subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 9

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, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / 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. inv-pow 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. +-commutative 99.9%

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

7. Applied egg-rr 99.9%

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

8. Final simplification 99.9%

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

Alternative 2: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+23} \lor \neg \left(n \leq 4.8 \cdot 10^{+23}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-f}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -2.1e+23) (not (<= n 4.8e+23)))
   (+ 1.0 (* 2.0 (/ f n)))
   (/ (- f) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -2.1e+23) || !(n <= 4.8e+23)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -f / (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 ((n <= (-2.1d+23)) .or. (.not. (n <= 4.8d+23))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = -f / (f - n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -2.1e+23) || !(n <= 4.8e+23)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -f / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -2.1e+23) or not (n <= 4.8e+23):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -f / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -2.1e+23) || !(n <= 4.8e+23))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(Float64(-f) / Float64(f - n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -2.1e+23) || ~((n <= 4.8e+23)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -f / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -2.1e+23], N[Not[LessEqual[n, 4.8e+23]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -2.1000000000000001e23 or 4.8e23 < 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 79.3%

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

   if -2.1000000000000001e23 < n < 4.8e23

   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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 63.0%

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

      4. associate-/r/ 62.9%

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

      5. unpow-prod-down 62.7%

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

      6. inv-pow 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. associate-*r/ 62.8%

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

      2. *-rgt-identity 62.8%

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

      3. unpow-1 62.8%

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

      4. associate-/r/ 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in n around 0 77.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   9. Step-by-step derivation

      1. mul-1-neg 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.1 \cdot 10^{+23} \lor \neg \left(n \leq 4.8 \cdot 10^{+23}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-f}{f - n}\\ \end{array} \]

Alternative 3: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+24} \lor \neg \left(n \leq 6.2 \cdot 10^{+23}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-f}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -1.8e+24) (not (<= n 6.2e+23)))
   (+ 1.0 (/ f n))
   (/ (- f) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -1.8e+24) || !(n <= 6.2e+23)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -f / (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 ((n <= (-1.8d+24)) .or. (.not. (n <= 6.2d+23))) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = -f / (f - n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -1.8e+24) || !(n <= 6.2e+23)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -f / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -1.8e+24) or not (n <= 6.2e+23):
		tmp = 1.0 + (f / n)
	else:
		tmp = -f / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -1.8e+24) || !(n <= 6.2e+23))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = Float64(Float64(-f) / Float64(f - n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -1.8e+24) || ~((n <= 6.2e+23)))
		tmp = 1.0 + (f / n);
	else
		tmp = -f / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -1.8e+24], N[Not[LessEqual[n, 6.2e+23]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.79999999999999992e24 or 6.19999999999999941e23 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. flip-+ 42.0%

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

      4. associate-/r/ 42.0%

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

      5. unpow-prod-down 41.9%

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

      6. inv-pow 41.9%

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

   5. Applied egg-rr 41.9%

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

      1. associate-*r/ 41.8%

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

      2. *-rgt-identity 41.8%

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

      3. unpow-1 41.8%

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

      4. associate-/r/ 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in n around inf 78.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   9. Step-by-step derivation

      1. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in n around inf 78.4%

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

   if -1.79999999999999992e24 < n < 6.19999999999999941e23

   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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 63.0%

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

      4. associate-/r/ 62.9%

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

      5. unpow-prod-down 62.7%

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

      6. inv-pow 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. associate-*r/ 62.8%

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

      2. *-rgt-identity 62.8%

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

      3. unpow-1 62.8%

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

      4. associate-/r/ 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in n around 0 77.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   9. Step-by-step derivation

      1. mul-1-neg 77.7%

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

   10. Simplified 77.7%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+24} \lor \neg \left(n \leq 6.2 \cdot 10^{+23}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-f}{f - n}\\ \end{array} \]

Alternative 4: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.5e+24)
   (+ 1.0 (/ f n))
   (if (<= n 3.1e+23) (/ (- f) (- f n)) (/ (- n) (- f n)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.5e+24) {
		tmp = 1.0 + (f / n);
	} else if (n <= 3.1e+23) {
		tmp = -f / (f - n);
	} else {
		tmp = -n / (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 (n <= (-3.5d+24)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 3.1d+23) then
        tmp = -f / (f - n)
    else
        tmp = -n / (f - n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.5e+24) {
		tmp = 1.0 + (f / n);
	} else if (n <= 3.1e+23) {
		tmp = -f / (f - n);
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.5e+24:
		tmp = 1.0 + (f / n)
	elif n <= 3.1e+23:
		tmp = -f / (f - n)
	else:
		tmp = -n / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.5e+24)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 3.1e+23)
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = Float64(Float64(-n) / Float64(f - n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.5e+24)
		tmp = 1.0 + (f / n);
	elseif (n <= 3.1e+23)
		tmp = -f / (f - n);
	else
		tmp = -n / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.5e+24], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.1e+23], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], N[((-n) / N[(f - n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.5000000000000002e24

