subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 5

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, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision] + N[(n / N[(n - f), $MachinePrecision]), $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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

5. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{{\left(\frac{n - f}{f + n}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 100.0%

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

   2. clear-num 100.0%

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

   3. *-un-lft-identity 100.0%

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

   4. associate-*l/ 99.7%

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

   5. distribute-rgt-in 99.7%

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

   6. un-div-inv 99.8%

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

   7. un-div-inv 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{+40} \lor \neg \left(n \leq 7.8 \cdot 10^{-21}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -2.85e+40) (not (<= n 7.8e-21))) (+ (* 2.0 (/ f n)) 1.0) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -2.85e+40) || !(n <= 7.8e-21)) {
		tmp = (2.0 * (f / n)) + 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 <= (-2.85d+40)) .or. (.not. (n <= 7.8d-21))) then
        tmp = (2.0d0 * (f / n)) + 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 <= -2.85e+40) || !(n <= 7.8e-21)) {
		tmp = (2.0 * (f / n)) + 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -2.85e+40) or not (n <= 7.8e-21):
		tmp = (2.0 * (f / n)) + 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -2.85e+40) || !(n <= 7.8e-21))
		tmp = Float64(Float64(2.0 * Float64(f / n)) + 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -2.85e+40) || ~((n <= 7.8e-21)))
		tmp = (2.0 * (f / n)) + 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -2.85e+40], N[Not[LessEqual[n, 7.8e-21]], $MachinePrecision]], N[(N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.8499999999999999e40 or 7.8000000000000001e-21 < 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 81.8%

      \[\leadsto \color{blue}{2 \cdot \frac{f}{n} + 1} \]

   if -2.8499999999999999e40 < n < 7.8000000000000001e-21

   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 75.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.85 \cdot 10^{+40} \lor \neg \left(n \leq 7.8 \cdot 10^{-21}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 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 4: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.7e+40) 1.0 (if (<= n 2e-20) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.7e+40) {
		tmp = 1.0;
	} else if (n <= 2e-20) {
		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 <= (-2.7d+40)) then
        tmp = 1.0d0
    else if (n <= 2d-20) 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 <= -2.7e+40) {
		tmp = 1.0;
	} else if (n <= 2e-20) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.7e+40:
		tmp = 1.0
	elif n <= 2e-20:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.7e+40)
		tmp = 1.0;
	elseif (n <= 2e-20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.7e+40)
		tmp = 1.0;
	elseif (n <= 2e-20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.7e+40], 1.0, If[LessEqual[n, 2e-20], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -2.70000000000000009e40 or 1.99999999999999989e-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 80.6%

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

   if -2.70000000000000009e40 < n < 1.99999999999999989e-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 f around inf 75.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+40}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 49.3% 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 47.8%

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

5. Final simplification 47.8%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023279 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))