2frac (problem 3.3.1)

Percentage Accurate: 77.5% → 99.9%

Time: 4.9s

Alternatives: 7

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: 77.5% 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

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

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(Float64(1.0 / x) / Float64(-1.0 - x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. clear-num 74.3%

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

   2. frac-sub 75.8%

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

   3. *-un-lft-identity 75.8%

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

   4. div-inv 75.8%

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

   5. metadata-eval 75.8%

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

   6. *-rgt-identity 75.8%

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

   7. *-rgt-identity 75.8%

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

   8. +-commutative 75.8%

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

   9. *-commutative 75.8%

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

   10. div-inv 75.8%

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

   11. metadata-eval 75.8%

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

   12. *-rgt-identity 75.8%

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

   13. +-commutative 75.8%

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

4. Applied egg-rr 75.8%

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

   1. associate-/r* 75.8%

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

   2. div-inv 75.8%

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

   3. +-commutative 75.8%

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

   4. associate--r+ 99.7%

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

   5. +-commutative 99.7%

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

6. Applied egg-rr 99.7%

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

   1. un-div-inv 99.8%

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

   2. +-inverses 99.8%

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

   3. metadata-eval 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto \frac{\frac{1}{x}}{-1 - x} \]
10. Add Preprocessing

Alternative 2: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 3.8e+61)))
   (+ (/ 1.0 x) (/ -1.0 x))
   (+ (- 1.0 x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.8e+61)) {
		tmp = (1.0 / x) + (-1.0 / x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 3.8d+61))) then
        tmp = (1.0d0 / x) + ((-1.0d0) / x)
    else
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 3.8e+61)) {
		tmp = (1.0 / x) + (-1.0 / x);
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 3.8e+61):
		tmp = (1.0 / x) + (-1.0 / x)
	else:
		tmp = (1.0 - x) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 3.8e+61))
		tmp = Float64(Float64(1.0 / x) + Float64(-1.0 / x));
	else
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 3.8e+61)))
		tmp = (1.0 / x) + (-1.0 / x);
	else
		tmp = (1.0 - x) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 3.8e+61]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 3.79999999999999995e61 < x

   1. Initial program 54.8%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 53.6%

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

   if -1 < x < 3.79999999999999995e61

   1. Initial program 91.8%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 89.0%

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

      1. neg-mul-1 89.0%

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

      2. unsub-neg 89.0%

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

   5. Simplified 89.0%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 3.8 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{1}{x} + \frac{-1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 9.5 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 9.5e+87)))
   (/ (/ 1.0 x) x)
   (+ (- 1.0 x) (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 9.5e+87)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 9.5d+87))) then
        tmp = (1.0d0 / x) / x
    else
        tmp = (1.0d0 - x) + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 9.5e+87)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = (1.0 - x) + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 9.5e+87):
		tmp = (1.0 / x) / x
	else:
		tmp = (1.0 - x) + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 9.5e+87))
		tmp = Float64(Float64(1.0 / x) / x);
	else
		tmp = Float64(Float64(1.0 - x) + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 9.5e+87)))
		tmp = (1.0 / x) / x;
	else
		tmp = (1.0 - x) + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 9.5e+87]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 9.4999999999999992e87 < x

