subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

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, 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. distribute-frac-neg 100.0%

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

   2. distribute-neg-frac2 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. remove-double-neg 100.0%

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

   7. 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. Add Preprocessing

Alternative 2: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -5.4e+25)
   (+ 1.0 (/ f n))
   (if (<= n 1.05e-54) (+ (* -2.0 (/ n f)) -1.0) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -5.4e+25) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.05e-54) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-5.4d+25)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 1.05d-54) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -5.4e+25) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.05e-54) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -5.4e+25:
		tmp = 1.0 + (f / n)
	elif n <= 1.05e-54:
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -5.4e+25)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 1.05e-54)
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -5.4e+25)
		tmp = 1.0 + (f / n);
	elseif (n <= 1.05e-54)
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -5.4e+25], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.05e-54], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -5.4e25

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 72.0%

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

   6. Taylor expanded in n around inf 72.3%

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

   if -5.4e25 < n < 1.05e-54

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 80.6%

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

   if 1.05e-54 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 76.5%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.4 \cdot 10^{+25}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{+18} \lor \neg \left(n \leq 1.05 \cdot 10^{-54}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -8.2e+18) (not (<= n 1.05e-54)))
   (+ 1.0 (/ f n))
   (- -1.0 (/ n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -8.2e+18) || !(n <= 1.05e-54)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((n <= (-8.2d+18)) .or. (.not. (n <= 1.05d-54))) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = (-1.0d0) - (n / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -8.2e+18) || !(n <= 1.05e-54)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -8.2e+18) or not (n <= 1.05e-54):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -8.2e+18) || !(n <= 1.05e-54))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -8.2e+18) || ~((n <= 1.05e-54)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -8.2e+18], N[Not[LessEqual[n, 1.05e-54]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.2e18 or 1.05e-54 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 74.5%

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

   6. Taylor expanded in n around inf 74.5%

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

   if -8.2e18 < n < 1.05e-54

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 79.6%

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

   6. Taylor expanded in f around inf 79.7%

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

      1. neg-mul-1 79.7%

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

      2. neg-sub0 79.7%

         \[\leadsto \color{blue}{\left(0 - \frac{n}{f}\right)} - 1 \]

      3. associate--r+ 79.7%

         \[\leadsto \color{blue}{0 - \left(\frac{n}{f} + 1\right)} \]

      4. +-commutative 79.7%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{n}{f}\right)} \]

      5. associate--r+ 79.7%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{n}{f}} \]

      6. metadata-eval 79.7%

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

   8. Simplified 79.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.2 \cdot 10^{+18} \lor \neg \left(n \leq 1.05 \cdot 10^{-54}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+30} \lor \neg \left(n \leq 4.8 \cdot 10^{-55}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -5e+30) (not (<= n 4.8e-55))) (+ 1.0 (/ f n)) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -5e+30) || !(n <= 4.8e-55)) {
		tmp = 1.0 + (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 <= (-5d+30)) .or. (.not. (n <= 4.8d-55))) then
        tmp = 1.0d0 + (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 <= -5e+30) || !(n <= 4.8e-55)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -5e+30) or not (n <= 4.8e-55):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -5e+30) || !(n <= 4.8e-55))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -5e+30) || ~((n <= 4.8e-55)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -5e+30], N[Not[LessEqual[n, 4.8e-55]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -4.9999999999999998e30 or 4.79999999999999983e-55 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 74.5%

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

   6. Taylor expanded in n around inf 74.5%

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

   if -4.9999999999999998e30 < n < 4.79999999999999983e-55

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 79.0%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+30} \lor \neg \left(n \leq 4.8 \cdot 10^{-55}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.8 \cdot 10^{+28}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-57}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.8e+28)
   (+ 1.0 (/ f n))
   (if (<= n 6e-57) (- -1.0 (/ n f)) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.8e+28) {
		tmp = 1.0 + (f / n);
	} else if (n <= 6e-57) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-3.8d+28)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 6d-57) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.8e+28) {
		tmp = 1.0 + (f / n);
	} else if (n <= 6e-57) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.8e+28:
		tmp = 1.0 + (f / n)
	elif n <= 6e-57:
		tmp = -1.0 - (n / f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.8e+28)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 6e-57)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.8e+28)
		tmp = 1.0 + (f / n);
	elseif (n <= 6e-57)
		tmp = -1.0 - (n / f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.8e+28], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 6e-57], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -3.7999999999999999e28

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 72.0%

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

   6. Taylor expanded in n around inf 72.3%

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

   if -3.7999999999999999e28 < n < 6.00000000000000001e-57

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 79.6%

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

   6. Taylor expanded in f around inf 79.7%

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

      1. neg-mul-1 79.7%

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

      2. neg-sub0 79.7%

         \[\leadsto \color{blue}{\left(0 - \frac{n}{f}\right)} - 1 \]

      3. associate--r+ 79.7%

         \[\leadsto \color{blue}{0 - \left(\frac{n}{f} + 1\right)} \]

      4. +-commutative 79.7%

         \[\leadsto 0 - \color{blue}{\left(1 + \frac{n}{f}\right)} \]

      5. associate--r+ 79.7%

         \[\leadsto \color{blue}{\left(0 - 1\right) - \frac{n}{f}} \]

      6. metadata-eval 79.7%

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

   8. Simplified 79.7%

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

   if 6.00000000000000001e-57 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 76.5%

      \[\leadsto \frac{\color{blue}{n}}{n - f} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 73.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.15 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 6 \cdot 10^{-57}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.15e+23) 1.0 (if (<= n 6e-57) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.15e+23) {
		tmp = 1.0;
	} else if (n <= 6e-57) {
		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 <= (-1.15d+23)) then
        tmp = 1.0d0
    else if (n <= 6d-57) 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 <= -1.15e+23) {
		tmp = 1.0;
	} else if (n <= 6e-57) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.15e+23:
		tmp = 1.0
	elif n <= 6e-57:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.15e+23)
		tmp = 1.0;
	elseif (n <= 6e-57)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.15e+23)
		tmp = 1.0;
	elseif (n <= 6e-57)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.15e+23], 1.0, If[LessEqual[n, 6e-57], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.15e23 or 6.00000000000000001e-57 < n

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 73.8%

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

   if -1.15e23 < n < 6.00000000000000001e-57

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. 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 79.0%

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

Alternative 7: 49.9% 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. distribute-frac-neg 100.0%

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

   2. distribute-neg-frac2 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. distribute-neg-in 100.0%

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

   6. remove-double-neg 100.0%

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

   7. 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 51.7%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024136 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))