3frac (problem 3.3.3)

Percentage Accurate: 84.1% → 99.6%

Time: 11.4s

Alternatives: 3

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 84.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))

C

double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((1.0d0 / (x + 1.0d0)) - (2.0d0 / x)) + (1.0d0 / (x - 1.0d0))
end function

Java

public static double code(double x) {
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
}

Python

def code(x):
	return ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0))

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / Float64(x + 1.0)) - Float64(2.0 / x)) + Float64(1.0 / Float64(x - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / (x + 1.0)) - (2.0 / x)) + (1.0 / (x - 1.0));
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] - N[(2.0 / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(x - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\left(x \cdot x - x\right) \cdot \left(x + 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* (- (* x x) x) (+ x 1.0))))

C

double code(double x) {
	return 2.0 / (((x * x) - x) * (x + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (((x * x) - x) * (x + 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (((x * x) - x) * (x + 1.0));
}

Python

def code(x):
	return 2.0 / (((x * x) - x) * (x + 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(Float64(Float64(x * x) - x) * Float64(x + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (((x * x) - x) * (x + 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] - x), $MachinePrecision] * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.3%

   \[\leadsto \color{blue}{\frac{1}{1 + x} - \left(\frac{2}{x} - \frac{1}{x + -1}\right)} \]
4. Step-by-step derivation

   1. frac-2neg 83.3%

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

   2. frac-2neg 83.3%

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

   3. metadata-eval 83.3%

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

   4. frac-sub 53.0%

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

   5. metadata-eval 53.0%

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

   6. +-commutative 53.0%

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

   7. distribute-neg-in 53.0%

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

   8. metadata-eval 53.0%

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

   9. sub-neg 53.0%

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

   10. +-commutative 53.0%

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

   11. distribute-neg-in 53.0%

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

   12. metadata-eval 53.0%

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

   13. sub-neg 53.0%

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

5. Applied egg-rr 53.0%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{-2 \cdot \left(1 - x\right) - \left(-x\right) \cdot -1}{\left(-x\right) \cdot \left(1 - x\right)}} \]
6. Step-by-step derivation

   1. cancel-sign-sub 53.0%

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

   2. *-commutative 53.0%

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

   3. neg-mul-1 53.0%

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

   4. unsub-neg 53.0%

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

   5. sub-neg 53.0%

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

   6. +-commutative 53.0%

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

   7. distribute-lft-in 53.1%

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

   8. sqr-neg 53.1%

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

   9. unpow2 53.1%

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

   10. *-rgt-identity 53.1%

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

   11. sub-neg 53.1%

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

   12. unpow2 53.1%

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

7. Simplified 53.1%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\frac{-2 \cdot \left(1 - x\right) - x}{x \cdot x - x}} \]
8. Step-by-step derivation

   1. div-inv 53.2%

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

9. Applied egg-rr 53.2%

   \[\leadsto \frac{1}{1 + x} - \color{blue}{\left(-2 \cdot \left(1 - x\right) - x\right) \cdot \frac{1}{x \cdot x - x}} \]
10. Step-by-step derivation

    1. un-div-inv 53.1%

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

    2. frac-sub 52.6%

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

    3. *-un-lft-identity 52.6%

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

11. Applied egg-rr 52.6%

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

    1. metadata-eval 52.6%

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

    2. remove-double-neg 52.6%

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

    3. distribute-neg-in 52.6%

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

    4. distribute-lft-neg-in 52.6%

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

    5. distribute-lft-neg-in 52.6%

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

    6. distribute-neg-in 52.6%

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

    7. metadata-eval 52.6%

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

    8. remove-double-neg 52.6%

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

    9. +-commutative 52.6%

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

    10. *-commutative 52.6%

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

    11. +-commutative 52.6%

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

13. Simplified 52.6%

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

14. Taylor expanded in x around 0 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 82.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(-1 - \frac{2}{x}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (- -1.0 (/ 2.0 x))))

C

double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-1.0d0) - (2.0d0 / x))
end function

Java

public static double code(double x) {
	return 1.0 + (-1.0 - (2.0 / x));
}

Python

def code(x):
	return 1.0 + (-1.0 - (2.0 / x))

Julia

function code(x)
	return Float64(1.0 + Float64(-1.0 - Float64(2.0 / x)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (-1.0 - (2.0 / x));
end

Wolfram

code[x_] := N[(1.0 + N[(-1.0 - N[(2.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.3%

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

4. Taylor expanded in x around 0 44.9%

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

5. Taylor expanded in x around 0 81.5%

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

6. Final simplification 81.5%

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

Alternative 3: 51.8% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{-2}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ -2.0 x))

C

double code(double x) {
	return -2.0 / x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (-2.0d0) / x
end function

Java

public static double code(double x) {
	return -2.0 / x;
}

Python

def code(x):
	return -2.0 / x

Julia

function code(x)
	return Float64(-2.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = -2.0 / x;
end

Wolfram

code[x_] := N[(-2.0 / x), $MachinePrecision]

Derivation

1. Initial program 83.3%

   \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1} \]

3. Simplified 83.3%

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

4. Taylor expanded in x around 0 45.9%

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

5. Final simplification 45.9%

   \[\leadsto \frac{-2}{x} \]

Developer target: 99.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{2}{x \cdot \left(x \cdot x - 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (* x (- (* x x) 1.0))))

C

double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 2.0d0 / (x * ((x * x) - 1.0d0))
end function

Java

public static double code(double x) {
	return 2.0 / (x * ((x * x) - 1.0));
}

Python

def code(x):
	return 2.0 / (x * ((x * x) - 1.0))

Julia

function code(x)
	return Float64(2.0 / Float64(x * Float64(Float64(x * x) - 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x * ((x * x) - 1.0));
end

Wolfram

code[x_] := N[(2.0 / N[(x * N[(N[(x * x), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023276 
(FPCore (x)
  :name "3frac (problem 3.3.3)"
  :precision binary64

  :herbie-target
  (/ 2.0 (* x (- (* x x) 1.0)))

  (+ (- (/ 1.0 (+ x 1.0)) (/ 2.0 x)) (/ 1.0 (- x 1.0))))