   1. Initial program 58.9%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.8%

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

      1. div-inv 97.8%

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

      2. pow-flip 99.2%

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

      3. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. neg-mul-1 99.2%

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

   7. Simplified 99.2%

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

      1. metadata-eval 99.2%

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

      2. pow-sqr 98.8%

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

      3. pow-prod-down 97.8%

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

      4. inv-pow 97.8%

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

      5. metadata-eval 97.8%

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

      6. frac-times 98.8%

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

      7. associate-*l/ 99.0%

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

      8. add-sqr-sqrt 61.4%

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

      9. sqrt-unprod 85.2%

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

      10. frac-times 85.3%

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

      11. metadata-eval 85.3%

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

      12. inv-pow 85.3%

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

      13. pow-prod-down 85.2%

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

      14. pow-sqr 85.3%

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

      15. metadata-eval 85.3%

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

      16. sqrt-pow1 56.9%

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

      17. metadata-eval 56.9%

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

      18. inv-pow 56.9%

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

      19. div-inv 56.9%

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

   9. Applied egg-rr 56.9%

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

   if -1 < x < 9.4999999999999992e87

   1. Initial program 86.4%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 83.7%

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

      1. neg-mul-1 83.7%

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

      2. unsub-neg 83.7%

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

   5. Simplified 83.7%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 9.5 \cdot 10^{+87}\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - x\right) + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.6))) (/ (/ 1.0 x) x) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.6)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = 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)) .or. (.not. (x <= 1.6d0))) then
        tmp = (1.0d0 / x) / x
    else
        tmp = 1.0d0 + ((-1.0d0) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.6)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = 1.0 + (-1.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.6):
		tmp = (1.0 / x) / x
	else:
		tmp = 1.0 + (-1.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.6))
		tmp = Float64(Float64(1.0 / x) / x);
	else
		tmp = Float64(1.0 + Float64(-1.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.6)))
		tmp = (1.0 / x) / x;
	else
		tmp = 1.0 + (-1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.6]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.6000000000000001 < x

   1. Initial program 51.7%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 96.6%

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

      1. div-inv 96.6%

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

      2. pow-flip 97.8%

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

      3. metadata-eval 97.8%

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

   5. Applied egg-rr 97.8%

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

      1. neg-mul-1 97.8%

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

   7. Simplified 97.8%

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

      1. metadata-eval 97.8%

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

      2. pow-sqr 97.4%

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

      3. pow-prod-down 96.6%

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

      4. inv-pow 96.6%

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

      5. metadata-eval 96.6%

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

      6. frac-times 97.4%

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

      7. associate-*l/ 97.5%

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

      8. add-sqr-sqrt 50.6%

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

      9. sqrt-unprod 70.6%

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

      10. frac-times 70.7%

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

      11. metadata-eval 70.7%

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

      12. inv-pow 70.7%

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

      13. pow-prod-down 70.6%

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

      14. pow-sqr 70.7%

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

      15. metadata-eval 70.7%

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

      16. sqrt-pow1 47.3%

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

      17. metadata-eval 47.3%

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

      18. inv-pow 47.3%

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

      19. div-inv 47.3%

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

   9. Applied egg-rr 47.3%

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

   if -1 < x < 1.6000000000000001

   1. Initial program 100.0%

      \[\frac{1}{x + 1} - \frac{1}{x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.7%

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

      1. div-sub 98.7%

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

      2. *-inverses 98.7%

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

   5. Simplified 98.7%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.6\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.3× speedup

Math

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

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(-1.0 / Float64(x * Float64(x + 1.0)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

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

   1. sub-neg 74.3%

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

   2. +-commutative 74.3%

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

   3. distribute-neg-frac 74.3%

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

   4. metadata-eval 74.3%

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

4. Applied egg-rr 74.3%

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

   1. metadata-eval 74.3%

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

   2. distribute-neg-frac 74.3%

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

   3. unsub-neg 74.3%

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

   4. *-inverses 74.3%

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

   5. associate-/r* 51.9%

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

   6. *-commutative 51.9%

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

   7. associate-/r* 74.3%

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

   8. div-sub 74.2%

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

   9. *-inverses 74.2%

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

   10. div-sub 75.8%

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

   11. associate-/l/ 75.8%

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

   12. +-commutative 75.8%

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

   13. associate--r+ 99.3%

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

   14. +-inverses 99.3%

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

   15. metadata-eval 99.3%

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

   16. distribute-lft-in 99.4%

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

   17. *-rgt-identity 99.4%

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

   18. unpow2 99.4%

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

6. Simplified 99.4%

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

   1. unpow2 99.4%

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

   2. distribute-rgt1-in 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 6: 52.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0 / x

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.3%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 48.6%

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

Alternative 7: 3.2% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

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

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 74.3%

   \[\frac{1}{x + 1} - \frac{1}{x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 47.9%

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

   1. neg-mul-1 47.9%

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

   2. unsub-neg 47.9%

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

5. Simplified 47.9%

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

6. Taylor expanded in x around inf 3.4%

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

   1. neg-mul-1 3.4%

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

8. Simplified 3.4%

   \[\leadsto \color{blue}{-x} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))