2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 5.0s

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: 77.3% 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 74.5%

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

   1. frac-sub 75.2%

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

   2. *-rgt-identity 75.2%

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

   3. metadata-eval 75.2%

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

   4. div-inv 75.2%

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

   5. associate-/r* 75.2%

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

   6. *-un-lft-identity 75.2%

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

   7. *-rgt-identity 75.2%

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

   8. +-commutative 75.2%

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

   9. div-inv 75.2%

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

   10. metadata-eval 75.2%

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

   11. *-rgt-identity 75.2%

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

   12. +-commutative 75.2%

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

3. Applied egg-rr 75.2%

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

   1. frac-2neg 75.2%

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

   2. div-inv 75.2%

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

   3. +-commutative 75.2%

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

   4. +-commutative 75.2%

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

   5. distribute-neg-in 75.2%

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

   6. neg-mul-1 75.2%

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

   7. metadata-eval 75.2%

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

   8. fma-def 75.2%

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

5. Applied egg-rr 75.2%

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

   1. associate-*r/ 75.2%

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

   2. *-rgt-identity 75.2%

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

   3. distribute-neg-frac 75.2%

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

   4. neg-mul-1 75.2%

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

   5. associate--r+ 99.9%

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

   6. +-inverses 99.9%

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

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

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

   9. associate-*r/ 99.9%

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

   10. associate-*r/ 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-*r/ 99.9%

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

   13. metadata-eval 99.9%

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

   14. fma-udef 99.9%

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

   15. neg-mul-1 99.9%

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

   16. +-commutative 99.9%

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

   17. unsub-neg 99.9%

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

7. Simplified 99.9%

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

   1. add-log-exp 33.0%

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

   2. *-un-lft-identity 33.0%

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

   3. log-prod 33.0%

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

   4. metadata-eval 33.0%

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

   5. add-log-exp 99.9%

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

   6. associate-/l/ 99.8%

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

   7. associate-/r* 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 98.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.0)) {
		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 <= 1.0d0))) 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 <= 1.0)) {
		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 <= 1.0):
		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 <= 1.0))
		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 <= 1.0)))
		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, 1.0]], $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 1 < x

   1. Initial program 54.7%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. metadata-eval 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. unpow2 98.1%

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

      4. associate-/r* 98.3%

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

      5. metadata-eval 98.3%

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

      6. associate-*l/ 98.3%

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

      7. associate-*r/ 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

      9. metadata-eval 98.1%

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

      10. distribute-neg-frac 98.1%

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

      11. distribute-neg-frac 98.1%

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

      12. metadata-eval 98.1%

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

      13. distribute-lft-neg-in 98.1%

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

      14. unpow-1 98.1%

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

      15. unpow-1 98.1%

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

      16. pow-sqr 98.5%

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

      17. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

      1. metadata-eval 98.5%

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

      2. pow-flip 98.1%

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

      3. pow2 98.1%

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

      4. associate-/r* 98.3%

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

   6. Applied egg-rr 98.3%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 99.1%

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

      1. neg-mul-1 99.1%

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

      2. sub-neg 99.1%

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

   4. Simplified 99.1%

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

4. Final simplification 98.7%

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

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ -1.0 (* x x)) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		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 <= 0.76d0))) 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 <= 0.76)) {
		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 <= 0.76):
		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 <= 0.76))
		tmp = Float64(-1.0 / Float64(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 <= 0.76)))
		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, 0.76]], $MachinePrecision]], N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.76000000000000001 < x

   1. Initial program 54.7%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. unpow2 98.1%

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

   4. Simplified 98.1%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{-1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

Alternative 4: 98.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.76))) (/ (/ -1.0 x) x) (+ 1.0 (/ -1.0 x))))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.76)) {
		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 <= 0.76d0))) 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 <= 0.76)) {
		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 <= 0.76):
		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 <= 0.76))
		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 <= 0.76)))
		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, 0.76]], $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 0.76000000000000001 < x

