subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 2.9s

Alternatives: 4

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{f + n}{n - f} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (+ f n) (- n f)))

C

double code(double f, double n) {
	return (f + n) / (n - f);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = (f + n) / (n - f)
end function

Java

public static double code(double f, double n) {
	return (f + n) / (n - f);
}

Python

def code(f, n):
	return (f + n) / (n - f)

Julia

function code(f, n)
	return Float64(Float64(f + n) / Float64(n - f))
end

MATLAB

function tmp = code(f, n)
	tmp = (f + n) / (n - f);
end

Wolfram

code[f_, n_] := N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

   2. remove-double-neg 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

   3. distribute-neg-in 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

   11. +-commutative 100.0%

       \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

   12. sub-neg 100.0%

       \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

4. Final simplification 100.0%

   \[\leadsto \frac{f + n}{n - f} \]

Alternative 2: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -4.2e+16) 1.0 (if (<= n 9e-20) (+ (* -2.0 (/ n f)) -1.0) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -4.2e+16) {
		tmp = 1.0;
	} else if (n <= 9e-20) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.2d+16)) then
        tmp = 1.0d0
    else if (n <= 9d-20) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -4.2e+16) {
		tmp = 1.0;
	} else if (n <= 9e-20) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -4.2e+16:
		tmp = 1.0
	elif n <= 9e-20:
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -4.2e+16)
		tmp = 1.0;
	elseif (n <= 9e-20)
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -4.2e+16)
		tmp = 1.0;
	elseif (n <= 9e-20)
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -4.2e+16], 1.0, If[LessEqual[n, 9e-20], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -4.2e16 or 9.0000000000000003e-20 < n

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

      11. +-commutative 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

      12. sub-neg 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around 0 78.1%

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

   if -4.2e16 < n < 9.0000000000000003e-20

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

      11. +-commutative 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

      12. sub-neg 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in n around 0 81.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.2 \cdot 10^{+16}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 73.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.2e+17) 1.0 (if (<= n 3.8e-41) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.2e+17) {
		tmp = 1.0;
	} else if (n <= 3.8e-41) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.2d+17)) then
        tmp = 1.0d0
    else if (n <= 3.8d-41) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -1.2e+17) {
		tmp = 1.0;
	} else if (n <= 3.8e-41) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.2e+17:
		tmp = 1.0
	elif n <= 3.8e-41:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.2e+17)
		tmp = 1.0;
	elseif (n <= 3.8e-41)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.2e+17)
		tmp = 1.0;
	elseif (n <= 3.8e-41)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.2e+17], 1.0, If[LessEqual[n, 3.8e-41], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2e17 or 3.79999999999999979e-41 < n

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

      11. +-commutative 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

      12. sub-neg 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around 0 76.8%

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

   if -1.2e17 < n < 3.79999999999999979e-41

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Step-by-step derivation

      1. sub-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

      2. remove-double-neg 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

      3. distribute-neg-in 100.0%

         \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

          \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

      11. +-commutative 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

      12. sub-neg 100.0%

          \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

   4. Taylor expanded in f around inf 82.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.8 \cdot 10^{-41}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 49.8% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (f n) :precision binary64 -1.0)

C

double code(double f, double n) {
	return -1.0;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -1.0d0
end function

Java

public static double code(double f, double n) {
	return -1.0;
}

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

function tmp = code(f, n)
	tmp = -1.0;
end

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 100.0%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Step-by-step derivation

   1. sub-neg 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]

   2. remove-double-neg 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{\left(-\left(-f\right)\right)} + \left(-n\right)} \]

   3. distribute-neg-in 100.0%

      \[\leadsto \frac{-\left(f + n\right)}{\color{blue}{-\left(\left(-f\right) + n\right)}} \]

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

       \[\leadsto \frac{\color{blue}{f + n}}{\left(-f\right) + n} \]

   11. +-commutative 100.0%

       \[\leadsto \frac{f + n}{\color{blue}{n + \left(-f\right)}} \]

   12. sub-neg 100.0%

       \[\leadsto \frac{f + n}{\color{blue}{n - f}} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]

4. Taylor expanded in f around inf 52.4%

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

5. Final simplification 52.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))