subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 9

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: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -7.5e-56)
   (/ f (- n f))
   (if (<= f 4.6e+14) (+ 1.0 (/ (* f 2.0) n)) (+ (* -2.0 (/ n f)) -1.0))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 4.6e+14) {
		tmp = 1.0 + ((f * 2.0) / 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 (f <= (-7.5d-56)) then
        tmp = f / (n - f)
    else if (f <= 4.6d+14) then
        tmp = 1.0d0 + ((f * 2.0d0) / 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 (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 4.6e+14) {
		tmp = 1.0 + ((f * 2.0) / n);
	} else {
		tmp = (-2.0 * (n / f)) + -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -7.5e-56:
		tmp = f / (n - f)
	elif f <= 4.6e+14:
		tmp = 1.0 + ((f * 2.0) / n)
	else:
		tmp = (-2.0 * (n / f)) + -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -7.5e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 4.6e+14)
		tmp = Float64(1.0 + Float64(Float64(f * 2.0) / 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 (f <= -7.5e-56)
		tmp = f / (n - f);
	elseif (f <= 4.6e+14)
		tmp = 1.0 + ((f * 2.0) / n);
	else
		tmp = (-2.0 * (n / f)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -7.5e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 4.6e+14], N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -7.50000000000000041e-56

   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 80.7%

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

   if -7.50000000000000041e-56 < f < 4.6e14

   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 80.5%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{f}{n}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.5%

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

   7. Simplified 80.5%

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

   if 4.6e14 < f

   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 n around 0 81.6%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -7.5e-56)
   (/ f (- n f))
   (if (<= f 8.6e+14) (+ 1.0 (/ (* f 2.0) n)) (- -1.0 (/ n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 8.6e+14) {
		tmp = 1.0 + ((f * 2.0) / 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 (f <= (-7.5d-56)) then
        tmp = f / (n - f)
    else if (f <= 8.6d+14) then
        tmp = 1.0d0 + ((f * 2.0d0) / 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 (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 8.6e+14) {
		tmp = 1.0 + ((f * 2.0) / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -7.5e-56:
		tmp = f / (n - f)
	elif f <= 8.6e+14:
		tmp = 1.0 + ((f * 2.0) / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -7.5e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 8.6e+14)
		tmp = Float64(1.0 + Float64(Float64(f * 2.0) / n));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -7.5e-56)
		tmp = f / (n - f);
	elseif (f <= 8.6e+14)
		tmp = 1.0 + ((f * 2.0) / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -7.5e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 8.6e+14], N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -7.50000000000000041e-56

   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 80.7%

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

   if -7.50000000000000041e-56 < f < 8.6e14

   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 80.5%

      \[\leadsto \color{blue}{1 + 2 \cdot \frac{f}{n}} \]
   6. Step-by-step derivation

      1. associate-*r/ 80.5%

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

   7. Simplified 80.5%

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

   if 8.6e14 < f

   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 81.1%

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

   6. Taylor expanded in f around inf 81.2%

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

      1. neg-mul-1 81.2%

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

      2. neg-sub0 81.2%

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

      3. associate--r+ 81.2%

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

      4. +-commutative 81.2%

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

      5. associate--r+ 81.2%

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

      6. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.05 \cdot 10^{-15} \lor \neg \left(f \leq 1.35 \cdot 10^{+16}\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -2.05e-15) (not (<= f 1.35e+16)))
   (- -1.0 (/ n f))
   (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -2.05e-15) || !(f <= 1.35e+16)) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (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 <= (-2.05d-15)) .or. (.not. (f <= 1.35d+16))) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -2.05e-15) || !(f <= 1.35e+16)) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -2.05e-15) or not (f <= 1.35e+16):
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -2.05e-15) || !(f <= 1.35e+16))
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -2.05e-15) || ~((f <= 1.35e+16)))
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -2.05e-15], N[Not[LessEqual[f, 1.35e+16]], $MachinePrecision]], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -2.05000000000000018e-15 or 1.35e16 < f

