2frac (problem 3.3.1)

Percentage Accurate: 76.6% → 99.9%

Time: 5.6s

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: 76.6% 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

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

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 78.1%

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

   1. clear-num 78.1%

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

   2. frac-sub 79.1%

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

   3. *-un-lft-identity 79.1%

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

   4. div-inv 79.1%

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

   5. metadata-eval 79.1%

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

   6. *-rgt-identity 79.1%

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

   7. *-rgt-identity 79.1%

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

   8. +-commutative 79.1%

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

   9. *-commutative 79.1%

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

   10. div-inv 79.1%

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

   11. metadata-eval 79.1%

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

   12. *-rgt-identity 79.1%

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

   13. +-commutative 79.1%

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

4. Applied egg-rr 79.1%

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

   1. *-un-lft-identity 79.1%

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

   2. associate-/l/ 79.1%

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

   3. div-sub 78.1%

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

   4. *-inverses 78.1%

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

   5. sub-neg 78.1%

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

   6. +-commutative 78.1%

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

   7. metadata-eval 78.1%

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

6. Applied egg-rr 78.1%

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

   1. *-lft-identity 78.1%

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

   2. metadata-eval 78.1%

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

   3. sub-neg 78.1%

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

   4. lft-mult-inverse 75.3%

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

   5. associate-*l/ 78.1%

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

   6. *-lft-identity 78.1%

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

   7. div-sub 79.1%

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

   8. rem-square-sqrt 24.1%

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

   9. associate-*r/ 24.1%

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

   10. associate-*l/ 24.1%

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

   11. associate-*r/ 24.1%

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

   12. associate-*l/ 24.1%

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

   13. rem-square-sqrt 79.1%

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

   14. associate--r+ 99.9%

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

   15. +-inverses 99.9%

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

   16. metadata-eval 99.9%

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

8. Simplified 99.9%

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

Alternative 2: 99.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

   1. sub-neg 78.1%

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

   2. +-commutative 78.1%

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

   3. distribute-neg-frac 78.1%

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

   4. metadata-eval 78.1%

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

4. Applied egg-rr 78.1%

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

   1. *-rgt-identity 78.1%

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

   2. cancel-sign-sub 78.1%

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

   3. distribute-neg-frac 78.1%

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

   4. metadata-eval 78.1%

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

   5. *-commutative 78.1%

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

   6. *-inverses 78.1%

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

   7. times-frac 56.7%

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

   8. *-rgt-identity 56.7%

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

   9. *-commutative 56.7%

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

   10. associate-/r* 78.0%

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

   11. div-sub 78.1%

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

   12. *-inverses 78.1%

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

   13. div-sub 79.1%

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

   14. associate-/r* 79.1%

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

   15. +-commutative 79.1%

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

   16. associate--r+ 99.7%

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

   17. +-inverses 99.7%

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

   18. metadata-eval 99.7%

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

   19. distribute-lft-in 99.7%

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

   20. unpow2 99.7%

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

   21. *-rgt-identity 99.7%

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

6. Simplified 99.7%

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

   1. unpow2 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 3: 49.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.1%

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

3. Taylor expanded in x around 0 54.0%

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

   1. div-sub 54.0%

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

   2. *-inverses 54.0%

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

   3. metadata-eval 54.0%

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

   4. distribute-neg-frac 54.0%

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

   5. neg-sub0 54.0%

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

   6. associate--r- 54.0%

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

   7. metadata-eval 54.0%

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

5. Simplified 54.0%

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

6. Final simplification 54.0%

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

Alternative 4: 2.9% accurate, 3.0× speedup

Math

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

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 78.1%

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

3. Taylor expanded in x around 0 53.9%

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

   1. neg-mul-1 53.9%

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

   2. sub-neg 53.9%

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

5. Simplified 53.9%

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

6. Taylor expanded in x around inf 2.7%

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

   1. sub-neg 2.7%

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

   2. metadata-eval 2.7%

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

   3. distribute-lft-in 2.7%

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

   4. rgt-mult-inverse 2.7%

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

   5. *-commutative 2.7%

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

   6. neg-mul-1 2.7%

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

   7. sub-neg 2.7%

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

8. Simplified 2.7%

   \[\leadsto \color{blue}{1 - x} \]
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 2024179 
(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)))