subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 4.2s

Alternatives: 7

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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{f}{n - f}\\ t_1 := \frac{n}{n - f}\\ \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1} \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ f (- n f))) (t_1 (/ n (- n f))))
   (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))

C

double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    t_0 = f / (n - f)
    t_1 = n / (n - f)
    code = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
end function

Java

public static double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Python

def code(f, n):
	t_0 = f / (n - f)
	t_1 = n / (n - f)
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)

Julia

function code(f, n)
	t_0 = Float64(f / Float64(n - f))
	t_1 = Float64(n / Float64(n - f))
	return Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1))
end

MATLAB

function tmp = code(f, n)
	t_0 = f / (n - f);
	t_1 = n / (n - f);
	tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. distribute-neg-in 99.9%

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

   4. neg-mul-1 99.9%

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

   5. associate-/r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-/l* 99.9%

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

   9. metadata-eval 99.9%

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

   10. /-rgt-identity 99.9%

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

   11. +-commutative 99.9%

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

   12. sub-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
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. associate-/r/ 99.7%

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

   3. distribute-lft-in 99.7%

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

   4. flip-+ 99.7%

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

   5. associate-*l/ 99.8%

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

   6. *-un-lft-identity 99.8%

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

   7. associate-*l/ 99.7%

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

   8. *-un-lft-identity 99.7%

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

   9. associate-*l/ 99.8%

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

   10. *-un-lft-identity 99.8%

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

   11. associate-*l/ 99.7%

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

   12. *-un-lft-identity 99.7%

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

   13. associate-*l/ 99.8%

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

   14. *-un-lft-identity 99.8%

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

   15. associate-*l/ 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 71.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+73} \lor \neg \left(n \leq -3.6 \cdot 10^{-22}\right) \land \left(n \leq -2.1 \cdot 10^{-72} \lor \neg \left(n \leq 3.2 \cdot 10^{-121} \lor \neg \left(n \leq 3.7 \cdot 10^{-30}\right) \land n \leq 4.5 \cdot 10^{-16}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -1.15e+73)
         (and (not (<= n -3.6e-22))
              (or (<= n -2.1e-72)
                  (not
                   (or (<= n 3.2e-121)
                       (and (not (<= n 3.7e-30)) (<= n 4.5e-16)))))))
   (+ 1.0 (* 2.0 (/ f n)))
   -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -1.15e+73) || (!(n <= -3.6e-22) && ((n <= -2.1e-72) || !((n <= 3.2e-121) || (!(n <= 3.7e-30) && (n <= 4.5e-16)))))) {
		tmp = 1.0 + (2.0 * (f / n));
	} 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 <= (-1.15d+73)) .or. (.not. (n <= (-3.6d-22))) .and. (n <= (-2.1d-72)) .or. (.not. (n <= 3.2d-121) .or. (.not. (n <= 3.7d-30)) .and. (n <= 4.5d-16))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -1.15e+73) || (!(n <= -3.6e-22) && ((n <= -2.1e-72) || !((n <= 3.2e-121) || (!(n <= 3.7e-30) && (n <= 4.5e-16)))))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -1.15e+73) or (not (n <= -3.6e-22) and ((n <= -2.1e-72) or not ((n <= 3.2e-121) or (not (n <= 3.7e-30) and (n <= 4.5e-16))))):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -1.15e+73) || (!(n <= -3.6e-22) && ((n <= -2.1e-72) || !((n <= 3.2e-121) || (!(n <= 3.7e-30) && (n <= 4.5e-16))))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -1.15e+73) || (~((n <= -3.6e-22)) && ((n <= -2.1e-72) || ~(((n <= 3.2e-121) || (~((n <= 3.7e-30)) && (n <= 4.5e-16)))))))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -1.15e+73], And[N[Not[LessEqual[n, -3.6e-22]], $MachinePrecision], Or[LessEqual[n, -2.1e-72], N[Not[Or[LessEqual[n, 3.2e-121], And[N[Not[LessEqual[n, 3.7e-30]], $MachinePrecision], LessEqual[n, 4.5e-16]]]], $MachinePrecision]]]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -1.15e73 or -3.5999999999999998e-22 < n < -2.1e-72 or 3.20000000000000019e-121 < n < 3.7000000000000003e-30 or 4.5000000000000002e-16 < 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%

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

   if -1.15e73 < n < -3.5999999999999998e-22 or -2.1e-72 < n < 3.20000000000000019e-121 or 3.7000000000000003e-30 < n < 4.5000000000000002e-16

