subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 7

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.55 \cdot 10^{-24} \lor \neg \left(f \leq 1.15 \cdot 10^{+49}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.55e-24) (not (<= f 1.15e+49)))
   (+ (* -2.0 (/ n f)) -1.0)
   (/ (- n) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.55e-24) || !(f <= 1.15e+49)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} 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 ((f <= (-1.55d-24)) .or. (.not. (f <= 1.15d+49))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = -n / (f - n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.55e-24) || !(f <= 1.15e+49)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.55e-24) or not (f <= 1.15e+49):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = -n / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.55e-24) || !(f <= 1.15e+49))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(Float64(-n) / Float64(f - n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.55e-24) || ~((f <= 1.15e+49)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = -n / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.55e-24], N[Not[LessEqual[f, 1.15e+49]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[((-n) / N[(f - n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.55e-24 or 1.15000000000000001e49 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in n around 0 85.6%

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

   if -1.55e-24 < f < 1.15000000000000001e49

   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-+ 62.2%

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

      4. associate-/r/ 62.2%

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

      5. unpow-prod-down 62.0%

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

      6. inv-pow 62.0%

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

   5. Applied egg-rr 62.0%

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

      1. associate-*r/ 61.9%

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

      2. *-rgt-identity 61.9%

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

      3. unpow-1 61.9%

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

      4. associate-/r/ 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in n around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   10. Simplified 77.8%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.55 \cdot 10^{-24} \lor \neg \left(f \leq 1.15 \cdot 10^{+49}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \]

Alternative 3: 73.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.56 \cdot 10^{-24} \lor \neg \left(f \leq 6.4 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.56e-24) (not (<= f 6.4e+47)))
   (/ (- f) (- f n))
   (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.56e-24) || !(f <= 6.4e+47)) {
		tmp = -f / (f - n);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-1.56d-24)) .or. (.not. (f <= 6.4d+47))) then
        tmp = -f / (f - n)
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.56e-24) || !(f <= 6.4e+47)) {
		tmp = -f / (f - n);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.56e-24) or not (f <= 6.4e+47):
		tmp = -f / (f - n)
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.56e-24) || !(f <= 6.4e+47))
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.56e-24) || ~((f <= 6.4e+47)))
		tmp = -f / (f - n);
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.56e-24], N[Not[LessEqual[f, 6.4e+47]], $MachinePrecision]], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.56e-24 or 6.4e47 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. 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-+ 43.0%

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

      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 42.9%

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

      6. inv-pow 42.9%

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

   5. Applied egg-rr 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-rgt-identity 42.9%

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

      3. unpow-1 42.9%

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

      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 0 85.1%

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

      1. neg-mul-1 85.1%

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

   10. Simplified 85.1%

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

   if -1.56e-24 < f < 6.4e47

   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-+ 62.2%

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

      4. associate-/r/ 62.2%

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

      5. unpow-prod-down 62.0%

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

      6. inv-pow 62.0%

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

   5. Applied egg-rr 62.0%

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

      1. associate-*r/ 61.9%

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

      2. *-rgt-identity 61.9%

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

      3. unpow-1 61.9%

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

      4. associate-/r/ 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in n around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   10. Simplified 77.8%

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

   11. Taylor expanded in n around inf 77.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.56 \cdot 10^{-24} \lor \neg \left(f \leq 6.4 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]

Alternative 4: 74.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.8 \cdot 10^{-25} \lor \neg \left(f \leq 6.8 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -2.8e-25) (not (<= f 6.8e+47)))
   (/ (- f) (- f n))
   (/ (- n) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -2.8e-25) || !(f <= 6.8e+47)) {
		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 ((f <= (-2.8d-25)) .or. (.not. (f <= 6.8d+47))) 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 ((f <= -2.8e-25) || !(f <= 6.8e+47)) {
		tmp = -f / (f - n);
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -2.8e-25) or not (f <= 6.8e+47):
		tmp = -f / (f - n)
	else:
		tmp = -n / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -2.8e-25) || !(f <= 6.8e+47))
		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 ((f <= -2.8e-25) || ~((f <= 6.8e+47)))
		tmp = -f / (f - n);
	else
		tmp = -n / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -2.8e-25], N[Not[LessEqual[f, 6.8e+47]], $MachinePrecision]], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], N[((-n) / N[(f - n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -2.79999999999999988e-25 or 6.7999999999999996e47 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. 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-+ 43.0%

