subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 2.3s

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, 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. sub-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

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

   11. +-commutative 100.0%

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

   12. 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. Final simplification 100.0%

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

Alternative 2: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{-24} \lor \neg \left(n \leq 6.5 \cdot 10^{-16} \lor \neg \left(n \leq 6 \cdot 10^{+17}\right) \land n \leq 1.1 \cdot 10^{+50}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -4.7e-24)
         (not (or (<= n 6.5e-16) (and (not (<= n 6e+17)) (<= n 1.1e+50)))))
   (+ 1.0 (* 2.0 (/ f n)))
   -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -4.7e-24) || !((n <= 6.5e-16) || (!(n <= 6e+17) && (n <= 1.1e+50)))) {
		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 ((n <= (-4.7d-24)) .or. (.not. (n <= 6.5d-16) .or. (.not. (n <= 6d+17)) .and. (n <= 1.1d+50))) 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 ((n <= -4.7e-24) || !((n <= 6.5e-16) || (!(n <= 6e+17) && (n <= 1.1e+50)))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -4.7e-24) or not ((n <= 6.5e-16) or (not (n <= 6e+17) and (n <= 1.1e+50))):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -4.7e-24) || !((n <= 6.5e-16) || (!(n <= 6e+17) && (n <= 1.1e+50))))
		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 ((n <= -4.7e-24) || ~(((n <= 6.5e-16) || (~((n <= 6e+17)) && (n <= 1.1e+50)))))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -4.7e-24], N[Not[Or[LessEqual[n, 6.5e-16], And[N[Not[LessEqual[n, 6e+17]], $MachinePrecision], LessEqual[n, 1.1e+50]]]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -4.69999999999999992e-24 or 6.50000000000000011e-16 < n < 6e17 or 1.10000000000000008e50 < n

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around 0 79.7%

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

   if -4.69999999999999992e-24 < n < 6.50000000000000011e-16 or 6e17 < n < 1.10000000000000008e50

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around inf 80.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.7 \cdot 10^{-24} \lor \neg \left(n \leq 6.5 \cdot 10^{-16} \lor \neg \left(n \leq 6 \cdot 10^{+17}\right) \land n \leq 1.1 \cdot 10^{+50}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{f}{n}\\ \mathbf{if}\;n \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+18} \lor \neg \left(n \leq 1.5 \cdot 10^{+49}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ f n)))))
   (if (<= n -6.5e-24)
     t_0
     (if (<= n 4.2e-20)
       (+ (* -2.0 (/ n f)) -1.0)
       (if (or (<= n 2.1e+18) (not (<= n 1.5e+49))) t_0 -1.0)))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -6.5e-24) {
		tmp = t_0;
	} else if (n <= 4.2e-20) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else if ((n <= 2.1e+18) || !(n <= 1.5e+49)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (2.0d0 * (f / n))
    if (n <= (-6.5d-24)) then
        tmp = t_0
    else if (n <= 4.2d-20) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else if ((n <= 2.1d+18) .or. (.not. (n <= 1.5d+49))) then
        tmp = t_0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -6.5e-24) {
		tmp = t_0;
	} else if (n <= 4.2e-20) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else if ((n <= 2.1e+18) || !(n <= 1.5e+49)) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (2.0 * (f / n))
	tmp = 0
	if n <= -6.5e-24:
		tmp = t_0
	elif n <= 4.2e-20:
		tmp = (-2.0 * (n / f)) + -1.0
	elif (n <= 2.1e+18) or not (n <= 1.5e+49):
		tmp = t_0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(f / n)))
	tmp = 0.0
	if (n <= -6.5e-24)
		tmp = t_0;
	elseif (n <= 4.2e-20)
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	elseif ((n <= 2.1e+18) || !(n <= 1.5e+49))
		tmp = t_0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (2.0 * (f / n));
	tmp = 0.0;
	if (n <= -6.5e-24)
		tmp = t_0;
	elseif (n <= 4.2e-20)
		tmp = (-2.0 * (n / f)) + -1.0;
	elseif ((n <= 2.1e+18) || ~((n <= 1.5e+49)))
		tmp = t_0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -6.5e-24], t$95$0, If[LessEqual[n, 4.2e-20], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[Or[LessEqual[n, 2.1e+18], N[Not[LessEqual[n, 1.5e+49]], $MachinePrecision]], t$95$0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if n < -6.5e-24 or 4.1999999999999998e-20 < n < 2.1e18 or 1.5000000000000001e49 < n

