subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 3.2s

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, 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. /-rgt-identity 99.9%

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

   2. metadata-eval 99.9%

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

   3. neg-mul-1 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-/l* 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. associate-/r* 99.9%

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

   9. neg-mul-1 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. distribute-neg-in 99.9%

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

   13. remove-double-neg 99.9%

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

   14. 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. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto \frac{f + n}{n - f} \]
6. Add Preprocessing

Alternative 2: 72.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{f}{n}\\ \mathbf{if}\;n \leq -2.9 \cdot 10^{+125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+65}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{+135}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ f n)))))
   (if (<= n -2.9e+125)
     t_0
     (if (<= n -6.4e+113)
       -1.0
       (if (<= n -3.6e-37)
         t_0
         (if (<= n 9.2e+40)
           -1.0
           (if (<= n 4e+65) 1.0 (if (<= n 5.6e+135) -1.0 t_0))))))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -2.9e+125) {
		tmp = t_0;
	} else if (n <= -6.4e+113) {
		tmp = -1.0;
	} else if (n <= -3.6e-37) {
		tmp = t_0;
	} else if (n <= 9.2e+40) {
		tmp = -1.0;
	} else if (n <= 4e+65) {
		tmp = 1.0;
	} else if (n <= 5.6e+135) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (f / n))
    if (n <= (-2.9d+125)) then
        tmp = t_0
    else if (n <= (-6.4d+113)) then
        tmp = -1.0d0
    else if (n <= (-3.6d-37)) then
        tmp = t_0
    else if (n <= 9.2d+40) then
        tmp = -1.0d0
    else if (n <= 4d+65) then
        tmp = 1.0d0
    else if (n <= 5.6d+135) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -2.9e+125) {
		tmp = t_0;
	} else if (n <= -6.4e+113) {
		tmp = -1.0;
	} else if (n <= -3.6e-37) {
		tmp = t_0;
	} else if (n <= 9.2e+40) {
		tmp = -1.0;
	} else if (n <= 4e+65) {
		tmp = 1.0;
	} else if (n <= 5.6e+135) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (2.0 * (f / n))
	tmp = 0
	if n <= -2.9e+125:
		tmp = t_0
	elif n <= -6.4e+113:
		tmp = -1.0
	elif n <= -3.6e-37:
		tmp = t_0
	elif n <= 9.2e+40:
		tmp = -1.0
	elif n <= 4e+65:
		tmp = 1.0
	elif n <= 5.6e+135:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(f / n)))
	tmp = 0.0
	if (n <= -2.9e+125)
		tmp = t_0;
	elseif (n <= -6.4e+113)
		tmp = -1.0;
	elseif (n <= -3.6e-37)
		tmp = t_0;
	elseif (n <= 9.2e+40)
		tmp = -1.0;
	elseif (n <= 4e+65)
		tmp = 1.0;
	elseif (n <= 5.6e+135)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (2.0 * (f / n));
	tmp = 0.0;
	if (n <= -2.9e+125)
		tmp = t_0;
	elseif (n <= -6.4e+113)
		tmp = -1.0;
	elseif (n <= -3.6e-37)
		tmp = t_0;
	elseif (n <= 9.2e+40)
		tmp = -1.0;
	elseif (n <= 4e+65)
		tmp = 1.0;
	elseif (n <= 5.6e+135)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -2.9e+125], t$95$0, If[LessEqual[n, -6.4e+113], -1.0, If[LessEqual[n, -3.6e-37], t$95$0, If[LessEqual[n, 9.2e+40], -1.0, If[LessEqual[n, 4e+65], 1.0, If[LessEqual[n, 5.6e+135], -1.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.89999999999999993e125 or -6.3999999999999996e113 < n < -3.60000000000000007e-37 or 5.60000000000000004e135 < n

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around 0 87.4%

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

   if -2.89999999999999993e125 < n < -6.3999999999999996e113 or -3.60000000000000007e-37 < n < 9.19999999999999975e40 or 4e65 < n < 5.60000000000000004e135