   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 100.0%

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

      2. inv-pow 100.0%

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

      3. flip-+ 43.1%

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

      4. associate-/r/ 43.0%

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

      5. unpow-prod-down 42.8%

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

      6. inv-pow 42.8%

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

   5. Applied egg-rr 42.8%

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

      1. associate-*r/ 42.8%

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

      2. *-rgt-identity 42.8%

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

      3. unpow-1 42.8%

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

      4. associate-/r/ 42.8%

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

   7. Simplified 42.8%

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

   8. Taylor expanded in n around inf 77.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   9. Step-by-step derivation

      1. neg-mul-1 77.2%

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

   10. Simplified 77.2%

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

   11. Taylor expanded in n around inf 77.7%

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

   if -3.5000000000000002e24 < n < 3.09999999999999971e23

   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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 63.0%

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

      4. associate-/r/ 62.9%

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

      5. unpow-prod-down 62.7%

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

      6. inv-pow 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. associate-*r/ 62.8%

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

      2. *-rgt-identity 62.8%

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

      3. unpow-1 62.8%

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

      4. associate-/r/ 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in n around 0 77.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   9. Step-by-step derivation

      1. mul-1-neg 77.7%

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

   10. Simplified 77.7%

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

   if 3.09999999999999971e23 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. flip-+ 41.0%

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

      4. associate-/r/ 41.0%

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

      5. unpow-prod-down 40.8%

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

      6. inv-pow 40.8%

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

   5. Applied egg-rr 40.8%

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

      1. associate-*r/ 40.8%

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

      2. *-rgt-identity 40.8%

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

      3. unpow-1 40.8%

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

      4. associate-/r/ 40.8%

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

   7. Simplified 40.8%

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

   8. Taylor expanded in n around inf 79.3%

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   9. Step-by-step derivation

      1. neg-mul-1 79.3%

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

   10. Simplified 79.3%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.5 \cdot 10^{+24}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 3.1 \cdot 10^{+23}:\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \]

Alternative 5: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+22} \lor \neg \left(n \leq 1.16 \cdot 10^{+24}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -2.3e+22) (not (<= n 1.16e+24))) (+ 1.0 (/ f n)) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -2.3e+22) || !(n <= 1.16e+24)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-2.3d+22)) .or. (.not. (n <= 1.16d+24))) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -2.3e+22) || !(n <= 1.16e+24)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -2.3e+22) or not (n <= 1.16e+24):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -2.3e+22) || !(n <= 1.16e+24))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -2.3e+22) || ~((n <= 1.16e+24)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -2.3e+22], N[Not[LessEqual[n, 1.16e+24]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -2.3000000000000002e22 or 1.16000000000000005e24 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. flip-+ 42.0%

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

      4. associate-/r/ 42.0%

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

      5. unpow-prod-down 41.9%

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

      6. inv-pow 41.9%

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

   5. Applied egg-rr 41.9%

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

      1. associate-*r/ 41.8%

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

      2. *-rgt-identity 41.8%

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

      3. unpow-1 41.8%

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

      4. associate-/r/ 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in n around inf 78.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   9. Step-by-step derivation

      1. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in n around inf 78.4%

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

   if -2.3000000000000002e22 < n < 1.16000000000000005e24

   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 76.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -2.3 \cdot 10^{+22} \lor \neg \left(n \leq 1.16 \cdot 10^{+24}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+23} \lor \neg \left(n \leq 6.8 \cdot 10^{+25}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -1.25e+23) (not (<= n 6.8e+25)))
   (+ 1.0 (/ f n))
   (- -1.0 (/ n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -1.25e+23) || !(n <= 6.8e+25)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	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+23)) .or. (.not. (n <= 6.8d+25))) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = (-1.0d0) - (n / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -1.25e+23) || !(n <= 6.8e+25)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -1.25e+23) or not (n <= 6.8e+25):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -1.25e+23) || !(n <= 6.8e+25))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -1.25e+23) || ~((n <= 6.8e+25)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -1.25e+23], N[Not[LessEqual[n, 6.8e+25]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.25e23 or 6.79999999999999967e25 < 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. Step-by-step derivation

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

      3. flip-+ 42.0%

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

      4. associate-/r/ 42.0%

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

      5. unpow-prod-down 41.9%

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

      6. inv-pow 41.9%

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

   5. Applied egg-rr 41.9%

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

      1. associate-*r/ 41.8%

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

      2. *-rgt-identity 41.8%

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

      3. unpow-1 41.8%

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

      4. associate-/r/ 41.9%

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

   7. Simplified 41.9%

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

   8. Taylor expanded in n around inf 78.2%

      \[\leadsto \frac{\color{blue}{-1 \cdot n}}{f - n} \]
   9. Step-by-step derivation

