *-commutative 54.3%

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

      6. un-div-inv 54.3%

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

      7. associate-/l/ 54.3%

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

      8. +-commutative 54.3%

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

      9. *-un-lft-identity 54.3%

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

      10. *-rgt-identity 54.3%

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

      11. sub-neg 54.3%

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

      12. neg-sub0 54.3%

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

      13. +-commutative 54.3%

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

      14. associate--r+ 54.3%

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

      15. metadata-eval 54.3%

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

   3. Applied egg-rr 54.3%

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

      1. associate-*l/ 54.3%

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

      2. *-lft-identity 54.3%

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

      3. associate-+r- 54.3%

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

      4. +-commutative 54.3%

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

      5. associate--l+ 97.7%

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

      6. +-commutative 97.7%

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

   5. Simplified 97.7%

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

   7. Applied egg-rr 47.2%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 1

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 98.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;1 + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0) 0.0 (if (<= x 4.5e+102) (/ -1.0 x) 0.0)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-1.0d0)) then
        tmp = 0.0d0
    else if (x <= 4.5d+102) then
        tmp = (-1.0d0) / x
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = 0.0;
	} else if (x <= 4.5e+102) {
		tmp = -1.0 / x;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.0:
		tmp = 0.0
	elif x <= 4.5e+102:
		tmp = -1.0 / x
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = Float64(-1.0 / x);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = 0.0;
	elseif (x <= 4.5e+102)
		tmp = -1.0 / x;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], 0.0, If[LessEqual[x, 4.5e+102], N[(-1.0 / x), $MachinePrecision], 0.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 4.50000000000000021e102 < x

   1. Initial program 59.8%

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

      1. frac-sub 60.5%

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

      2. div-inv 60.5%

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

      3. metadata-eval 60.5%

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

      4. frac-times 60.4%

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

      5. *-commutative 60.4%

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

      6. un-div-inv 60.4%

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

      7. associate-/l/ 60.5%

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

      8. +-commutative 60.5%

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

      9. *-un-lft-identity 60.5%

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

      10. *-rgt-identity 60.5%

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

      11. sub-neg 60.5%

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

      12. neg-sub0 60.5%

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

      13. +-commutative 60.5%

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

      14. associate--r+ 60.5%

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

      15. metadata-eval 60.5%

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

   3. Applied egg-rr 60.5%

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

      1. associate-*l/ 60.5%

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

      2. *-lft-identity 60.5%

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

      3. associate-+r- 60.5%

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

      4. +-commutative 60.5%

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

      5. associate--l+ 97.4%

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

      6. +-commutative 97.4%

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

   5. Simplified 97.4%

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

   7. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{0} \]

   if -1 < x < 4.50000000000000021e102

   1. Initial program 89.3%

      \[\frac{1}{x + 1} - \frac{1}{x} \]

   2. Taylor expanded in x around 0 85.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;0\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.7%

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

   1. frac-sub 78.4%

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

   2. *-commutative 78.4%

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

   3. associate-/r* 78.4%

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

   4. *-un-lft-identity 78.4%

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

   5. *-rgt-identity 78.4%

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

   6. sub-neg 78.4%

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

   7. neg-sub0 78.4%

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

   8. +-commutative 78.4%

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

   9. associate--r+ 78.4%

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

   10. metadata-eval 78.4%

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

   11. +-commutative 78.4%

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

3. Applied egg-rr 78.4%

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

4. Taylor expanded in x around 0 99.9%

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

   1. frac-2neg 99.9%

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

   2. distribute-frac-neg 99.9%

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

   3. neg-sub0 99.9%

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

   4. associate--r+ 99.9%

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

   5. metadata-eval 99.9%

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

6. Applied egg-rr 99.9%

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

   1. neg-mul-1 99.9%

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

   2. associate-/l/ 98.9%

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

   3. associate-*r/ 98.9%

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

   4. metadata-eval 98.9%

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

   5. *-commutative 98.9%

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

8. Simplified 98.9%

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

9. Final simplification 98.9%

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

Alternative 6: 26.5% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 77.7%

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

   1. frac-sub 78.4%

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

   2. div-inv 78.4%

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

   3. metadata-eval 78.4%

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

   4. frac-times 78.4%

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

   5. *-commutative 78.4%

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

   6. un-div-inv 78.4%

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

   7. associate-/l/ 78.4%

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

   8. +-commutative 78.4%

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

   9. *-un-lft-identity 78.4%

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

   10. *-rgt-identity 78.4%

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

   11. sub-neg 78.4%

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

   12. neg-sub0 78.4%

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

   13. +-commutative 78.4%

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

   14. associate--r+ 78.4%

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

   15. metadata-eval 78.4%

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

3. Applied egg-rr 78.4%

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

   1. associate-*l/ 78.4%

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

   2. *-lft-identity 78.4%

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

   3. associate-+r- 78.4%

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

   4. +-commutative 78.4%

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

   5. associate--l+ 98.9%

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

   6. +-commutative 98.9%

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

5. Simplified 98.9%

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

7. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{0} \]

8. Final simplification 23.5%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))