2frac (problem 3.3.1)

Percentage Accurate: 77.1% → 99.9%

Time: 3.9s

Alternatives: 4

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: 77.1% 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 79.3%

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

   1. sub-neg 79.3%

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

   2. +-commutative 79.3%

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

   3. distribute-neg-frac 79.3%

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

   4. metadata-eval 79.3%

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

3. Applied egg-rr 79.3%

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

5. Simplified 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{--1}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 0.75)))
   (/ (- -1.0) (* x (- x)))
   (+ (/ -1.0 x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		tmp = -(-1.0) / (x * -x);
	} else {
		tmp = (-1.0 / x) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 0.75d0))) then
        tmp = -(-1.0d0) / (x * -x)
    else
        tmp = ((-1.0d0) / x) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 0.75)) {
		tmp = -(-1.0) / (x * -x);
	} else {
		tmp = (-1.0 / x) + 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 0.75):
		tmp = -(-1.0) / (x * -x)
	else:
		tmp = (-1.0 / x) + 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 0.75))
		tmp = Float64(Float64(-(-1.0)) / Float64(x * Float64(-x)));
	else
		tmp = Float64(Float64(-1.0 / x) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 0.75)))
		tmp = -(-1.0) / (x * -x);
	else
		tmp = (-1.0 / x) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 0.75]], $MachinePrecision]], N[((--1.0) / N[(x * (-x)), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 0.75 < x

   1. Initial program 57.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. unpow2 95.7%

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

      2. associate-/r* 96.1%

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

      3. *-lft-identity 96.1%

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

      4. associate-*l/ 96.0%

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

      5. metadata-eval 96.0%

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

      6. distribute-neg-frac 96.0%

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

      7. distribute-rgt-neg-in 96.0%

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

      8. unpow-1 96.0%

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

      9. unpow-1 96.0%

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

      10. pow-sqr 96.3%

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

      11. metadata-eval 96.3%

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

   4. Simplified 96.3%

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

      1. *-un-lft-identity 96.3%

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

      2. metadata-eval 96.3%

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

      3. metadata-eval 96.3%

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

      4. pow-prod-up 96.0%

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

      5. inv-pow 96.0%

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

      6. inv-pow 96.0%

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

      7. swap-sqr 96.0%

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

      8. div-inv 96.0%

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

      9. div-inv 96.0%

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

      10. frac-2neg 96.0%

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

      11. metadata-eval 96.0%

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

      12. frac-times 95.7%

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

      13. metadata-eval 95.7%

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

   6. Applied egg-rr 95.7%

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

   if -1 < x < 0.75

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 0.75\right):\\ \;\;\;\;\frac{--1}{x \cdot \left(-x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \]

Alternative 3: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -1.0) (not (<= x 1.65))) (/ (/ 1.0 x) x) (+ (/ -1.0 x) 1.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.65)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = (-1.0 / x) + 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-1.0d0)) .or. (.not. (x <= 1.65d0))) then
        tmp = (1.0d0 / x) / x
    else
        tmp = ((-1.0d0) / x) + 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -1.0) || !(x <= 1.65)) {
		tmp = (1.0 / x) / x;
	} else {
		tmp = (-1.0 / x) + 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -1.0) or not (x <= 1.65):
		tmp = (1.0 / x) / x
	else:
		tmp = (-1.0 / x) + 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -1.0) || !(x <= 1.65))
		tmp = Float64(Float64(1.0 / x) / x);
	else
		tmp = Float64(Float64(-1.0 / x) + 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -1.0) || ~((x <= 1.65)))
		tmp = (1.0 / x) / x;
	else
		tmp = (-1.0 / x) + 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -1.0], N[Not[LessEqual[x, 1.65]], $MachinePrecision]], N[(N[(1.0 / x), $MachinePrecision] / x), $MachinePrecision], N[(N[(-1.0 / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1.6499999999999999 < x

   1. Initial program 57.7%

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

   2. Taylor expanded in x around inf 95.7%

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

      1. unpow2 95.7%

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

      2. associate-/r* 96.1%

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

      3. *-lft-identity 96.1%

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

      4. associate-*l/ 96.0%

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

      5. metadata-eval 96.0%

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

      6. distribute-neg-frac 96.0%

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

      7. distribute-rgt-neg-in 96.0%

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

      8. unpow-1 96.0%

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

      9. unpow-1 96.0%

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

      10. pow-sqr 96.3%

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

      11. metadata-eval 96.3%

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

   4. Simplified 96.3%

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

      1. *-un-lft-identity 96.3%

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

      2. metadata-eval 96.3%

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

      3. pow-prod-up 96.0%

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

      4. inv-pow 96.0%

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

      5. inv-pow 96.0%

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

      6. metadata-eval 96.0%

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

      7. swap-sqr 96.0%

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

      8. div-inv 96.0%

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

      9. div-inv 96.0%

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

      10. associate-*l/ 96.1%

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

   6. Applied egg-rr 51.8%

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

   if -1 < x < 1.6499999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0 98.8%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \lor \neg \left(x \leq 1.65\right):\\ \;\;\;\;\frac{\frac{1}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{x} + 1\\ \end{array} \]

Alternative 4: 51.8% 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 79.3%

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

2. Taylor expanded in x around 0 52.4%

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

3. Final simplification 52.4%

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))