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. /-rgt-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 81.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+73} \lor \neg \left(n \leq -3.6 \cdot 10^{-22}\right) \land \left(n \leq -2.1 \cdot 10^{-72} \lor \neg \left(n \leq 3.2 \cdot 10^{-121} \lor \neg \left(n \leq 3.7 \cdot 10^{-30}\right) \land n \leq 4.5 \cdot 10^{-16}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 73.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{+72} \lor \neg \left(n \leq -5.6 \cdot 10^{-24} \lor \neg \left(n \leq -2.9 \cdot 10^{-72}\right) \land n \leq 9 \cdot 10^{-15}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -7e+72)
         (not (or (<= n -5.6e-24) (and (not (<= n -2.9e-72)) (<= n 9e-15)))))
   (+ 1.0 (* 2.0 (/ f n)))
   (+ (* -2.0 (/ n f)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -7e+72) || !((n <= -5.6e-24) || (!(n <= -2.9e-72) && (n <= 9e-15)))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -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 <= (-7d+72)) .or. (.not. (n <= (-5.6d-24)) .or. (.not. (n <= (-2.9d-72))) .and. (n <= 9d-15))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -7e+72) || !((n <= -5.6e-24) || (!(n <= -2.9e-72) && (n <= 9e-15)))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -7e+72) or not ((n <= -5.6e-24) or (not (n <= -2.9e-72) and (n <= 9e-15))):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = (-2.0 * (n / f)) + -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -7e+72) || !((n <= -5.6e-24) || (!(n <= -2.9e-72) && (n <= 9e-15))))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -7e+72) || ~(((n <= -5.6e-24) || (~((n <= -2.9e-72)) && (n <= 9e-15)))))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = (-2.0 * (n / f)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -7e+72], N[Not[Or[LessEqual[n, -5.6e-24], And[N[Not[LessEqual[n, -2.9e-72]], $MachinePrecision], LessEqual[n, 9e-15]]]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -7.0000000000000002e72 or -5.6000000000000003e-24 < n < -2.89999999999999998e-72 or 8.9999999999999995e-15 < 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 82.2%

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

   if -7.0000000000000002e72 < n < -5.6000000000000003e-24 or -2.89999999999999998e-72 < n < 8.9999999999999995e-15

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. /-rgt-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in n around 0 78.1%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -7 \cdot 10^{+72} \lor \neg \left(n \leq -5.6 \cdot 10^{-24} \lor \neg \left(n \leq -2.9 \cdot 10^{-72}\right) \land n \leq 9 \cdot 10^{-15}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]

Alternative 4: 72.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -4e+73)
   1.0
   (if (<= n -1.65e-20)
     -1.0
     (if (<= n -4.5e-72) 1.0 (if (<= n 9e-15) -1.0 1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -4e+73) {
		tmp = 1.0;
	} else if (n <= -1.65e-20) {
		tmp = -1.0;
	} else if (n <= -4.5e-72) {
		tmp = 1.0;
	} else if (n <= 9e-15) {
		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 <= (-4d+73)) then
        tmp = 1.0d0
    else if (n <= (-1.65d-20)) then
        tmp = -1.0d0
    else if (n <= (-4.5d-72)) then
        tmp = 1.0d0
    else if (n <= 9d-15) 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 <= -4e+73) {
		tmp = 1.0;
	} else if (n <= -1.65e-20) {
		tmp = -1.0;
	} else if (n <= -4.5e-72) {
		tmp = 1.0;
	} else if (n <= 9e-15) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -4e+73:
		tmp = 1.0
	elif n <= -1.65e-20:
		tmp = -1.0
	elif n <= -4.5e-72:
		tmp = 1.0
	elif n <= 9e-15:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -4e+73)
		tmp = 1.0;
	elseif (n <= -1.65e-20)
		tmp = -1.0;
	elseif (n <= -4.5e-72)
		tmp = 1.0;
	elseif (n <= 9e-15)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -4e+73)
		tmp = 1.0;
	elseif (n <= -1.65e-20)
		tmp = -1.0;
	elseif (n <= -4.5e-72)
		tmp = 1.0;
	elseif (n <= 9e-15)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -4e+73], 1.0, If[LessEqual[n, -1.65e-20], -1.0, If[LessEqual[n, -4.5e-72], 1.0, If[LessEqual[n, 9e-15], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.99999999999999993e73 or -1.65e-20 < n < -4.5e-72 or 8.9999999999999995e-15 < 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 81.1%

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

   if -3.99999999999999993e73 < n < -1.65e-20 or -4.5e-72 < n < 8.9999999999999995e-15

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. distribute-neg-in 99.9%

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

      4. neg-mul-1 99.9%

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

      5. associate-/r* 99.9%

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

      6. neg-mul-1 99.9%

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

      7. *-commutative 99.9%

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

      8. associate-/l* 99.9%

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

      9. metadata-eval 99.9%

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

      10. /-rgt-identity 99.9%

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

      11. +-commutative 99.9%

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

      12. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 76.4%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.65 \cdot 10^{-20}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -4.5 \cdot 10^{-72}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return 1.0 / ((n - f) / (f + n))

Julia

function code(f, n)
	return Float64(1.0 / Float64(Float64(n - f) / Float64(f + n)))
end

MATLAB

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

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / N[(f + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. sub-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. distribute-neg-in 99.9%

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

   4. neg-mul-1 99.9%

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

   5. associate-/r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-/l* 99.9%

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

   9. metadata-eval 99.9%

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

   10. /-rgt-identity 99.9%

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

   11. +-commutative 99.9%

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

   12. sub-neg 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
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. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 6: 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 99.9%

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

   1. sub-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. distribute-neg-in 99.9%

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

   4. neg-mul-1 99.9%

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

   5. associate-/r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-/l* 99.9%

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

   9. metadata-eval 99.9%

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

   10. /-rgt-identity 99.9%

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

   11. +-commutative 99.9%

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

   12. sub-neg 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 7: 48.7% 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 99.9%

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

   1. sub-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. distribute-neg-in 99.9%

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

   4. neg-mul-1 99.9%

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

   5. associate-/r* 99.9%

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

   6. neg-mul-1 99.9%

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

   7. *-commutative 99.9%

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

   8. associate-/l* 99.9%

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

   9. metadata-eval 99.9%

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

   10. /-rgt-identity 99.9%

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

   11. +-commutative 99.9%

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

   12. sub-neg 99.9%

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

3. Simplified 99.9%

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

4. Taylor expanded in f around inf 47.0%

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

5. Final simplification 47.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023306 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))