subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 8

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: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -4.5e-29)
   (+ (/ f n) 1.0)
   (if (<= n 1.25e-5) (/ f (- n f)) (+ 1.0 (/ (* f 2.0) n)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -4.5e-29) {
		tmp = (f / n) + 1.0;
	} else if (n <= 1.25e-5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f * 2.0) / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.5d-29)) then
        tmp = (f / n) + 1.0d0
    else if (n <= 1.25d-5) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0 + ((f * 2.0d0) / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -4.5e-29) {
		tmp = (f / n) + 1.0;
	} else if (n <= 1.25e-5) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f * 2.0) / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -4.5e-29:
		tmp = (f / n) + 1.0
	elif n <= 1.25e-5:
		tmp = f / (n - f)
	else:
		tmp = 1.0 + ((f * 2.0) / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -4.5e-29)
		tmp = Float64(Float64(f / n) + 1.0);
	elseif (n <= 1.25e-5)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(1.0 + Float64(Float64(f * 2.0) / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -4.5e-29)
		tmp = (f / n) + 1.0;
	elseif (n <= 1.25e-5)
		tmp = f / (n - f);
	else
		tmp = 1.0 + ((f * 2.0) / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -4.5e-29], N[(N[(f / n), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[n, 1.25e-5], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.4999999999999998e-29

   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 75.6%

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

   6. Taylor expanded in n around inf 75.6%

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

      1. +-commutative 75.6%

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

   8. Simplified 75.6%

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

   if -4.4999999999999998e-29 < n < 1.25000000000000006e-5

   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 77.2%

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

   if 1.25000000000000006e-5 < 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 87.6%

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

      1. associate-*r/ 87.6%

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

   7. Simplified 87.6%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{elif}\;n \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-29} \lor \neg \left(n \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -8.8e-29) (not (<= n 4.7e-5))) (+ (/ f n) 1.0) (/ f (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -8.8e-29) || !(n <= 4.7e-5)) {
		tmp = (f / n) + 1.0;
	} 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 <= (-8.8d-29)) .or. (.not. (n <= 4.7d-5))) then
        tmp = (f / n) + 1.0d0
    else
        tmp = f / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((n <= -8.8e-29) || !(n <= 4.7e-5)) {
		tmp = (f / n) + 1.0;
	} else {
		tmp = f / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -8.8e-29) or not (n <= 4.7e-5):
		tmp = (f / n) + 1.0
	else:
		tmp = f / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -8.8e-29) || !(n <= 4.7e-5))
		tmp = Float64(Float64(f / n) + 1.0);
	else
		tmp = Float64(f / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -8.8e-29) || ~((n <= 4.7e-5)))
		tmp = (f / n) + 1.0;
	else
		tmp = f / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -8.8e-29], N[Not[LessEqual[n, 4.7e-5]], $MachinePrecision]], N[(N[(f / n), $MachinePrecision] + 1.0), $MachinePrecision], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -8.79999999999999961e-29 or 4.69999999999999972e-5 < 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 81.9%

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

   6. Taylor expanded in n around inf 81.7%

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

      1. +-commutative 81.7%

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

   8. Simplified 81.7%

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

   if -8.79999999999999961e-29 < n < 4.69999999999999972e-5

   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 77.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -8.8 \cdot 10^{-29} \lor \neg \left(n \leq 4.7 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5.1 \cdot 10^{-27} \lor \neg \left(n \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -5.1e-27) (not (<= n 8.5e-6)))
   (+ (/ f n) 1.0)
   (- -1.0 (/ n f))))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -5.1e-27) || !(n <= 8.5e-6)) {
		tmp = (f / n) + 1.0;
	} 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 <= (-5.1d-27)) .or. (.not. (n <= 8.5d-6))) then
        tmp = (f / n) + 1.0d0
    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 <= -5.1e-27) || !(n <= 8.5e-6)) {
		tmp = (f / n) + 1.0;
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -5.1e-27) or not (n <= 8.5e-6):
		tmp = (f / n) + 1.0
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -5.1e-27) || !(n <= 8.5e-6))
		tmp = Float64(Float64(f / n) + 1.0);
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -5.1e-27) || ~((n <= 8.5e-6)))
		tmp = (f / n) + 1.0;
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -5.1e-27], N[Not[LessEqual[n, 8.5e-6]], $MachinePrecision]], N[(N[(f / n), $MachinePrecision] + 1.0), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if n < -5.0999999999999999e-27 or 8.4999999999999999e-6 < 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 81.9%