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

      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 42.9%

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

      6. inv-pow 42.9%

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

   5. Applied egg-rr 42.9%

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

      1. associate-*r/ 42.9%

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

      2. *-rgt-identity 42.9%

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

      3. unpow-1 42.9%

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

      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 0 85.1%

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

      1. neg-mul-1 85.1%

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

   10. Simplified 85.1%

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

   if -2.79999999999999988e-25 < f < 6.7999999999999996e47

   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-+ 62.2%

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

      4. associate-/r/ 62.2%

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

      5. unpow-prod-down 62.0%

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

      6. inv-pow 62.0%

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

   5. Applied egg-rr 62.0%

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

      1. associate-*r/ 61.9%

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

      2. *-rgt-identity 61.9%

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

      3. unpow-1 61.9%

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

      4. associate-/r/ 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in n around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   10. Simplified 77.8%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.8 \cdot 10^{-25} \lor \neg \left(f \leq 6.8 \cdot 10^{+47}\right):\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \]

Alternative 5: 73.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.6 \cdot 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.6e-24) -1.0 (if (<= f 1.45e+48) (+ 1.0 (/ f n)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.6e-24) {
		tmp = -1.0;
	} else if (f <= 1.45e+48) {
		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 (f <= (-1.6d-24)) then
        tmp = -1.0d0
    else if (f <= 1.45d+48) 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 (f <= -1.6e-24) {
		tmp = -1.0;
	} else if (f <= 1.45e+48) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.6e-24:
		tmp = -1.0
	elif f <= 1.45e+48:
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.6e-24)
		tmp = -1.0;
	elseif (f <= 1.45e+48)
		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 (f <= -1.6e-24)
		tmp = -1.0;
	elseif (f <= 1.45e+48)
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.6e-24], -1.0, If[LessEqual[f, 1.45e+48], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.60000000000000006e-24 or 1.4499999999999999e48 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 84.7%

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

   if -1.60000000000000006e-24 < f < 1.4499999999999999e48

   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-+ 62.2%

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

      4. associate-/r/ 62.2%

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

      5. unpow-prod-down 62.0%

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

      6. inv-pow 62.0%

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

   5. Applied egg-rr 62.0%

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

      1. associate-*r/ 61.9%

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

      2. *-rgt-identity 61.9%

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

      3. unpow-1 61.9%

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

      4. associate-/r/ 62.0%

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

   7. Simplified 62.0%

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

   8. Taylor expanded in n around inf 77.8%

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

      1. neg-mul-1 77.8%

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

   10. Simplified 77.8%

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

   11. Taylor expanded in n around inf 77.5%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.6 \cdot 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.45 \cdot 10^{+48}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 73.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 7 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.35e-25) -1.0 (if (<= f 7e+47) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.35e-25) {
		tmp = -1.0;
	} else if (f <= 7e+47) {
		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 (f <= (-1.35d-25)) then
        tmp = -1.0d0
    else if (f <= 7d+47) 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 (f <= -1.35e-25) {
		tmp = -1.0;
	} else if (f <= 7e+47) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.35e-25:
		tmp = -1.0
	elif f <= 7e+47:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.35e-25)
		tmp = -1.0;
	elseif (f <= 7e+47)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.35e-25)
		tmp = -1.0;
	elseif (f <= 7e+47)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.35e-25], -1.0, If[LessEqual[f, 7e+47], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.35000000000000008e-25 or 7.00000000000000031e47 < f

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-/l* 99.9%

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

      4. div-sub 99.9%

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

      5. metadata-eval 99.9%

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

      6. metadata-eval 99.9%

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

      7. associate-/l* 99.9%

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

      8. *-commutative 99.9%

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

      9. neg-mul-1 99.9%

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

      10. metadata-eval 99.9%

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

      11. metadata-eval 99.9%

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

      12. associate-/l* 99.9%

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

      13. *-commutative 99.9%

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

      14. neg-mul-1 99.9%

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

      15. div-sub 99.9%

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

      16. unsub-neg 99.9%

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

      17. remove-double-neg 99.9%

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

      18. +-commutative 99.9%

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

      19. sub-neg 99.9%

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

      20. metadata-eval 99.9%

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

      21. /-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around inf 84.7%

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

   if -1.35000000000000008e-25 < f < 7.00000000000000031e47

   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 77.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 7 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 7: 49.6% 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 52.0%

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

5. Final simplification 52.0%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023201 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))