subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 7.7s

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, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{n}{n + f} - \frac{f}{n + f}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return 1.0 / ((n / (n + f)) - (f / (n + f)))

Julia

function code(f, n)
	return Float64(1.0 / Float64(Float64(n / Float64(n + f)) - Float64(f / Float64(n + f))))
end

MATLAB

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

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n / N[(n + f), $MachinePrecision]), $MachinePrecision] - N[(f / N[(n + f), $MachinePrecision]), $MachinePrecision]), $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. Taylor expanded in n around inf 86.3%

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

   1. clear-num 86.3%

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

   2. inv-pow 86.3%

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

7. Applied egg-rr 86.3%

   \[\leadsto \color{blue}{{\left(\frac{n - f}{n \cdot \left(1 + \frac{f}{n}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 86.3%

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

   2. distribute-lft-in 86.2%

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

   3. *-rgt-identity 86.2%

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

   4. associate-*r/ 83.9%

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

   5. associate-*l/ 99.9%

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

   6. *-inverses 99.9%

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

   7. *-lft-identity 99.9%

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

9. Simplified 99.9%

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

    1. div-sub 100.0%

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

11. Applied egg-rr 100.0%

    \[\leadsto \frac{1}{\color{blue}{\frac{n}{n + f} - \frac{f}{n + f}}} \]
12. Add Preprocessing

Alternative 2: 75.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{-47} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.35e-47) (not (<= f 5e+32)))
   (+ (* -2.0 (/ n f)) -1.0)
   (/ n (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.35e-47) || !(f <= 5e+32)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-1.35d-47)) .or. (.not. (f <= 5d+32))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.35e-47) || !(f <= 5e+32)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.35e-47) or not (f <= 5e+32):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.35e-47) || !(f <= 5e+32))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.35e-47) || ~((f <= 5e+32)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.35e-47], N[Not[LessEqual[f, 5e+32]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.3499999999999999e-47 or 4.9999999999999997e32 < f

   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 81.1%

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

   if -1.3499999999999999e-47 < f < 4.9999999999999997e32

   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 78.5%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{-47} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.45 \cdot 10^{-48} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{n + f}{-f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -2.45e-48) (not (<= f 5e+32))) (/ (+ n f) (- f)) (/ n (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -2.45e-48) || !(f <= 5e+32)) {
		tmp = (n + f) / -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 ((f <= (-2.45d-48)) .or. (.not. (f <= 5d+32))) then
        tmp = (n + f) / -f
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -2.45e-48) || !(f <= 5e+32)) {
		tmp = (n + f) / -f;
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -2.45e-48) or not (f <= 5e+32):
		tmp = (n + f) / -f
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -2.45e-48) || !(f <= 5e+32))
		tmp = Float64(Float64(n + f) / Float64(-f));
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -2.45e-48) || ~((f <= 5e+32)))
		tmp = (n + f) / -f;
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -2.45e-48], N[Not[LessEqual[f, 5e+32]], $MachinePrecision]], N[(N[(n + f), $MachinePrecision] / (-f)), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -2.4500000000000001e-48 or 4.9999999999999997e32 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in n around 0 80.3%

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

      1. neg-mul-1 80.3%

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

   7. Simplified 80.3%

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

   if -2.4500000000000001e-48 < f < 4.9999999999999997e32

   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 78.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.45 \cdot 10^{-48} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{n + f}{-f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -9 \cdot 10^{-49} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -9e-49) (not (<= f 5e+32))) (/ f (- n f)) (/ n (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -9e-49) || !(f <= 5e+32)) {
		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 ((f <= (-9d-49)) .or. (.not. (f <= 5d+32))) 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 ((f <= -9e-49) || !(f <= 5e+32)) {
		tmp = f / (n - f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -9e-49) or not (f <= 5e+32):
		tmp = f / (n - f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -9e-49) || !(f <= 5e+32))
		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 ((f <= -9e-49) || ~((f <= 5e+32)))
		tmp = f / (n - f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -9e-49], N[Not[LessEqual[f, 5e+32]], $MachinePrecision]], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -9.0000000000000004e-49 or 4.9999999999999997e32 < f