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around inf 82.1%

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

   if 9.19999999999999975e40 < n < 4e65

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around 0 100.0%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.9 \cdot 10^{+125}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{+40}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 4 \cdot 10^{+65}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 5.6 \cdot 10^{+135}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{f}{n}\\ t_1 := -2 \cdot \frac{n}{f} + -1\\ \mathbf{if}\;n \leq -1.25 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq -28000000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ f n)))) (t_1 (+ (* -2.0 (/ n f)) -1.0)))
   (if (<= n -1.25e+124)
     t_0
     (if (<= n -6.4e+113)
       t_1
       (if (<= n -28000000000000.0)
         t_0
         (if (<= n 1.95e+42)
           t_1
           (if (<= n 1.45e+70) 1.0 (if (<= n 1.65e+131) t_1 t_0))))))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double t_1 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (n <= -1.25e+124) {
		tmp = t_0;
	} else if (n <= -6.4e+113) {
		tmp = t_1;
	} else if (n <= -28000000000000.0) {
		tmp = t_0;
	} else if (n <= 1.95e+42) {
		tmp = t_1;
	} else if (n <= 1.45e+70) {
		tmp = 1.0;
	} else if (n <= 1.65e+131) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

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
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (f / n))
    t_1 = ((-2.0d0) * (n / f)) + (-1.0d0)
    if (n <= (-1.25d+124)) then
        tmp = t_0
    else if (n <= (-6.4d+113)) then
        tmp = t_1
    else if (n <= (-28000000000000.0d0)) then
        tmp = t_0
    else if (n <= 1.95d+42) then
        tmp = t_1
    else if (n <= 1.45d+70) then
        tmp = 1.0d0
    else if (n <= 1.65d+131) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double t_1 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (n <= -1.25e+124) {
		tmp = t_0;
	} else if (n <= -6.4e+113) {
		tmp = t_1;
	} else if (n <= -28000000000000.0) {
		tmp = t_0;
	} else if (n <= 1.95e+42) {
		tmp = t_1;
	} else if (n <= 1.45e+70) {
		tmp = 1.0;
	} else if (n <= 1.65e+131) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (2.0 * (f / n))
	t_1 = (-2.0 * (n / f)) + -1.0
	tmp = 0
	if n <= -1.25e+124:
		tmp = t_0
	elif n <= -6.4e+113:
		tmp = t_1
	elif n <= -28000000000000.0:
		tmp = t_0
	elif n <= 1.95e+42:
		tmp = t_1
	elif n <= 1.45e+70:
		tmp = 1.0
	elif n <= 1.65e+131:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(f / n)))
	t_1 = Float64(Float64(-2.0 * Float64(n / f)) + -1.0)
	tmp = 0.0
	if (n <= -1.25e+124)
		tmp = t_0;
	elseif (n <= -6.4e+113)
		tmp = t_1;
	elseif (n <= -28000000000000.0)
		tmp = t_0;
	elseif (n <= 1.95e+42)
		tmp = t_1;
	elseif (n <= 1.45e+70)
		tmp = 1.0;
	elseif (n <= 1.65e+131)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (2.0 * (f / n));
	t_1 = (-2.0 * (n / f)) + -1.0;
	tmp = 0.0;
	if (n <= -1.25e+124)
		tmp = t_0;
	elseif (n <= -6.4e+113)
		tmp = t_1;
	elseif (n <= -28000000000000.0)
		tmp = t_0;
	elseif (n <= 1.95e+42)
		tmp = t_1;
	elseif (n <= 1.45e+70)
		tmp = 1.0;
	elseif (n <= 1.65e+131)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[n, -1.25e+124], t$95$0, If[LessEqual[n, -6.4e+113], t$95$1, If[LessEqual[n, -28000000000000.0], t$95$0, If[LessEqual[n, 1.95e+42], t$95$1, If[LessEqual[n, 1.45e+70], 1.0, If[LessEqual[n, 1.65e+131], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.2499999999999999e124 or -6.3999999999999996e113 < n < -2.8e13 or 1.6499999999999999e131 < n

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around 0 89.4%

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

   if -1.2499999999999999e124 < n < -6.3999999999999996e113 or -2.8e13 < n < 1.94999999999999985e42 or 1.4499999999999999e70 < n < 1.6499999999999999e131

