subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 7.0s

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: 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: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5.5 \cdot 10^{+29} \lor \neg \left(f \leq 4.1 \cdot 10^{-88}\right) \land \left(f \leq 7.8 \cdot 10^{-46} \lor \neg \left(f \leq 70000\right)\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -5.5e+29)
         (and (not (<= f 4.1e-88)) (or (<= f 7.8e-46) (not (<= f 70000.0)))))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (/ (* f 2.0) n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -5.5e+29) || (!(f <= 4.1e-88) && ((f <= 7.8e-46) || !(f <= 70000.0)))) {
		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 <= (-5.5d+29)) .or. (.not. (f <= 4.1d-88)) .and. (f <= 7.8d-46) .or. (.not. (f <= 70000.0d0))) 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 <= -5.5e+29) || (!(f <= 4.1e-88) && ((f <= 7.8e-46) || !(f <= 70000.0)))) {
		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 <= -5.5e+29) or (not (f <= 4.1e-88) and ((f <= 7.8e-46) or not (f <= 70000.0))):
		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 <= -5.5e+29) || (!(f <= 4.1e-88) && ((f <= 7.8e-46) || !(f <= 70000.0))))
		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 <= -5.5e+29) || (~((f <= 4.1e-88)) && ((f <= 7.8e-46) || ~((f <= 70000.0)))))
		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, -5.5e+29], And[N[Not[LessEqual[f, 4.1e-88]], $MachinePrecision], Or[LessEqual[f, 7.8e-46], N[Not[LessEqual[f, 70000.0]], $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 < -5.5e29 or 4.1000000000000001e-88 < f < 7.8000000000000005e-46 or 7e4 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in n around 0 80.0%

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

   if -5.5e29 < f < 4.1000000000000001e-88 or 7.8000000000000005e-46 < f < 7e4

   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 80.2%

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

      1. associate-*r/ 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5.5 \cdot 10^{+29} \lor \neg \left(f \leq 4.1 \cdot 10^{-88}\right) \land \left(f \leq 7.8 \cdot 10^{-46} \lor \neg \left(f \leq 70000\right)\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ t_1 := 1 + \frac{f \cdot 2}{n}\\ \mathbf{if}\;f \leq -6.8 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;f \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 1150000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))) (t_1 (+ 1.0 (/ (* f 2.0) n))))
   (if (<= f -6.8e+23)
     t_0
     (if (<= f 4.2e-88)
       t_1
       (if (<= f 8e-46) (/ f (- n f)) (if (<= f 1150000.0) t_1 t_0))))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = 1.0 + ((f * 2.0) / n);
	double tmp;
	if (f <= -6.8e+23) {
		tmp = t_0;
	} else if (f <= 4.2e-88) {
		tmp = t_1;
	} else if (f <= 8e-46) {
		tmp = f / (n - f);
	} else if (f <= 1150000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) - (n / f)
    t_1 = 1.0d0 + ((f * 2.0d0) / n)
    if (f <= (-6.8d+23)) then
        tmp = t_0
    else if (f <= 4.2d-88) then
        tmp = t_1
    else if (f <= 8d-46) then
        tmp = f / (n - f)
    else if (f <= 1150000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = 1.0 + ((f * 2.0) / n);
	double tmp;
	if (f <= -6.8e+23) {
		tmp = t_0;
	} else if (f <= 4.2e-88) {
		tmp = t_1;
	} else if (f <= 8e-46) {
		tmp = f / (n - f);
	} else if (f <= 1150000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	t_1 = 1.0 + ((f * 2.0) / n)
	tmp = 0
	if f <= -6.8e+23:
		tmp = t_0
	elif f <= 4.2e-88:
		tmp = t_1
	elif f <= 8e-46:
		tmp = f / (n - f)
	elif f <= 1150000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	t_1 = Float64(1.0 + Float64(Float64(f * 2.0) / n))
	tmp = 0.0
	if (f <= -6.8e+23)
		tmp = t_0;
	elseif (f <= 4.2e-88)
		tmp = t_1;
	elseif (f <= 8e-46)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 1150000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	t_1 = 1.0 + ((f * 2.0) / n);
	tmp = 0.0;
	if (f <= -6.8e+23)
		tmp = t_0;
	elseif (f <= 4.2e-88)
		tmp = t_1;
	elseif (f <= 8e-46)
		tmp = f / (n - f);
	elseif (f <= 1150000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(f * 2.0), $MachinePrecision] / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -6.8e+23], t$95$0, If[LessEqual[f, 4.2e-88], t$95$1, If[LessEqual[f, 8e-46], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 1150000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if f < -6.79999999999999983e23 or 1.15e6 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 80.3%

