2frac (problem 3.3.1)

Percentage Accurate: 77.3% → 99.9%

Time: 3.6s

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: 77.3% 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

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

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 78.0%

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

   1. sub-neg 78.0%

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

   2. +-commutative 78.0%

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

   3. distribute-neg-frac 78.0%

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

   4. metadata-eval 78.0%

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

4. Applied egg-rr 78.0%

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

6. Simplified 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 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 78.0%

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

   1. sub-neg 78.0%

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

   2. +-commutative 78.0%

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

   3. distribute-neg-frac 78.0%

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

   4. metadata-eval 78.0%

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

4. Applied egg-rr 78.0%

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

6. Simplified 99.9%

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

   1. div-inv 99.8%

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

   2. div-inv 99.8%

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

   3. associate-*l* 99.8%

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

   4. add-sqr-sqrt 53.2%

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

   5. sqrt-unprod 53.6%

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

   6. frac-times 53.3%

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

   7. metadata-eval 53.3%

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

   8. metadata-eval 53.3%

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

   9. frac-times 53.6%

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

   10. sqrt-unprod 13.6%

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

   11. add-sqr-sqrt 27.7%

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

   12. clear-num 27.7%

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

   13. frac-times 27.7%

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

   14. metadata-eval 27.7%

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

8. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.4%

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

   2. metadata-eval 99.4%

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

   3. +-commutative 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 3: 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 78.0%

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

3. Taylor expanded in x around 0 50.7%

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

4. Final simplification 50.7%

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

Alternative 4: 3.0% 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 78.0%

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

3. Taylor expanded in x around 0 49.3%

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

4. Taylor expanded in x around inf 2.8%

   \[\leadsto \color{blue}{1} \]

5. Final simplification 2.8%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))