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around 0 79.7%

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

   if -6.5e-24 < n < 4.1999999999999998e-20

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in n around 0 80.4%

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

   if 2.1e18 < n < 1.5000000000000001e49

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around inf 100.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.5 \cdot 10^{-24}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{-20}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;n \leq 2.1 \cdot 10^{+18} \lor \neg \left(n \leq 1.5 \cdot 10^{+49}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 73.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.15 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.15e-24)
   1.0
   (if (<= n 6.4e-12)
     -1.0
     (if (<= n 6.6e+17) 1.0 (if (<= n 1.5e+49) -1.0 1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.15e-24) {
		tmp = 1.0;
	} else if (n <= 6.4e-12) {
		tmp = -1.0;
	} else if (n <= 6.6e+17) {
		tmp = 1.0;
	} else if (n <= 1.5e+49) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.15d-24)) then
        tmp = 1.0d0
    else if (n <= 6.4d-12) then
        tmp = -1.0d0
    else if (n <= 6.6d+17) then
        tmp = 1.0d0
    else if (n <= 1.5d+49) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.15e-24) {
		tmp = 1.0;
	} else if (n <= 6.4e-12) {
		tmp = -1.0;
	} else if (n <= 6.6e+17) {
		tmp = 1.0;
	} else if (n <= 1.5e+49) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.15e-24:
		tmp = 1.0
	elif n <= 6.4e-12:
		tmp = -1.0
	elif n <= 6.6e+17:
		tmp = 1.0
	elif n <= 1.5e+49:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.15e-24)
		tmp = 1.0;
	elseif (n <= 6.4e-12)
		tmp = -1.0;
	elseif (n <= 6.6e+17)
		tmp = 1.0;
	elseif (n <= 1.5e+49)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.15e-24)
		tmp = 1.0;
	elseif (n <= 6.4e-12)
		tmp = -1.0;
	elseif (n <= 6.6e+17)
		tmp = 1.0;
	elseif (n <= 1.5e+49)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.15e-24], 1.0, If[LessEqual[n, 6.4e-12], -1.0, If[LessEqual[n, 6.6e+17], 1.0, If[LessEqual[n, 1.5e+49], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.1499999999999999e-24 or 6.4000000000000002e-12 < n < 6.6e17 or 1.5000000000000001e49 < n

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around 0 78.6%

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

   if -3.1499999999999999e-24 < n < 6.4000000000000002e-12 or 6.6e17 < n < 1.5000000000000001e49

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. distribute-neg-in 100.0%

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

      4. neg-mul-1 100.0%

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

      5. associate-/r* 100.0%

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

      6. neg-mul-1 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-/l* 100.0%

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

      9. metadata-eval 100.0%

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

      10. /-rgt-identity 100.0%

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

      11. +-commutative 100.0%

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

      12. 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. Taylor expanded in f around inf 80.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -3.15 \cdot 10^{-24}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 6.4 \cdot 10^{-12}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 6.6 \cdot 10^{+17}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.5 \cdot 10^{+49}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 50.3% 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. sub-neg 100.0%

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

   2. remove-double-neg 100.0%

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

   3. distribute-neg-in 100.0%

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

   4. neg-mul-1 100.0%

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

   5. associate-/r* 100.0%

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

   6. neg-mul-1 100.0%

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

   7. *-commutative 100.0%

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

   8. associate-/l* 100.0%

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

   9. metadata-eval 100.0%

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

   10. /-rgt-identity 100.0%

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

   11. +-commutative 100.0%

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

   12. 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. Taylor expanded in f around inf 50.5%

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

5. Final simplification 50.5%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))