subtraction fraction

Percentage Accurate: 100.0% → 99.9%

Time: 21.0s

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: 99.9% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(\frac{f + n}{n - f}\right)}^{3}} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (cbrt (pow (/ (+ f n) (- n f)) 3.0)))

C

double code(double f, double n) {
	return cbrt(pow(((f + n) / (n - f)), 3.0));
}

Java

public static double code(double f, double n) {
	return Math.cbrt(Math.pow(((f + n) / (n - f)), 3.0));
}

Julia

function code(f, n)
	return cbrt((Float64(Float64(f + n) / Float64(n - f)) ^ 3.0))
end

Wolfram

code[f_, n_] := N[Power[N[Power[N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $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. Step-by-step derivation

   1. add-cbrt-cube 99.9%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\frac{f + n}{n - f} \cdot \frac{f + n}{n - f}\right) \cdot \frac{f + n}{n - f}}} \]

   2. pow3 99.9%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]
7. Add Preprocessing

Alternative 2: 75.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+26} \lor \neg \left(f \leq 6.5 \cdot 10^{+37}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -4.2e+26) (not (<= f 6.5e+37)))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (/ (* f 2.0) n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -4.2e+26) || !(f <= 6.5e+37)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} 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 ((f <= (-4.2d+26)) .or. (.not. (f <= 6.5d+37))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    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 ((f <= -4.2e+26) || !(f <= 6.5e+37)) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + ((f * 2.0) / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -4.2e+26) or not (f <= 6.5e+37):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + ((f * 2.0) / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -4.2e+26) || !(f <= 6.5e+37))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	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 ((f <= -4.2e+26) || ~((f <= 6.5e+37)))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + ((f * 2.0) / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -4.2e+26], N[Not[LessEqual[f, 6.5e+37]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -4.2000000000000002e26 or 6.4999999999999998e37 < 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 80.7%

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

   if -4.2000000000000002e26 < f < 6.4999999999999998e37

   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 73.3%

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

      1. associate-*r/ 73.3%

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

   7. Simplified 73.3%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+26} \lor \neg \left(f \leq 6.5 \cdot 10^{+37}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3.6 \cdot 10^{+25} \lor \neg \left(f \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -3.6e+25) (not (<= f 2.8e+37)))
   (/ f (- n f))
   (+ 1.0 (/ (* f 2.0) n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -3.6e+25) || !(f <= 2.8e+37)) {
		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 ((f <= (-3.6d+25)) .or. (.not. (f <= 2.8d+37))) 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 ((f <= -3.6e+25) || !(f <= 2.8e+37)) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0 + ((f * 2.0) / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -3.6e+25) or not (f <= 2.8e+37):
		tmp = f / (n - f)
	else:
		tmp = 1.0 + ((f * 2.0) / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -3.6e+25) || !(f <= 2.8e+37))
		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 ((f <= -3.6e+25) || ~((f <= 2.8e+37)))
		tmp = f / (n - f);
	else
		tmp = 1.0 + ((f * 2.0) / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -3.6e+25], N[Not[LessEqual[f, 2.8e+37]], $MachinePrecision]], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -3.60000000000000015e25 or 2.7999999999999998e37 < 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 80.5%

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

   if -3.60000000000000015e25 < f < 2.7999999999999998e37

   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 73.3%

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

      1. associate-*r/ 73.3%

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

   7. Simplified 73.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -3.6 \cdot 10^{+25} \lor \neg \left(f \leq 2.8 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.4 \cdot 10^{+26} \lor \neg \left(f \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{f + n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.4e+26) (not (<= f 5.2e+37))) (/ f (- n f)) (/ (+ f n) n)))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.4e+26) || !(f <= 5.2e+37)) {
		tmp = f / (n - f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-1.4d+26)) .or. (.not. (f <= 5.2d+37))) then
        tmp = f / (n - f)
    else
        tmp = (f + n) / n
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -1.4e+26) || !(f <= 5.2e+37)) {
		tmp = f / (n - f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -1.4e+26) or not (f <= 5.2e+37):
		tmp = f / (n - f)
	else:
		tmp = (f + n) / n
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -1.4e+26) || !(f <= 5.2e+37))
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(Float64(f + n) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -1.4e+26) || ~((f <= 5.2e+37)))
		tmp = f / (n - f);
	else
		tmp = (f + n) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -1.4e+26], N[Not[LessEqual[f, 5.2e+37]], $MachinePrecision]], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(N[(f + n), $MachinePrecision] / n), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -1.4e26 or 5.1999999999999998e37 < 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 80.5%