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in n around 0 82.2%

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

   if 1.94999999999999985e42 < n < 1.4499999999999999e70

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around 0 100.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+124}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{elif}\;n \leq -6.4 \cdot 10^{+113}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;n \leq -28000000000000:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{+42}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;n \leq 1.45 \cdot 10^{+70}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+124}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{+111}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 10^{+68}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.25e+124)
   1.0
   (if (<= n -5e+111)
     -1.0
     (if (<= n -3.6e-37)
       1.0
       (if (<= n 2.5e+41)
         -1.0
         (if (<= n 1e+68) 1.0 (if (<= n 1.65e+131) -1.0 1.0)))))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.25e+124) {
		tmp = 1.0;
	} else if (n <= -5e+111) {
		tmp = -1.0;
	} else if (n <= -3.6e-37) {
		tmp = 1.0;
	} else if (n <= 2.5e+41) {
		tmp = -1.0;
	} else if (n <= 1e+68) {
		tmp = 1.0;
	} else if (n <= 1.65e+131) {
		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 <= (-1.25d+124)) then
        tmp = 1.0d0
    else if (n <= (-5d+111)) then
        tmp = -1.0d0
    else if (n <= (-3.6d-37)) then
        tmp = 1.0d0
    else if (n <= 2.5d+41) then
        tmp = -1.0d0
    else if (n <= 1d+68) then
        tmp = 1.0d0
    else if (n <= 1.65d+131) 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 <= -1.25e+124) {
		tmp = 1.0;
	} else if (n <= -5e+111) {
		tmp = -1.0;
	} else if (n <= -3.6e-37) {
		tmp = 1.0;
	} else if (n <= 2.5e+41) {
		tmp = -1.0;
	} else if (n <= 1e+68) {
		tmp = 1.0;
	} else if (n <= 1.65e+131) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.25e+124:
		tmp = 1.0
	elif n <= -5e+111:
		tmp = -1.0
	elif n <= -3.6e-37:
		tmp = 1.0
	elif n <= 2.5e+41:
		tmp = -1.0
	elif n <= 1e+68:
		tmp = 1.0
	elif n <= 1.65e+131:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.25e+124)
		tmp = 1.0;
	elseif (n <= -5e+111)
		tmp = -1.0;
	elseif (n <= -3.6e-37)
		tmp = 1.0;
	elseif (n <= 2.5e+41)
		tmp = -1.0;
	elseif (n <= 1e+68)
		tmp = 1.0;
	elseif (n <= 1.65e+131)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.25e+124)
		tmp = 1.0;
	elseif (n <= -5e+111)
		tmp = -1.0;
	elseif (n <= -3.6e-37)
		tmp = 1.0;
	elseif (n <= 2.5e+41)
		tmp = -1.0;
	elseif (n <= 1e+68)
		tmp = 1.0;
	elseif (n <= 1.65e+131)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.25e+124], 1.0, If[LessEqual[n, -5e+111], -1.0, If[LessEqual[n, -3.6e-37], 1.0, If[LessEqual[n, 2.5e+41], -1.0, If[LessEqual[n, 1e+68], 1.0, If[LessEqual[n, 1.65e+131], -1.0, 1.0]]]]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.2499999999999999e124 or -4.9999999999999997e111 < n < -3.60000000000000007e-37 or 2.50000000000000011e41 < n < 9.99999999999999953e67 or 1.6499999999999999e131 < n

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around 0 86.7%

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

   if -1.2499999999999999e124 < n < -4.9999999999999997e111 or -3.60000000000000007e-37 < n < 2.50000000000000011e41 or 9.99999999999999953e67 < n < 1.6499999999999999e131

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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. Add Preprocessing

   5. Taylor expanded in f around inf 82.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+124}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -5 \cdot 10^{+111}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -3.6 \cdot 10^{-37}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.5 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 10^{+68}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.65 \cdot 10^{+131}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.9% 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. /-rgt-identity 99.9%

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

   2. metadata-eval 99.9%

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

   3. neg-mul-1 99.9%

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

   4. *-commutative 99.9%

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

   5. associate-/l* 99.9%

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

   6. metadata-eval 99.9%

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

   7. metadata-eval 99.9%

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

   8. associate-/r* 99.9%

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

   9. neg-mul-1 99.9%

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

   10. sub-neg 99.9%

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

   11. +-commutative 99.9%

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

   12. distribute-neg-in 99.9%

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

   13. remove-double-neg 99.9%

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

   14. 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. Add Preprocessing

5. Taylor expanded in f around inf 50.9%

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

6. Final simplification 50.9%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))