      1. neg-mul-1 78.2%

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

   10. Simplified 78.2%

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

   11. Taylor expanded in n around inf 78.4%

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

   if -1.25e23 < n < 6.79999999999999967e25

   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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

      3. flip-+ 63.0%

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

      4. associate-/r/ 62.9%

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

      5. unpow-prod-down 62.7%

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

      6. inv-pow 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. associate-*r/ 62.8%

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

      2. *-rgt-identity 62.8%

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

      3. unpow-1 62.8%

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

      4. associate-/r/ 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in n around 0 77.7%

      \[\leadsto \frac{\color{blue}{-1 \cdot f}}{f - n} \]
   9. Step-by-step derivation

      1. mul-1-neg 77.7%

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

   10. Simplified 77.7%

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

   11. Taylor expanded in f around inf 77.4%

       \[\leadsto \color{blue}{-1 \cdot \frac{n}{f} - 1} \]
   12. Step-by-step derivation

       1. fma-neg 77.4%

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

       2. metadata-eval 77.4%

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

       3. *-rgt-identity 77.4%

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

       4. metadata-eval 77.4%

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

       5. rem-square-sqrt 0.0%

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

       6. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{n}{f \cdot \left(\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot -1\right)}, -1\right) \]

       7. rem-square-sqrt 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{n}{f \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)}\right)}, -1\right) \]

       8. unpow2 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{n}{f \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{2}}\right)}, -1\right) \]

       9. pow-sqr 0.0%

          \[\leadsto \mathsf{fma}\left(-1, \frac{n}{f \cdot \color{blue}{{\left(\sqrt{-1}\right)}^{\left(2 \cdot 2\right)}}}, -1\right) \]

       10. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{n}{f \cdot {\left(\sqrt{-1}\right)}^{\color{blue}{4}}}, -1\right) \]

       11. *-commutative 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{n}{\color{blue}{{\left(\sqrt{-1}\right)}^{4} \cdot f}}, -1\right) \]

       12. metadata-eval 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}, \color{blue}{\frac{1}{-1}}\right) \]

       13. rem-square-sqrt 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}, \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}}\right) \]

       14. unpow2 0.0%

           \[\leadsto \mathsf{fma}\left(-1, \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}, \frac{1}{\color{blue}{{\left(\sqrt{-1}\right)}^{2}}}\right) \]

       15. fma-def 0.0%

           \[\leadsto \color{blue}{-1 \cdot \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f} + \frac{1}{{\left(\sqrt{-1}\right)}^{2}}} \]

       16. +-commutative 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} + -1 \cdot \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}} \]

       17. mul-1-neg 0.0%

           \[\leadsto \frac{1}{{\left(\sqrt{-1}\right)}^{2}} + \color{blue}{\left(-\frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}\right)} \]

       18. unsub-neg 0.0%

           \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt{-1}\right)}^{2}} - \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f}} \]

       19. unpow2 0.0%

           \[\leadsto \frac{1}{\color{blue}{\sqrt{-1} \cdot \sqrt{-1}}} - \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f} \]

       20. rem-square-sqrt 0.0%

           \[\leadsto \frac{1}{\color{blue}{-1}} - \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f} \]

       21. metadata-eval 0.0%

           \[\leadsto \color{blue}{-1} - \frac{n}{{\left(\sqrt{-1}\right)}^{4} \cdot f} \]

   13. Simplified 77.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.25 \cdot 10^{+23} \lor \neg \left(n \leq 6.8 \cdot 10^{+25}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]

Alternative 7: 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 8: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.8e+24) 1.0 (if (<= n 2e+27) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.8e+24) {
		tmp = 1.0;
	} else if (n <= 2e+27) {
		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 <= (-1.8d+24)) then
        tmp = 1.0d0
    else if (n <= 2d+27) 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 <= -1.8e+24) {
		tmp = 1.0;
	} else if (n <= 2e+27) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.8e+24:
		tmp = 1.0
	elif n <= 2e+27:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.8e+24)
		tmp = 1.0;
	elseif (n <= 2e+27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.8e+24)
		tmp = 1.0;
	elseif (n <= 2e+27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.8e+24], 1.0, If[LessEqual[n, 2e+27], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.79999999999999992e24 or 2e27 < 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 77.7%

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

   if -1.79999999999999992e24 < n < 2e27

   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 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1.8 \cdot 10^{+24}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 50.1% 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 48.5%

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

5. Final simplification 48.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023172 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))