subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 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 99.9%

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

   2. associate-/r/ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. add-cbrt-cube 32.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{1}{n - f} \cdot \frac{1}{n - f}\right) \cdot \frac{1}{n - f}}} \cdot \left(f + n\right) \]

   2. pow3 32.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{n - f}\right)}^{3}}} \cdot \left(f + n\right) \]

7. Applied egg-rr 32.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{1}{n - f}\right)}^{3}}} \cdot \left(f + n\right) \]
8. Step-by-step derivation

   1. rem-cbrt-cube 99.7%

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

   2. distribute-lft-in 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-*l/ 99.8%

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

   5. *-un-lft-identity 99.8%

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

   6. associate-*l/ 100.0%

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

   7. *-un-lft-identity 100.0%

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

9. Applied egg-rr 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 73.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -8.4 \cdot 10^{+94} \lor \neg \left(f \leq -14.5 \lor \neg \left(f \leq -5.5 \cdot 10^{-64}\right) \land f \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -8.4e+94)
         (not (or (<= f -14.5) (and (not (<= f -5.5e-64)) (<= f 4e-12)))))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -8.4e+94) || !((f <= -14.5) || (!(f <= -5.5e-64) && (f <= 4e-12)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-8.4d+94)) .or. (.not. (f <= (-14.5d0)) .or. (.not. (f <= (-5.5d-64))) .and. (f <= 4d-12))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -8.4e+94) || !((f <= -14.5) || (!(f <= -5.5e-64) && (f <= 4e-12)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -8.4e+94) or not ((f <= -14.5) or (not (f <= -5.5e-64) and (f <= 4e-12))):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -8.4e+94) || !((f <= -14.5) || (!(f <= -5.5e-64) && (f <= 4e-12))))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -8.4e+94) || ~(((f <= -14.5) || (~((f <= -5.5e-64)) && (f <= 4e-12)))))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -8.4e+94], N[Not[Or[LessEqual[f, -14.5], And[N[Not[LessEqual[f, -5.5e-64]], $MachinePrecision], LessEqual[f, 4e-12]]]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -8.39999999999999958e94 or -14.5 < f < -5.4999999999999999e-64 or 3.99999999999999992e-12 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 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 78.6%

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

   if -8.39999999999999958e94 < f < -14.5 or -5.4999999999999999e-64 < f < 3.99999999999999992e-12

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. div-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. div-sub 100.0%

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

      16. unsub-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

      19. sub-neg 100.0%

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

      20. metadata-eval 100.0%

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

      21. /-rgt-identity 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%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -8.4 \cdot 10^{+94} \lor \neg \left(f \leq -14.5 \lor \neg \left(f \leq -5.5 \cdot 10^{-64}\right) \land f \leq 4 \cdot 10^{-12}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 3: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -6000 \lor \neg \left(f \leq -7 \cdot 10^{-64}\right) \land f \leq 5 \cdot 10^{-6}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.1e+136)
   -1.0
   (if (or (<= f -6000.0) (and (not (<= f -7e-64)) (<= f 5e-6)))
     (+ 1.0 (* 2.0 (/ f n)))
     -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.1e+136) {
		tmp = -1.0;
	} else if ((f <= -6000.0) || (!(f <= -7e-64) && (f <= 5e-6))) {
		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 (f <= (-1.1d+136)) then
        tmp = -1.0d0
    else if ((f <= (-6000.0d0)) .or. (.not. (f <= (-7d-64))) .and. (f <= 5d-6)) 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 (f <= -1.1e+136) {
		tmp = -1.0;
	} else if ((f <= -6000.0) || (!(f <= -7e-64) && (f <= 5e-6))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.1e+136:
		tmp = -1.0
	elif (f <= -6000.0) or (not (f <= -7e-64) and (f <= 5e-6)):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.1e+136)
		tmp = -1.0;
	elseif ((f <= -6000.0) || (!(f <= -7e-64) && (f <= 5e-6)))
		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 (f <= -1.1e+136)
		tmp = -1.0;
	elseif ((f <= -6000.0) || (~((f <= -7e-64)) && (f <= 5e-6)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.1e+136], -1.0, If[Or[LessEqual[f, -6000.0], And[N[Not[LessEqual[f, -7e-64]], $MachinePrecision], LessEqual[f, 5e-6]]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.1e136 or -6e3 < f < -7.0000000000000006e-64 or 5.00000000000000041e-6 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 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 79.2%

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

   if -1.1e136 < f < -6e3 or -7.0000000000000006e-64 < f < 5.00000000000000041e-6

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. div-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. div-sub 100.0%

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

      16. unsub-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

      19. sub-neg 100.0%

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

      20. metadata-eval 100.0%

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

      21. /-rgt-identity 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 79.6%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.1 \cdot 10^{+136}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -6000 \lor \neg \left(f \leq -7 \cdot 10^{-64}\right) \land f \leq 5 \cdot 10^{-6}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(N[(n + f), $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. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 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%

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

Alternative 5: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2e-51) 1.0 (if (<= n 3.5e-46) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2e-51) {
		tmp = 1.0;
	} else if (n <= 3.5e-46) {
		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 <= (-2d-51)) then
        tmp = 1.0d0
    else if (n <= 3.5d-46) 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 <= -2e-51) {
		tmp = 1.0;
	} else if (n <= 3.5e-46) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2e-51:
		tmp = 1.0
	elif n <= 3.5e-46:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2e-51)
		tmp = 1.0;
	elseif (n <= 3.5e-46)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2e-51)
		tmp = 1.0;
	elseif (n <= 3.5e-46)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2e-51], 1.0, If[LessEqual[n, 3.5e-46], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -2e-51 or 3.5000000000000002e-46 < n

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 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 0 75.9%

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

   if -2e-51 < n < 3.5000000000000002e-46

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. div-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. div-sub 100.0%

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

      16. unsub-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

      19. sub-neg 100.0%

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

      20. metadata-eval 100.0%

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

      21. /-rgt-identity 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 76.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2 \cdot 10^{-51}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.5 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 49.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

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

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. neg-mul-1 99.9%

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

   2. *-commutative 99.9%

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

   3. associate-/l* 99.9%

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

   4. div-sub 99.9%

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

   5. metadata-eval 99.9%

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

   6. metadata-eval 99.9%

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

   7. associate-/l* 99.9%

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

   8. *-commutative 99.9%

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

   9. neg-mul-1 99.9%

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

   10. metadata-eval 99.9%

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

   11. metadata-eval 99.9%

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

   12. associate-/l* 99.9%

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

   13. *-commutative 99.9%

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

   14. neg-mul-1 99.9%

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

   15. div-sub 99.9%

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

   16. unsub-neg 99.9%

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

   17. remove-double-neg 99.9%

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

   18. +-commutative 99.9%

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

   19. sub-neg 99.9%

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

   20. metadata-eval 99.9%

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

   21. /-rgt-identity 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 46.9%

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

5. Final simplification 46.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023258 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))