subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

Alternatives: 6

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3 \cdot 10^{+19} \lor \neg \left(f \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -3e+19) (not (<= f 3e+14)))
   (- -1.0 (/ n f))
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -3e+19) || !(f <= 3e+14)) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-3d+19)) .or. (.not. (f <= 3d+14))) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -3e+19) || !(f <= 3e+14)) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -3e+19) or not (f <= 3e+14):
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -3e+19) || !(f <= 3e+14))
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -3e+19) || ~((f <= 3e+14)))
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -3e+19], N[Not[LessEqual[f, 3e+14]], $MachinePrecision]], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -3e19 or 3e14 < f

   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. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. add-sqr-sqrt 50.6%

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

      4. associate-*r* 50.6%

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

      5. fma-def 50.6%

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

      6. associate-*l/ 50.7%

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

      7. *-un-lft-identity 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. associate-*l/ 50.7%

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

      2. *-lft-identity 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in n around 0 79.8%

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

       1. sub-neg 79.8%

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

       2. metadata-eval 79.8%

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

       3. +-commutative 79.8%

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

       4. mul-1-neg 79.8%

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

       5. sub-neg 79.8%

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

   12. Simplified 79.8%

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

   if -3e19 < f < 3e14

   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 77.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -3 \cdot 10^{+19} \lor \neg \left(f \leq 3 \cdot 10^{+14}\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 3: 75.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.02 \cdot 10^{+15} \lor \neg \left(f \leq 2100000000000\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.02e+15) (not (<= f 2100000000000.0)))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.02e+15) || !(f <= 2100000000000.0)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-1.02d+15)) .or. (.not. (f <= 2100000000000.0d0))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.02e+15) || !(f <= 2100000000000.0)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.02e+15) or not (f <= 2100000000000.0):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.02e+15) || !(f <= 2100000000000.0))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.02e+15) || ~((f <= 2100000000000.0)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.02e+15], N[Not[LessEqual[f, 2100000000000.0]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.02e15 or 2.1e12 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-/l* 100.0%

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

      4. div-sub 100.0%

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

      5. metadata-eval 100.0%

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

      6. metadata-eval 100.0%

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

      7. associate-/l* 100.0%

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

      8. *-commutative 100.0%

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

      9. neg-mul-1 100.0%

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

      10. metadata-eval 100.0%

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

      11. metadata-eval 100.0%

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

      12. associate-/l* 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. div-sub 100.0%

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

      16. unsub-neg 100.0%

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

      17. remove-double-neg 100.0%

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

      18. +-commutative 100.0%

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

      19. sub-neg 100.0%

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

      20. metadata-eval 100.0%

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

      21. /-rgt-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in n around 0 80.2%

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

   if -1.02e15 < f < 2.1e12

   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 77.7%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.02 \cdot 10^{+15} \lor \neg \left(f \leq 2100000000000\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 4: 74.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.02 \cdot 10^{+15} \lor \neg \left(f \leq 400000000000\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.02e+15) (not (<= f 400000000000.0))) (- -1.0 (/ n f)) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.02e+15) || !(f <= 400000000000.0)) {
		tmp = -1.0 - (n / f);
	} 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 <= (-1.02d+15)) .or. (.not. (f <= 400000000000.0d0))) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.02e+15) || !(f <= 400000000000.0)) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.02e+15) or not (f <= 400000000000.0):
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.02e+15) || !(f <= 400000000000.0))
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.02e+15) || ~((f <= 400000000000.0)))
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.02e+15], N[Not[LessEqual[f, 400000000000.0]], $MachinePrecision]], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if f < -1.02e15 or 4e11 < f

   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. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. distribute-lft-in 99.7%

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

      3. add-sqr-sqrt 50.6%

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

      4. associate-*r* 50.6%

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

      5. fma-def 50.6%

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

      6. associate-*l/ 50.7%

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

      7. *-un-lft-identity 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. associate-*l/ 50.7%

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

      2. *-lft-identity 50.7%

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

   9. Simplified 50.7%

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

   10. Taylor expanded in n around 0 79.8%

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

       1. sub-neg 79.8%

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

       2. metadata-eval 79.8%

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

       3. +-commutative 79.8%

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

       4. mul-1-neg 79.8%

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

       5. sub-neg 79.8%

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

   12. Simplified 79.8%

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

   if -1.02e15 < f < 4e11

   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 76.7%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.02 \cdot 10^{+15} \lor \neg \left(f \leq 400000000000\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2e+24) -1.0 (if (<= f 100000000000.0) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2e+24) {
		tmp = -1.0;
	} else if (f <= 100000000000.0) {
		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 <= (-2d+24)) then
        tmp = -1.0d0
    else if (f <= 100000000000.0d0) 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 <= -2e+24) {
		tmp = -1.0;
	} else if (f <= 100000000000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2e+24:
		tmp = -1.0
	elif f <= 100000000000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2e+24)
		tmp = -1.0;
	elseif (f <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2e+24)
		tmp = -1.0;
	elseif (f <= 100000000000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2e+24], -1.0, If[LessEqual[f, 100000000000.0], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -2e24 or 1e11 < f

   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 79.1%

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

   if -2e24 < f < 1e11

   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 76.7%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 100000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 49.2% 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 50.1%

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

5. Final simplification 50.1%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023181 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))