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

   6. Taylor expanded in n around inf 81.7%

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

      1. +-commutative 81.7%

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

   8. Simplified 81.7%

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

   if -5.0999999999999999e-27 < n < 8.4999999999999999e-6

   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 77.2%

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

   6. Taylor expanded in f around inf 77.2%

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

      1. mul-1-neg 77.2%

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

      2. neg-sub0 77.2%

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

      3. associate--r+ 77.2%

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

      4. +-commutative 77.2%

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

      5. associate--r+ 77.2%

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

      6. metadata-eval 77.2%

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

   8. Simplified 77.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -5.1 \cdot 10^{-27} \lor \neg \left(n \leq 8.5 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-27} \lor \neg \left(n \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= n -4.6e-27) (not (<= n 2.7e-6))) (+ (/ f n) 1.0) -1.0))

C

double code(double f, double n) {
	double tmp;
	if ((n <= -4.6e-27) || !(n <= 2.7e-6)) {
		tmp = (f / n) + 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 <= (-4.6d-27)) .or. (.not. (n <= 2.7d-6))) then
        tmp = (f / n) + 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 <= -4.6e-27) || !(n <= 2.7e-6)) {
		tmp = (f / n) + 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (n <= -4.6e-27) or not (n <= 2.7e-6):
		tmp = (f / n) + 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((n <= -4.6e-27) || !(n <= 2.7e-6))
		tmp = Float64(Float64(f / n) + 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((n <= -4.6e-27) || ~((n <= 2.7e-6)))
		tmp = (f / n) + 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[n, -4.6e-27], N[Not[LessEqual[n, 2.7e-6]], $MachinePrecision]], N[(N[(f / n), $MachinePrecision] + 1.0), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if n < -4.5999999999999999e-27 or 2.69999999999999998e-6 < 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 81.9%

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

   6. Taylor expanded in n around inf 81.7%

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

      1. +-commutative 81.7%

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

   8. Simplified 81.7%

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

   if -4.5999999999999999e-27 < n < 2.69999999999999998e-6

   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 76.7%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;n \leq -4.6 \cdot 10^{-27} \lor \neg \left(n \leq 2.7 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{elif}\;n \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.05e-29)
   (+ (/ f n) 1.0)
   (if (<= n 1.05e-6) (/ f (- n f)) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.05e-29) {
		tmp = (f / n) + 1.0;
	} else if (n <= 1.05e-6) {
		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 <= (-1.05d-29)) then
        tmp = (f / n) + 1.0d0
    else if (n <= 1.05d-6) 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 <= -1.05e-29) {
		tmp = (f / n) + 1.0;
	} else if (n <= 1.05e-6) {
		tmp = f / (n - f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.05e-29:
		tmp = (f / n) + 1.0
	elif n <= 1.05e-6:
		tmp = f / (n - f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.05e-29)
		tmp = Float64(Float64(f / n) + 1.0);
	elseif (n <= 1.05e-6)
		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 <= -1.05e-29)
		tmp = (f / n) + 1.0;
	elseif (n <= 1.05e-6)
		tmp = f / (n - f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.05e-29], N[(N[(f / n), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[n, 1.05e-6], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.04999999999999995e-29

   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 75.6%

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

   6. Taylor expanded in n around inf 75.6%

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

      1. +-commutative 75.6%

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

   8. Simplified 75.6%

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

   if -1.04999999999999995e-29 < n < 1.0499999999999999e-6

   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 77.2%

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

   if 1.0499999999999999e-6 < 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 87.3%

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

Alternative 7: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 0.00019:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -5e-27) 1.0 (if (<= n 0.00019) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -5e-27) {
		tmp = 1.0;
	} else if (n <= 0.00019) {
		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 <= (-5d-27)) then
        tmp = 1.0d0
    else if (n <= 0.00019d0) 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 <= -5e-27) {
		tmp = 1.0;
	} else if (n <= 0.00019) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -5e-27:
		tmp = 1.0
	elif n <= 0.00019:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -5e-27)
		tmp = 1.0;
	elseif (n <= 0.00019)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -5e-27)
		tmp = 1.0;
	elseif (n <= 0.00019)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -5e-27], 1.0, If[LessEqual[n, 0.00019], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -5.0000000000000002e-27 or 1.9000000000000001e-4 < 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 81.4%

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

   if -5.0000000000000002e-27 < n < 1.9000000000000001e-4

   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 76.7%

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

Alternative 8: 49.0% 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 44.0%

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

Reproduce

herbie shell --seed 2024132 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))