   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 82.5%

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

   6. Taylor expanded in f around inf 82.2%

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

      1. neg-mul-1 82.2%

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

      2. neg-sub0 82.2%

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

      3. associate--r+ 82.2%

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

      4. +-commutative 82.2%

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

      5. associate--r+ 82.2%

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

      6. metadata-eval 82.2%

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

   8. Simplified 82.2%

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

   if -2.05000000000000018e-15 < f < 1.35e16

   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 78.5%

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

   6. Taylor expanded in n around inf 77.9%

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

      1. +-commutative 77.9%

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

   8. Simplified 77.9%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.05 \cdot 10^{-15} \lor \neg \left(f \leq 1.35 \cdot 10^{+16}\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.55 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.55e-56)
   (/ f (- n f))
   (if (<= f 2.1e+14) (/ n (- n f)) (- -1.0 (/ n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.55e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.1e+14) {
		tmp = n / (n - f);
	} 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 (f <= (-1.55d-56)) then
        tmp = f / (n - f)
    else if (f <= 2.1d+14) then
        tmp = n / (n - f)
    else
        tmp = (-1.0d0) - (n / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -1.55e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.1e+14) {
		tmp = n / (n - f);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.55e-56:
		tmp = f / (n - f)
	elif f <= 2.1e+14:
		tmp = n / (n - f)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.55e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 2.1e+14)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.55e-56)
		tmp = f / (n - f);
	elseif (f <= 2.1e+14)
		tmp = n / (n - f);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.55e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 2.1e+14], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -1.54999999999999994e-56

   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 80.7%

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

   if -1.54999999999999994e-56 < f < 2.1e14

   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 80.5%

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

   if 2.1e14 < f

   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 81.1%

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

   6. Taylor expanded in f around inf 81.2%

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

      1. neg-mul-1 81.2%

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

      2. neg-sub0 81.2%

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

      3. associate--r+ 81.2%

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

      4. +-commutative 81.2%

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

      5. associate--r+ 81.2%

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

      6. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

Alternative 6: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -6.6e-56)
   (/ f (- n f))
   (if (<= f 5.4e+14) (+ 1.0 (/ f n)) (- -1.0 (/ n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -6.6e-56) {
		tmp = f / (n - f);
	} else if (f <= 5.4e+14) {
		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 (f <= (-6.6d-56)) then
        tmp = f / (n - f)
    else if (f <= 5.4d+14) 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 (f <= -6.6e-56) {
		tmp = f / (n - f);
	} else if (f <= 5.4e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -6.6e-56:
		tmp = f / (n - f)
	elif f <= 5.4e+14:
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -6.6e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 5.4e+14)
		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 (f <= -6.6e-56)
		tmp = f / (n - f);
	elseif (f <= 5.4e+14)
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -6.6e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 5.4e+14], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -6.59999999999999967e-56

   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 80.7%

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

   if -6.59999999999999967e-56 < f < 5.4e14

   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 80.5%

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

   6. Taylor expanded in n around inf 79.9%

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

      1. +-commutative 79.9%

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

   8. Simplified 79.9%

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

   if 5.4e14 < f

   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 81.1%

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

   6. Taylor expanded in f around inf 81.2%

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

      1. neg-mul-1 81.2%

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

      2. neg-sub0 81.2%

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

      3. associate--r+ 81.2%

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

      4. +-commutative 81.2%

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

      5. associate--r+ 81.2%

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

      6. metadata-eval 81.2%

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

   8. Simplified 81.2%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.12 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.12e-29) -1.0 (if (<= f 2e+14) (+ 1.0 (/ f n)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.12e-29) {
		tmp = -1.0;
	} else if (f <= 2e+14) {
		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 (f <= (-1.12d-29)) then
        tmp = -1.0d0
    else if (f <= 2d+14) 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 (f <= -1.12e-29) {
		tmp = -1.0;
	} else if (f <= 2e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.12e-29:
		tmp = -1.0
	elif f <= 2e+14:
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.12e-29)
		tmp = -1.0;
	elseif (f <= 2e+14)
		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 (f <= -1.12e-29)
		tmp = -1.0;
	elseif (f <= 2e+14)
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.12e-29], -1.0, If[LessEqual[f, 2e+14], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.11999999999999995e-29 or 2e14 < f

   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 81.4%

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

   if -1.11999999999999995e-29 < f < 2e14

   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 78.9%

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

   6. Taylor expanded in n around inf 78.3%

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

      1. +-commutative 78.3%

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

   8. Simplified 78.3%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.12 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.9e-15) -1.0 (if (<= f 1.95e+14) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.9e-15) {
		tmp = -1.0;
	} else if (f <= 1.95e+14) {
		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 <= (-2.9d-15)) then
        tmp = -1.0d0
    else if (f <= 1.95d+14) 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 <= -2.9e-15) {
		tmp = -1.0;
	} else if (f <= 1.95e+14) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.9e-15:
		tmp = -1.0
	elif f <= 1.95e+14:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.9e-15)
		tmp = -1.0;
	elseif (f <= 1.95e+14)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.9e-15)
		tmp = -1.0;
	elseif (f <= 1.95e+14)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.9e-15], -1.0, If[LessEqual[f, 1.95e+14], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -2.90000000000000019e-15 or 1.95e14 < f

   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 81.9%

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

   if -2.90000000000000019e-15 < f < 1.95e14

   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 77.7%

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

Alternative 9: 49.6% 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 54.9%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))