subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 4.8s

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. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1800 \lor \neg \left(n \leq 8.5 \cdot 10^{+66}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -1800.0) (not (<= n 8.5e+66))) (+ 1.0 (* 2.0 (/ f n))) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -1800.0) || !(n <= 8.5e+66)) {
		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 <= (-1800.0d0)) .or. (.not. (n <= 8.5d+66))) 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 <= -1800.0) || !(n <= 8.5e+66)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -1800.0) or not (n <= 8.5e+66):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -1800.0) || !(n <= 8.5e+66))
		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 <= -1800.0) || ~((n <= 8.5e+66)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -1800.0], N[Not[LessEqual[n, 8.5e+66]], $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 < -1800 or 8.5000000000000004e66 < n

   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. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 81.9%

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

   if -1800 < n < 8.5000000000000004e66

   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. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 78.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1800 \lor \neg \left(n \leq 8.5 \cdot 10^{+66}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1800000 \lor \neg \left(n \leq 1.6 \cdot 10^{+68}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -1800000.0) (not (<= n 1.6e+68)))
   (+ 1.0 (* 2.0 (/ f n)))
   (+ (* -2.0 (/ n f)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -1800000.0) || !(n <= 1.6e+68)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -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 <= (-1800000.0d0)) .or. (.not. (n <= 1.6d+68))) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -1800000.0) || !(n <= 1.6e+68)) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = (-2.0 * (n / f)) + -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -1800000.0) or not (n <= 1.6e+68):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = (-2.0 * (n / f)) + -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -1800000.0) || !(n <= 1.6e+68))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -1800000.0) || ~((n <= 1.6e+68)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = (-2.0 * (n / f)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -1800000.0], N[Not[LessEqual[n, 1.6e+68]], $MachinePrecision]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -1.8e6 or 1.59999999999999997e68 < n

   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. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 81.9%

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

   if -1.8e6 < n < 1.59999999999999997e68

   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. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in n around 0 79.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -1800000 \lor \neg \left(n \leq 1.6 \cdot 10^{+68}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -6.7 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 10^{+53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -6.7e+22) 1.0 (if (<= n 1e+53) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -6.7e+22) {
		tmp = 1.0;
	} else if (n <= 1e+53) {
		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 <= (-6.7d+22)) then
        tmp = 1.0d0
    else if (n <= 1d+53) 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 <= -6.7e+22) {
		tmp = 1.0;
	} else if (n <= 1e+53) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -6.7e+22:
		tmp = 1.0
	elif n <= 1e+53:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -6.7e+22)
		tmp = 1.0;
	elseif (n <= 1e+53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -6.7e+22)
		tmp = 1.0;
	elseif (n <= 1e+53)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -6.7e+22], 1.0, If[LessEqual[n, 1e+53], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -6.7000000000000002e22 or 9.9999999999999999e52 < n

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 81.1%

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

   if -6.7000000000000002e22 < n < 9.9999999999999999e52

   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. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 77.9%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -6.7 \cdot 10^{+22}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 10^{+53}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

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. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in f around inf 49.0%

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

6. Final simplification 49.0%

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

Reproduce

herbie shell --seed 2024036 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))