2frac (problem 3.3.1)

Percentage Accurate: 77.6% → 99.9%

Time: 5.2s

Alternatives: 9

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: 77.6% 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 / Float64(x + 1.0)) / x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 77.0%

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

   1. clear-num 77.0%

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

   2. frac-sub 77.5%

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

   3. *-commutative 77.5%

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

   4. *-un-lft-identity 77.5%

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

   5. *-un-lft-identity 77.5%

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

   6. div-inv 77.5%

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

   7. metadata-eval 77.5%

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

   8. *-rgt-identity 77.5%

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

   9. +-commutative 77.5%

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

   10. *-commutative 77.5%

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

   11. div-inv 77.5%

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

   12. metadata-eval 77.5%

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

   13. *-rgt-identity 77.5%

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

   14. +-commutative 77.5%

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

4. Applied egg-rr 77.5%

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

5. Taylor expanded in x around 0 99.4%

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

   1. associate-/r* 99.9%

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

   2. div-inv 99.8%

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

   3. +-commutative 99.8%

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

7. Applied egg-rr 99.8%

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

   1. associate-*l/ 99.9%

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

   2. un-div-inv 99.9%

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

9. Applied egg-rr 99.9%

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

Alternative 2: 98.5% accurate, 0.5× 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 55.0%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. unpow2 95.6%

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

      2. associate-/r* 96.5%

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

      3. *-lft-identity 96.5%

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

      4. associate-*l/ 96.4%

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

      5. metadata-eval 96.4%

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

      6. distribute-neg-frac 96.4%

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

      7. distribute-rgt-neg-out 96.4%

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

      8. unpow-1 96.4%

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

      9. unpow-1 96.4%

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

      10. pow-sqr 96.7%

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

      11. metadata-eval 96.7%

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

   5. Simplified 96.7%

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

      1. neg-mul-1 96.7%

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

      2. metadata-eval 96.7%

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

      3. pow-flip 95.6%

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

      4. pow2 95.6%

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

      5. div-inv 95.6%

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

      6. associate-/r* 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.6%

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

      1. neg-mul-1 98.6%

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

      2. sub-neg 98.6%

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

   5. Simplified 98.6%

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

4. Final simplification 97.5%

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

Alternative 3: 98.3% accurate, 0.6× 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 55.0%

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

   3. Taylor expanded in x around inf 95.6%

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

      1. unpow2 95.6%

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

      2. associate-/r* 96.5%

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

      3. *-lft-identity 96.5%

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

      4. associate-*l/ 96.4%

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

      5. metadata-eval 96.4%

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

      6. distribute-neg-frac 96.4%

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

      7. distribute-rgt-neg-out 96.4%

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

      8. unpow-1 96.4%

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

      9. unpow-1 96.4%

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

      10. pow-sqr 96.7%

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

      11. metadata-eval 96.7%

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

   5. Simplified 96.7%

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

      1. neg-mul-1 96.7%

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

      2. metadata-eval 96.7%

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

      3. pow-flip 95.6%

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

      4. pow2 95.6%

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

      5. div-inv 95.6%

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

      6. associate-/r* 96.5%

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

   7. Applied egg-rr 96.5%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 97.3%

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

Alternative 4: 97.8% accurate, 0.6× 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 55.0%

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

      1. clear-num 55.0%

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

      2. frac-sub 56.0%

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

      3. *-commutative 56.0%

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

      4. *-un-lft-identity 56.0%

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

      5. *-un-lft-identity 56.0%

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

      6. div-inv 56.0%

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

      7. metadata-eval 56.0%

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

      8. *-rgt-identity 56.0%

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

      9. +-commutative 56.0%

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

      10. *-commutative 56.0%

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

      11. div-inv 56.0%

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

      12. metadata-eval 56.0%

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

      13. *-rgt-identity 56.0%

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

      14. +-commutative 56.0%

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

   4. Applied egg-rr 56.0%

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

   5. Taylor expanded in x around 0 98.9%

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

   6. Taylor expanded in x around inf 95.6%

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

   if -1 < x < 0.76000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 96.8%

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

Alternative 5: 75.5% 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 55.0%

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

   3. Taylor expanded in x around inf 50.8%

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

   4. Taylor expanded in x around 0 50.8%

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

   if -1 < x < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 98.2%

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

4. Final simplification 73.9%

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

Alternative 6: 75.3% 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 61.8%

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

   3. Taylor expanded in x around inf 57.8%

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

   4. Taylor expanded in x around 0 57.8%

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

   if -1 < x < 4.50000000000000021e102

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0 86.7%

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

Alternative 7: 99.4% 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 77.0%

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

   1. clear-num 77.0%

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

   2. frac-sub 77.5%

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

   3. *-commutative 77.5%

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

   4. *-un-lft-identity 77.5%

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

   5. *-un-lft-identity 77.5%

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

   6. div-inv 77.5%

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

   7. metadata-eval 77.5%

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

   8. *-rgt-identity 77.5%

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

   9. +-commutative 77.5%

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

   10. *-commutative 77.5%

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

   11. div-inv 77.5%

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

   12. metadata-eval 77.5%

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

   13. *-rgt-identity 77.5%

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

   14. +-commutative 77.5%

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

4. Applied egg-rr 77.5%

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

5. Taylor expanded in x around 0 99.4%

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

6. Final simplification 99.4%

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

Alternative 8: 26.9% 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 77.0%

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

3. Taylor expanded in x around inf 27.1%

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

4. Taylor expanded in x around 0 27.1%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Alternative 9: 3.7% accurate, 9.0× 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 77.0%

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

3. Taylor expanded in x around 0 49.3%

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

   1. frac-2neg 49.3%

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

   2. div-inv 49.3%

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

   3. sub-neg 49.3%

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

   4. metadata-eval 49.3%

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

   5. distribute-neg-in 49.3%

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

   6. metadata-eval 49.3%

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

   7. metadata-eval 49.3%

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

   8. frac-2neg 49.3%

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

   9. add-sqr-sqrt 25.4%

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

   10. sqrt-unprod 37.4%

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

   11. frac-times 38.1%

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

   12. metadata-eval 38.1%

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

   13. metadata-eval 38.1%

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

   14. frac-times 37.4%

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

   15. sqrt-prod 1.2%

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

   16. add-sqr-sqrt 2.6%

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

5. Applied egg-rr 2.6%

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

   1. +-commutative 2.6%

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

   2. sub-neg 2.6%

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

   3. associate-*r/ 2.6%

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

   4. *-rgt-identity 2.6%

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

   5. div-sub 2.6%

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

   6. *-inverses 2.6%

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

   7. sub-neg 2.6%

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

   8. metadata-eval 2.6%

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

   9. +-commutative 2.6%

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

7. Simplified 2.6%

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

8. Taylor expanded in x around inf 3.7%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Developer Target 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]

Developer Target 2: 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]

Reproduce

herbie shell --seed 2024177 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ -1 x) (+ x 1)))

  :alt
  (! :herbie-platform default (/ 1 (* x (- -1 x))))

  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))