subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 2.2s

Alternatives: 4

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. 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. Final simplification 100.0%

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

Alternative 2: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -110000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-59} \lor \neg \left(n \leq 3.2 \cdot 10^{-22}\right) \land n \leq 9 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -110000000000.0)
   1.0
   (if (or (<= n 2.15e-59) (and (not (<= n 3.2e-22)) (<= n 9e+62)))
     (+ (* -2.0 (/ n f)) -1.0)
     1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -110000000000.0) {
		tmp = 1.0;
	} else if ((n <= 2.15e-59) || (!(n <= 3.2e-22) && (n <= 9e+62))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-110000000000.0d0)) then
        tmp = 1.0d0
    else if ((n <= 2.15d-59) .or. (.not. (n <= 3.2d-22)) .and. (n <= 9d+62)) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -110000000000.0) {
		tmp = 1.0;
	} else if ((n <= 2.15e-59) || (!(n <= 3.2e-22) && (n <= 9e+62))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -110000000000.0:
		tmp = 1.0
	elif (n <= 2.15e-59) or (not (n <= 3.2e-22) and (n <= 9e+62)):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -110000000000.0)
		tmp = 1.0;
	elseif ((n <= 2.15e-59) || (!(n <= 3.2e-22) && (n <= 9e+62)))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -110000000000.0)
		tmp = 1.0;
	elseif ((n <= 2.15e-59) || (~((n <= 3.2e-22)) && (n <= 9e+62)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -110000000000.0], 1.0, If[Or[LessEqual[n, 2.15e-59], And[N[Not[LessEqual[n, 3.2e-22]], $MachinePrecision], LessEqual[n, 9e+62]]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.1e11 or 2.1500000000000001e-59 < n < 3.19999999999999987e-22 or 8.99999999999999997e62 < n

   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 83.0%

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

   if -1.1e11 < n < 2.1500000000000001e-59 or 3.19999999999999987e-22 < n < 8.99999999999999997e62

   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 n around 0 79.1%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -110000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.15 \cdot 10^{-59} \lor \neg \left(n \leq 3.2 \cdot 10^{-22}\right) \land n \leq 9 \cdot 10^{+62}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3: 71.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -45000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-82}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.95 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -45000000000.0)
   1.0
   (if (<= n -1.5e-82)
     -1.0
     (if (<= n -1.3e-110)
       1.0
       (if (<= n 4.8e-83)
         -1.0
         (if (<= n 1.5e-21) 1.0 (if (<= n 2.95e+41) -1.0 1.0)))))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -45000000000.0) {
		tmp = 1.0;
	} else if (n <= -1.5e-82) {
		tmp = -1.0;
	} else if (n <= -1.3e-110) {
		tmp = 1.0;
	} else if (n <= 4.8e-83) {
		tmp = -1.0;
	} else if (n <= 1.5e-21) {
		tmp = 1.0;
	} else if (n <= 2.95e+41) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-45000000000.0d0)) then
        tmp = 1.0d0
    else if (n <= (-1.5d-82)) then
        tmp = -1.0d0
    else if (n <= (-1.3d-110)) then
        tmp = 1.0d0
    else if (n <= 4.8d-83) then
        tmp = -1.0d0
    else if (n <= 1.5d-21) then
        tmp = 1.0d0
    else if (n <= 2.95d+41) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -45000000000.0) {
		tmp = 1.0;
	} else if (n <= -1.5e-82) {
		tmp = -1.0;
	} else if (n <= -1.3e-110) {
		tmp = 1.0;
	} else if (n <= 4.8e-83) {
		tmp = -1.0;
	} else if (n <= 1.5e-21) {
		tmp = 1.0;
	} else if (n <= 2.95e+41) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -45000000000.0:
		tmp = 1.0
	elif n <= -1.5e-82:
		tmp = -1.0
	elif n <= -1.3e-110:
		tmp = 1.0
	elif n <= 4.8e-83:
		tmp = -1.0
	elif n <= 1.5e-21:
		tmp = 1.0
	elif n <= 2.95e+41:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -45000000000.0)
		tmp = 1.0;
	elseif (n <= -1.5e-82)
		tmp = -1.0;
	elseif (n <= -1.3e-110)
		tmp = 1.0;
	elseif (n <= 4.8e-83)
		tmp = -1.0;
	elseif (n <= 1.5e-21)
		tmp = 1.0;
	elseif (n <= 2.95e+41)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -45000000000.0)
		tmp = 1.0;
	elseif (n <= -1.5e-82)
		tmp = -1.0;
	elseif (n <= -1.3e-110)
		tmp = 1.0;
	elseif (n <= 4.8e-83)
		tmp = -1.0;
	elseif (n <= 1.5e-21)
		tmp = 1.0;
	elseif (n <= 2.95e+41)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -45000000000.0], 1.0, If[LessEqual[n, -1.5e-82], -1.0, If[LessEqual[n, -1.3e-110], 1.0, If[LessEqual[n, 4.8e-83], -1.0, If[LessEqual[n, 1.5e-21], 1.0, If[LessEqual[n, 2.95e+41], -1.0, 1.0]]]]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.5e10 or -1.4999999999999999e-82 < n < -1.29999999999999995e-110 or 4.8000000000000002e-83 < n < 1.49999999999999996e-21 or 2.95e41 < n

   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 80.9%

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

   if -4.5e10 < n < -1.4999999999999999e-82 or -1.29999999999999995e-110 < n < 4.8000000000000002e-83 or 1.49999999999999996e-21 < n < 2.95e41

   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 84.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -45000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.5 \cdot 10^{-82}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -1.3 \cdot 10^{-110}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.95 \cdot 10^{+41}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 50.5% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

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

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. 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 46.3%

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

5. Final simplification 46.3%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023175 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))