   1. Initial program 54.7%

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

   2. Taylor expanded in x around inf 98.1%

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

      1. metadata-eval 98.1%

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

      2. distribute-neg-frac 98.1%

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

      3. unpow2 98.1%

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

      4. associate-/r* 98.3%

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

      5. metadata-eval 98.3%

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

      6. associate-*l/ 98.3%

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

      7. associate-*r/ 98.1%

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

      8. distribute-rgt-neg-in 98.1%

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

      9. metadata-eval 98.1%

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

      10. distribute-neg-frac 98.1%

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

      11. distribute-neg-frac 98.1%

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

      12. metadata-eval 98.1%

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

      13. distribute-lft-neg-in 98.1%

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

      14. unpow-1 98.1%

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

      15. unpow-1 98.1%

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

      16. pow-sqr 98.5%

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

      17. metadata-eval 98.5%

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

   4. Simplified 98.5%

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

      1. metadata-eval 98.5%

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

      2. pow-flip 98.1%

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

      3. pow2 98.1%

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

      4. associate-/r* 98.3%

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

   6. Applied egg-rr 98.3%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.76\right):\\ \;\;\;\;\frac{\frac{-1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-1}{x}\\ \end{array} \]

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 74.5%

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

   1. sub-neg 74.5%

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

   2. +-commutative 74.5%

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

   3. distribute-neg-frac 74.5%

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

   4. metadata-eval 74.5%

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

3. Applied egg-rr 74.5%

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

   1. +-commutative 74.5%

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

   2. *-rgt-identity 74.5%

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

   3. associate-*l/ 74.5%

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

   4. associate-*r/ 74.5%

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

   5. neg-mul-1 74.5%

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

   6. neg-sub0 74.5%

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

   7. associate-+l- 74.5%

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

   8. sub0-neg 74.5%

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

   9. sub-neg 74.5%

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

   10. distribute-neg-out 74.5%

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

   11. distribute-neg-frac 74.5%

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

   12. metadata-eval 74.5%

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

   13. remove-double-neg 74.5%

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

   14. remove-double-neg 74.5%

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

   15. +-commutative 74.5%

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

   16. distribute-neg-in 74.5%

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

   17. neg-mul-1 74.5%

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

   18. metadata-eval 74.5%

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

   19. fma-udef 74.5%

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

   20. neg-mul-1 74.5%

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

   21. associate-/r* 74.5%

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

   22. metadata-eval 74.5%

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

   23. metadata-eval 74.5%

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

   24. associate-*r/ 74.5%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 6: 99.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.5%

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

   1. frac-sub 75.2%

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

   2. *-rgt-identity 75.2%

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

   3. metadata-eval 75.2%

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

   4. div-inv 75.2%

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

   5. associate-/r* 75.2%

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

   6. *-un-lft-identity 75.2%

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

   7. *-rgt-identity 75.2%

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

   8. +-commutative 75.2%

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

   9. div-inv 75.2%

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

   10. metadata-eval 75.2%

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

   11. *-rgt-identity 75.2%

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

   12. +-commutative 75.2%

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

3. Applied egg-rr 75.2%

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

   1. frac-2neg 75.2%

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

   2. div-inv 75.2%

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

   3. +-commutative 75.2%

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

   4. +-commutative 75.2%

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

   5. distribute-neg-in 75.2%

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

   6. neg-mul-1 75.2%

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

   7. metadata-eval 75.2%

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

   8. fma-def 75.2%

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

5. Applied egg-rr 75.2%

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

   1. associate-*r/ 75.2%

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

   2. *-rgt-identity 75.2%

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

   3. distribute-neg-frac 75.2%

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

   4. neg-mul-1 75.2%

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

   5. associate--r+ 99.9%

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

   6. +-inverses 99.9%

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

   7. metadata-eval 99.9%

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

   8. metadata-eval 99.9%

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

   9. associate-*r/ 99.9%

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

   10. associate-*r/ 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-*r/ 99.9%

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

   13. metadata-eval 99.9%

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

   14. fma-udef 99.9%

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

   15. neg-mul-1 99.9%

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

   16. +-commutative 99.9%

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

   17. unsub-neg 99.9%

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

7. Simplified 99.9%

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

8. Final simplification 99.9%

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

Alternative 7: 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 74.5%

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

2. Taylor expanded in x around 0 46.0%

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

3. Final simplification 46.0%

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

Reproduce

herbie shell --seed 2023172 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))