2frac (problem 3.3.1)

Percentage Accurate: 78.4% → 99.4%

Time: 5.4s

Alternatives: 3

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: 78.4% 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: 99.4% 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 76.5%

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

   1. sub-neg 76.5%

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

   2. +-commutative 76.5%

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

   3. distribute-neg-frac 76.5%

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

   4. metadata-eval 76.5%

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

4. Applied egg-rr 76.5%

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

6. Simplified 99.9%

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

   1. flip3-+ 66.5%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)}}} \]

   2. associate-/r/ 64.2%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{{x}^{3} + {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right)} \]

   3. metadata-eval 64.2%

      \[\leadsto \frac{\frac{-1}{x}}{{x}^{3} + \color{blue}{1}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) \]

   4. +-commutative 64.2%

      \[\leadsto \frac{\frac{-1}{x}}{\color{blue}{1 + {x}^{3}}} \cdot \left(x \cdot x + \left(1 \cdot 1 - x \cdot 1\right)\right) \]

   5. fma-def 64.2%

      \[\leadsto \frac{\frac{-1}{x}}{1 + {x}^{3}} \cdot \color{blue}{\mathsf{fma}\left(x, x, 1 \cdot 1 - x \cdot 1\right)} \]

   6. metadata-eval 64.2%

      \[\leadsto \frac{\frac{-1}{x}}{1 + {x}^{3}} \cdot \mathsf{fma}\left(x, x, \color{blue}{1} - x \cdot 1\right) \]

   7. *-rgt-identity 64.2%

      \[\leadsto \frac{\frac{-1}{x}}{1 + {x}^{3}} \cdot \mathsf{fma}\left(x, x, 1 - \color{blue}{x}\right) \]

8. Applied egg-rr 64.2%

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

   1. associate-*l/ 66.5%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x} \cdot \mathsf{fma}\left(x, x, 1 - x\right)}{1 + {x}^{3}}} \]

   2. associate-/l* 66.5%

      \[\leadsto \color{blue}{\frac{\frac{-1}{x}}{\frac{1 + {x}^{3}}{\mathsf{fma}\left(x, x, 1 - x\right)}}} \]

   3. div-inv 66.5%

      \[\leadsto \frac{\color{blue}{-1 \cdot \frac{1}{x}}}{\frac{1 + {x}^{3}}{\mathsf{fma}\left(x, x, 1 - x\right)}} \]

   4. +-commutative 66.5%

      \[\leadsto \frac{-1 \cdot \frac{1}{x}}{\frac{\color{blue}{{x}^{3} + 1}}{\mathsf{fma}\left(x, x, 1 - x\right)}} \]

   5. metadata-eval 66.5%

      \[\leadsto \frac{-1 \cdot \frac{1}{x}}{\frac{{x}^{3} + \color{blue}{{1}^{3}}}{\mathsf{fma}\left(x, x, 1 - x\right)}} \]

   6. fma-udef 66.5%

      \[\leadsto \frac{-1 \cdot \frac{1}{x}}{\frac{{x}^{3} + {1}^{3}}{\color{blue}{x \cdot x + \left(1 - x\right)}}} \]

   7. *-rgt-identity 66.5%

      \[\leadsto \frac{-1 \cdot \frac{1}{x}}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(1 - \color{blue}{x \cdot 1}\right)}} \]

   8. metadata-eval 66.5%

      \[\leadsto \frac{-1 \cdot \frac{1}{x}}{\frac{{x}^{3} + {1}^{3}}{x \cdot x + \left(\color{blue}{1 \cdot 1} - x \cdot 1\right)}} \]

   9. flip3-+ 99.9%

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

   10. +-commutative 99.9%

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

   11. associate-*l/ 99.8%

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

   12. frac-2neg 99.8%

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

   13. metadata-eval 99.8%

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

   14. frac-times 99.9%

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

   15. metadata-eval 99.9%

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

   16. distribute-neg-in 99.9%

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

   17. metadata-eval 99.9%

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

10. Applied egg-rr 99.9%

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

11. Taylor expanded in x around 0 99.9%

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

    1. mul-1-neg 99.9%

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

    2. mul-1-neg 99.9%

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

    3. distribute-neg-in 99.9%

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

    4. *-rgt-identity 99.9%

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

    5. unpow2 99.9%

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

    6. distribute-lft-in 99.9%

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

    7. distribute-rgt-neg-in 99.9%

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

    8. distribute-neg-in 99.9%

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

    9. metadata-eval 99.9%

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

    10. unsub-neg 99.9%

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

13. Simplified 99.9%

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

14. Final simplification 99.9%

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

Alternative 2: 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 76.5%

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

3. Taylor expanded in x around 0 52.1%

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

4. Final simplification 52.1%

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

Alternative 3: 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 76.5%

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

3. Taylor expanded in x around 0 51.5%

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

4. Taylor expanded in x around inf 3.1%

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

5. Final simplification 3.1%

   \[\leadsto 1 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(FPCore (x)
  :name "2frac (problem 3.3.1)"
  :precision binary64
  (- (/ 1.0 (+ x 1.0)) (/ 1.0 x)))