subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.6s

Alternatives: 5

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. /-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. neg-mul-1 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-/l* 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/r* 100.0%

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

   9. neg-mul-1 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. remove-double-neg 100.0%

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

   14. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 99.9%

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

   2. pow3 99.9%

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

6. Applied egg-rr 99.9%

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

   1. rem-cbrt-cube 100.0%

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

   2. clear-num 100.0%

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

8. Applied egg-rr 100.0%

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

   1. clear-num 99.9%

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

   2. associate-/r/ 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*l/ 100.0%

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

    2. *-un-lft-identity 100.0%

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

    3. associate-/r/ 99.7%

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

    4. +-commutative 99.7%

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

    5. distribute-lft-out 99.7%

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

    6. associate-*l/ 99.8%

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

    7. *-un-lft-identity 99.8%

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

    8. associate-*l/ 100.0%

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

    9. *-un-lft-identity 100.0%

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

12. Applied egg-rr 100.0%

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

13. Final simplification 100.0%

    \[\leadsto \frac{n}{n - f} + \frac{f}{n - f} \]
14. Add Preprocessing

Alternative 2: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3.4 \cdot 10^{-69} \lor \neg \left(f \leq 2.6 \cdot 10^{-32}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -3.4e-69) (not (<= f 2.6e-32))) (+ (* -2.0 (/ n f)) -1.0) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -3.4e-69) || !(f <= 2.6e-32)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-3.4d-69)) .or. (.not. (f <= 2.6d-32))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -3.4e-69) || !(f <= 2.6e-32)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -3.4e-69) or not (f <= 2.6e-32):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -3.4e-69) || !(f <= 2.6e-32))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -3.4e-69) || ~((f <= 2.6e-32)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -3.4e-69], N[Not[LessEqual[f, 2.6e-32]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if f < -3.40000000000000008e-69 or 2.5999999999999997e-32 < f

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. sub-neg 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. Add Preprocessing

   5. Taylor expanded in n around 0 79.6%

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

   if -3.40000000000000008e-69 < f < 2.5999999999999997e-32

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 84.0%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -3.4 \cdot 10^{-69} \lor \neg \left(f \leq 2.6 \cdot 10^{-32}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.9 \cdot 10^{-62}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 6 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -4.9e-62) -1.0 (if (<= f 6e-33) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -4.9e-62) {
		tmp = -1.0;
	} else if (f <= 6e-33) {
		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 <= (-4.9d-62)) then
        tmp = -1.0d0
    else if (f <= 6d-33) 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 <= -4.9e-62) {
		tmp = -1.0;
	} else if (f <= 6e-33) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -4.9e-62:
		tmp = -1.0
	elif f <= 6e-33:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -4.9e-62)
		tmp = -1.0;
	elseif (f <= 6e-33)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -4.9e-62)
		tmp = -1.0;
	elseif (f <= 6e-33)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -4.9e-62], -1.0, If[LessEqual[f, 6e-33], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.9000000000000004e-62 or 6.0000000000000003e-33 < f

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. sub-neg 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around inf 79.2%

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

   if -4.9000000000000004e-62 < f < 6.0000000000000003e-33

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. sub-neg 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   4. Add Preprocessing

   5. Taylor expanded in f around 0 83.4%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.9 \cdot 10^{-62}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 6 \cdot 10^{-33}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. /-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. neg-mul-1 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-/l* 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/r* 100.0%

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

   9. neg-mul-1 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. remove-double-neg 100.0%

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

   14. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
4. Add Preprocessing

5. Final simplification 100.0%

   \[\leadsto \frac{n + f}{n - f} \]
6. Add Preprocessing

Alternative 5: 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. /-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. neg-mul-1 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-/l* 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/r* 100.0%

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

   9. neg-mul-1 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. remove-double-neg 100.0%

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

   14. sub-neg 100.0%

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

3. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
4. Add Preprocessing

5. Taylor expanded in f around inf 51.0%

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

6. Final simplification 51.0%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024022 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))