subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

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 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. Add Preprocessing

Alternative 2: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -59:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -59.0)
   (+ 1.0 (/ f n))
   (if (<= n 1.08e-11) (+ (* -2.0 (/ n f)) -1.0) (+ 1.0 (* (/ f n) 2.0)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -59.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.08e-11) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + ((f / n) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-59.0d0)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 1.08d-11) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0 + ((f / n) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -59.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.08e-11) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + ((f / n) * 2.0);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -59.0:
		tmp = 1.0 + (f / n)
	elif n <= 1.08e-11:
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + ((f / n) * 2.0)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -59.0)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 1.08e-11)
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(Float64(f / n) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -59.0)
		tmp = 1.0 + (f / n);
	elseif (n <= 1.08e-11)
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + ((f / n) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -59.0], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.08e-11], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(f / n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -59

   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 75.0%

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

   6. Taylor expanded in n around inf 75.1%

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

   if -59 < n < 1.07999999999999992e-11

   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 83.4%

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

   if 1.07999999999999992e-11 < n

   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 82.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -59:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.08 \cdot 10^{-11}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -59:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n} \cdot 2\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -59.0)
   (+ 1.0 (/ f n))
   (if (<= n 1.18e-7) (/ f (- n f)) (+ 1.0 (* (/ f n) 2.0)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -59.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.18e-7) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f / n) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-59.0d0)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 1.18d-7) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0 + ((f / n) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -59.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 1.18e-7) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f / n) * 2.0);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -59.0:
		tmp = 1.0 + (f / n)
	elif n <= 1.18e-7:
		tmp = f / (n - f)
	else:
		tmp = 1.0 + ((f / n) * 2.0)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -59.0)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 1.18e-7)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(1.0 + Float64(Float64(f / n) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -59.0)
		tmp = 1.0 + (f / n);
	elseif (n <= 1.18e-7)
		tmp = f / (n - f);
	else
		tmp = 1.0 + ((f / n) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -59.0], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 1.18e-7], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(f / n), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -59

   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 75.0%

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

   6. Taylor expanded in n around inf 75.1%

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

   if -59 < n < 1.18e-7

   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 82.4%

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

   if 1.18e-7 < n

   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 82.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -59:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 1.18 \cdot 10^{-7}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n} \cdot 2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -20.5 \lor \neg \left(n \leq 1.76 \cdot 10^{-7}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -20.5) (not (<= n 1.76e-7))) (+ 1.0 (/ f n)) (/ f (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -20.5) || !(n <= 1.76e-7)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = f / (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 <= (-20.5d0)) .or. (.not. (n <= 1.76d-7))) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = f / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -20.5) || !(n <= 1.76e-7)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = f / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -20.5) or not (n <= 1.76e-7):
		tmp = 1.0 + (f / n)
	else:
		tmp = f / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -20.5) || !(n <= 1.76e-7))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = Float64(f / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -20.5) || ~((n <= 1.76e-7)))
		tmp = 1.0 + (f / n);
	else
		tmp = f / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -20.5], N[Not[LessEqual[n, 1.76e-7]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -20.5 or 1.7599999999999999e-7 < n

   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 78.4%

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

   6. Taylor expanded in n around inf 78.2%

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

   if -20.5 < n < 1.7599999999999999e-7

   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 82.4%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -20.5 \lor \neg \left(n \leq 1.76 \cdot 10^{-7}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -76 \lor \neg \left(n \leq 7 \cdot 10^{-12}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -76.0) (not (<= n 7e-12))) (+ 1.0 (/ f n)) (- -1.0 (/ n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -76.0) || !(n <= 7e-12)) {
		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 <= (-76.0d0)) .or. (.not. (n <= 7d-12))) 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 <= -76.0) || !(n <= 7e-12)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -76.0) or not (n <= 7e-12):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -76.0) || !(n <= 7e-12))
		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 <= -76.0) || ~((n <= 7e-12)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -76.0], N[Not[LessEqual[n, 7e-12]], $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 < -76 or 7.0000000000000001e-12 < n

   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 78.0%

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

   6. Taylor expanded in n around inf 77.8%

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

   if -76 < n < 7.0000000000000001e-12

   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 82.8%

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

   6. Taylor expanded in f around inf 82.8%

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

      1. neg-mul-1 82.8%

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

      2. neg-sub0 82.8%

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

      3. associate--r+ 82.8%

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

      4. +-commutative 82.8%

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

      5. associate--r+ 82.8%

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

      6. metadata-eval 82.8%

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

   8. Simplified 82.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -76 \lor \neg \left(n \leq 7 \cdot 10^{-12}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -13 \lor \neg \left(n \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -13.0) (not (<= n 1.5e-7))) (+ 1.0 (/ f n)) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -13.0) || !(n <= 1.5e-7)) {
		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 <= (-13.0d0)) .or. (.not. (n <= 1.5d-7))) 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 <= -13.0) || !(n <= 1.5e-7)) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -13.0) or not (n <= 1.5e-7):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -13.0) || !(n <= 1.5e-7))
		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 <= -13.0) || ~((n <= 1.5e-7)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -13.0], N[Not[LessEqual[n, 1.5e-7]], $MachinePrecision]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -13 or 1.4999999999999999e-7 < n

   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 78.4%

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

   6. Taylor expanded in n around inf 78.2%

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

   if -13 < n < 1.4999999999999999e-7

   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 81.8%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -13 \lor \neg \left(n \leq 1.5 \cdot 10^{-7}\right):\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -50:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 5.8 \cdot 10^{-12}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -50.0)
   (+ 1.0 (/ f n))
   (if (<= n 5.8e-12) (/ f (- n f)) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -50.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 5.8e-12) {
		tmp = f / (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 <= (-50.0d0)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 5.8d-12) then
        tmp = f / (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 <= -50.0) {
		tmp = 1.0 + (f / n);
	} else if (n <= 5.8e-12) {
		tmp = f / (n - f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -50.0:
		tmp = 1.0 + (f / n)
	elif n <= 5.8e-12:
		tmp = f / (n - f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -50.0)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 5.8e-12)
		tmp = Float64(f / 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 <= -50.0)
		tmp = 1.0 + (f / n);
	elseif (n <= 5.8e-12)
		tmp = f / (n - f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -50.0], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 5.8e-12], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -50

   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 75.0%

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

   6. Taylor expanded in n around inf 75.1%

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

   if -50 < n < 5.8000000000000003e-12

   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 82.8%

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

   if 5.8000000000000003e-12 < n

   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 79.8%

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

Alternative 8: 74.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -20.5:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -20.5) 1.0 (if (<= n 1.4e-11) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -20.5) {
		tmp = 1.0;
	} else if (n <= 1.4e-11) {
		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 <= (-20.5d0)) then
        tmp = 1.0d0
    else if (n <= 1.4d-11) 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 <= -20.5) {
		tmp = 1.0;
	} else if (n <= 1.4e-11) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -20.5:
		tmp = 1.0
	elif n <= 1.4e-11:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -20.5)
		tmp = 1.0;
	elseif (n <= 1.4e-11)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -20.5)
		tmp = 1.0;
	elseif (n <= 1.4e-11)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -20.5], 1.0, If[LessEqual[n, 1.4e-11], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -20.5 or 1.4e-11 < n

   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 77.2%

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

   if -20.5 < n < 1.4e-11

   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 82.3%

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

Alternative 9: 48.4% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

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

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 99.9%

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

   1. 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 51.0%

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

Reproduce

herbie shell --seed 2024172 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))