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

   if -1.4e26 < f < 5.1999999999999998e37

   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 99.9%

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

      1. mul-1-neg 99.9%

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

      2. unsub-neg 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in n around inf 72.7%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.4 \cdot 10^{+26} \lor \neg \left(f \leq 5.2 \cdot 10^{+37}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{f + n}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+25} \lor \neg \left(f \leq 3 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -4.2e+25) (not (<= f 3e-43))) (/ f (- n f)) (/ n (- n f))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -4.2e+25) || !(f <= 3e-43)) {
		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 <= (-4.2d+25)) .or. (.not. (f <= 3d-43))) 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 <= -4.2e+25) || !(f <= 3e-43)) {
		tmp = f / (n - f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -4.2e+25) or not (f <= 3e-43):
		tmp = f / (n - f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -4.2e+25) || !(f <= 3e-43))
		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 <= -4.2e+25) || ~((f <= 3e-43)))
		tmp = f / (n - f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -4.2e+25], N[Not[LessEqual[f, 3e-43]], $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 < -4.1999999999999998e25 or 3.00000000000000003e-43 < 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 77.9%

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

   if -4.1999999999999998e25 < f < 3.00000000000000003e-43

   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 74.7%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+25} \lor \neg \left(f \leq 3 \cdot 10^{-43}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -8.8 \cdot 10^{+26} \lor \neg \left(f \leq 1.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -8.8e+26) (not (<= f 1.5e-44))) (/ f (- n f)) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -8.8e+26) || !(f <= 1.5e-44)) {
		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 <= (-8.8d+26)) .or. (.not. (f <= 1.5d-44))) 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 <= -8.8e+26) || !(f <= 1.5e-44)) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -8.8e+26) or not (f <= 1.5e-44):
		tmp = f / (n - f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -8.8e+26) || !(f <= 1.5e-44))
		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 <= -8.8e+26) || ~((f <= 1.5e-44)))
		tmp = f / (n - f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -8.8e+26], N[Not[LessEqual[f, 1.5e-44]], $MachinePrecision]], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if f < -8.80000000000000028e26 or 1.5000000000000001e-44 < 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 77.9%

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

   if -8.80000000000000028e26 < f < 1.5000000000000001e-44

   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 73.9%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -8.8 \cdot 10^{+26} \lor \neg \left(f \leq 1.5 \cdot 10^{-44}\right):\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3.5 \cdot 10^{+25}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 10^{-44}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -3.5e+25) -1.0 (if (<= f 1e-44) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -3.5e+25) {
		tmp = -1.0;
	} else if (f <= 1e-44) {
		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 <= (-3.5d+25)) then
        tmp = -1.0d0
    else if (f <= 1d-44) 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 <= -3.5e+25) {
		tmp = -1.0;
	} else if (f <= 1e-44) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -3.5e+25:
		tmp = -1.0
	elif f <= 1e-44:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -3.5e+25)
		tmp = -1.0;
	elseif (f <= 1e-44)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -3.5e+25)
		tmp = -1.0;
	elseif (f <= 1e-44)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -3.5e+25], -1.0, If[LessEqual[f, 1e-44], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -3.49999999999999999e25 or 9.99999999999999953e-45 < 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 77.3%

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

   if -3.49999999999999999e25 < f < 9.99999999999999953e-45

   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 73.9%

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

Alternative 8: 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 9: 49.8% 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 52.5%

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

Reproduce

herbie shell --seed 2024116 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))