subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.4s

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, 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 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

6. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

   2. associate-/r/ 99.8%

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

   3. +-commutative 99.8%

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

   4. distribute-rgt-in 99.8%

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

   5. un-div-inv 99.9%

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

   6. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.6 \cdot 10^{+110} \lor \neg \left(f \leq 4.4 \cdot 10^{+15}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -2.6e+110) (not (<= f 4.4e+15)))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -2.6e+110) || !(f <= 4.4e+15)) {
		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 <= (-2.6d+110)) .or. (.not. (f <= 4.4d+15))) 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 <= -2.6e+110) || !(f <= 4.4e+15)) {
		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 <= -2.6e+110) or not (f <= 4.4e+15):
		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 <= -2.6e+110) || !(f <= 4.4e+15))
		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 <= -2.6e+110) || ~((f <= 4.4e+15)))
		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, -2.6e+110], N[Not[LessEqual[f, 4.4e+15]], $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 < -2.6e110 or 4.4e15 < f

   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 n around 0 82.0%

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

   if -2.6e110 < f < 4.4e15

   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 0 75.6%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.6 \cdot 10^{+110} \lor \neg \left(f \leq 4.4 \cdot 10^{+15}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+121}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -4.2e+121) -1.0 (if (<= f 1.9e+16) (+ 1.0 (* 2.0 (/ f n))) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -4.2e+121) {
		tmp = -1.0;
	} else if (f <= 1.9e+16) {
		tmp = 1.0 + (2.0 * (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 <= (-4.2d+121)) then
        tmp = -1.0d0
    else if (f <= 1.9d+16) then
        tmp = 1.0d0 + (2.0d0 * (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 <= -4.2e+121) {
		tmp = -1.0;
	} else if (f <= 1.9e+16) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -4.2e+121:
		tmp = -1.0
	elif f <= 1.9e+16:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -4.2e+121)
		tmp = -1.0;
	elseif (f <= 1.9e+16)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -4.2e+121)
		tmp = -1.0;
	elseif (f <= 1.9e+16)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -4.2e+121], -1.0, If[LessEqual[f, 1.9e+16], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.2000000000000003e121 or 1.9e16 < f

   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 81.9%

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

   if -4.2000000000000003e121 < f < 1.9e16

   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 0 75.3%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+121}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.9 \cdot 10^{+16}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+109}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 11500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2e+109) -1.0 (if (<= f 11500.0) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2e+109) {
		tmp = -1.0;
	} else if (f <= 11500.0) {
		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+109)) then
        tmp = -1.0d0
    else if (f <= 11500.0d0) 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+109) {
		tmp = -1.0;
	} else if (f <= 11500.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2e+109:
		tmp = -1.0
	elif f <= 11500.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2e+109)
		tmp = -1.0;
	elseif (f <= 11500.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2e+109)
		tmp = -1.0;
	elseif (f <= 11500.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2e+109], -1.0, If[LessEqual[f, 11500.0], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.99999999999999996e109 or 11500 < f

   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 80.0%

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

   if -1.99999999999999996e109 < f < 11500

   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 0 74.8%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+109}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 11500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / N[(n + f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. /-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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

6. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

   2. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \frac{1}{\frac{n - f}{n + f}} \]
10. Add Preprocessing

Alternative 6: 100.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. /-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. Final simplification 99.9%

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

Alternative 7: 49.8% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

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

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. /-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 46.4%

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

6. Final simplification 46.4%

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

Reproduce

herbie shell --seed 2024014 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))