   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 80.2%

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

   if -9.0000000000000004e-49 < f < 4.9999999999999997e32

   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 78.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -9 \cdot 10^{-49} \lor \neg \left(f \leq 5 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.3 \cdot 10^{-55} \lor \neg \left(f \leq 5.1 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -2.3e-55) (not (<= f 5.1e+32))) (/ f (- n f)) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -2.3e-55) || !(f <= 5.1e+32)) {
		tmp = f / (n - f);
	} 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 <= (-2.3d-55)) .or. (.not. (f <= 5.1d+32))) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -2.3e-55) || !(f <= 5.1e+32)) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -2.3e-55) or not (f <= 5.1e+32):
		tmp = f / (n - f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -2.3e-55) || !(f <= 5.1e+32))
		tmp = Float64(f / Float64(n - f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -2.3e-55) || ~((f <= 5.1e+32)))
		tmp = f / (n - f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -2.3e-55], N[Not[LessEqual[f, 5.1e+32]], $MachinePrecision]], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if f < -2.30000000000000011e-55 or 5.10000000000000004e32 < f

   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 80.2%

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

   if -2.30000000000000011e-55 < f < 5.10000000000000004e32

   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 77.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.3 \cdot 10^{-55} \lor \neg \left(f \leq 5.1 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{-47}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5 \cdot 10^{+32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.35e-47) -1.0 (if (<= f 5e+32) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.35e-47) {
		tmp = -1.0;
	} else if (f <= 5e+32) {
		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 <= (-1.35d-47)) then
        tmp = -1.0d0
    else if (f <= 5d+32) 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 <= -1.35e-47) {
		tmp = -1.0;
	} else if (f <= 5e+32) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.35e-47:
		tmp = -1.0
	elif f <= 5e+32:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.35e-47)
		tmp = -1.0;
	elseif (f <= 5e+32)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.35e-47)
		tmp = -1.0;
	elseif (f <= 5e+32)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.35e-47], -1.0, If[LessEqual[f, 5e+32], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.3499999999999999e-47 or 4.9999999999999997e32 < f

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around inf 79.7%

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

   if -1.3499999999999999e-47 < f < 4.9999999999999997e32

   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 77.8%

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

Alternative 7: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{n - f}{n + f}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return 1.0 / ((n - f) / (n + f))

Julia

function code(f, n)
	return Float64(1.0 / Float64(Float64(n - f) / Float64(n + f)))
end

MATLAB

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

Wolfram

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / N[(n + f), $MachinePrecision]), $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. Taylor expanded in n around inf 86.3%

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

   1. clear-num 86.3%

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

   2. inv-pow 86.3%

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

7. Applied egg-rr 86.3%

   \[\leadsto \color{blue}{{\left(\frac{n - f}{n \cdot \left(1 + \frac{f}{n}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 86.3%

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

   2. distribute-lft-in 86.2%

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

   3. *-rgt-identity 86.2%

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

   4. associate-*r/ 83.9%

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

   5. associate-*l/ 99.9%

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

   6. *-inverses 99.9%

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

   7. *-lft-identity 99.9%

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

9. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{n + f}}} \]
10. Add Preprocessing

Alternative 8: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{n + f}{n - f} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (+ n f) (- n f)))

C

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

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = (n + f) / (n - f)
end function

Java

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

Python

def code(f, n):
	return (n + f) / (n - f)

Julia

function code(f, n)
	return Float64(Float64(n + f) / Float64(n - f))
end

MATLAB

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

Wolfram

code[f_, n_] := N[(N[(n + f), $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. Final simplification 99.9%

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

Alternative 9: 49.7% 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 55.6%

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

Reproduce

herbie shell --seed 2024146 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))