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

   6. Taylor expanded in f around inf 80.6%

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

      1. sub-neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. +-commutative 80.6%

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

      4. mul-1-neg 80.6%

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

      5. sub-neg 80.6%

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

   8. Simplified 80.6%

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

   if -6.79999999999999983e23 < f < 4.1999999999999999e-88 or 8.00000000000000018e-46 < f < 1.15e6

   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 80.2%

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

      1. associate-*r/ 80.2%

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

   7. Simplified 80.2%

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

   if 4.1999999999999999e-88 < f < 8.00000000000000018e-46

   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 71.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -6.8 \cdot 10^{+23}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{elif}\;f \leq 8 \cdot 10^{-46}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 1150000:\\ \;\;\;\;1 + \frac{f \cdot 2}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -6 \cdot 10^{+27} \lor \neg \left(f \leq 4 \cdot 10^{-88}\right) \land \left(f \leq 7.8 \cdot 10^{-46} \lor \neg \left(f \leq 1550\right)\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -6e+27)
         (and (not (<= f 4e-88)) (or (<= f 7.8e-46) (not (<= f 1550.0)))))
   (- -1.0 (/ n f))
   (+ 1.0 (/ f n))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -6e+27) || (!(f <= 4e-88) && ((f <= 7.8e-46) || !(f <= 1550.0)))) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((f <= (-6d+27)) .or. (.not. (f <= 4d-88)) .and. (f <= 7.8d-46) .or. (.not. (f <= 1550.0d0))) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0 + (f / n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -6e+27) || (!(f <= 4e-88) && ((f <= 7.8e-46) || !(f <= 1550.0)))) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0 + (f / n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -6e+27) or (not (f <= 4e-88) and ((f <= 7.8e-46) or not (f <= 1550.0))):
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0 + (f / n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -6e+27) || (!(f <= 4e-88) && ((f <= 7.8e-46) || !(f <= 1550.0))))
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(1.0 + Float64(f / n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -6e+27) || (~((f <= 4e-88)) && ((f <= 7.8e-46) || ~((f <= 1550.0)))))
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0 + (f / n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -6e+27], And[N[Not[LessEqual[f, 4e-88]], $MachinePrecision], Or[LessEqual[f, 7.8e-46], N[Not[LessEqual[f, 1550.0]], $MachinePrecision]]]], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -5.99999999999999953e27 or 3.99999999999999974e-88 < f < 7.8000000000000005e-46 or 1550 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 79.1%

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

   6. Taylor expanded in f around inf 79.3%

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

      1. sub-neg 79.3%

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

      2. metadata-eval 79.3%

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

      3. +-commutative 79.3%

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

      4. mul-1-neg 79.3%

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

      5. sub-neg 79.3%

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

   8. Simplified 79.3%

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

   if -5.99999999999999953e27 < f < 3.99999999999999974e-88 or 7.8000000000000005e-46 < f < 1550

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 79.7%

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

   6. Taylor expanded in n around inf 79.2%

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

      1. +-commutative 79.2%

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

   8. Simplified 79.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -6 \cdot 10^{+27} \lor \neg \left(f \leq 4 \cdot 10^{-88}\right) \land \left(f \leq 7.8 \cdot 10^{-46} \lor \neg \left(f \leq 1550\right)\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.5 \cdot 10^{+31}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88} \lor \neg \left(f \leq 6.8 \cdot 10^{-46}\right) \land f \leq 280000:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.5e+31)
   -1.0
   (if (or (<= f 4.2e-88) (and (not (<= f 6.8e-46)) (<= f 280000.0)))
     (+ 1.0 (/ f n))
     -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.5e+31) {
		tmp = -1.0;
	} else if ((f <= 4.2e-88) || (!(f <= 6.8e-46) && (f <= 280000.0))) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-2.5d+31)) then
        tmp = -1.0d0
    else if ((f <= 4.2d-88) .or. (.not. (f <= 6.8d-46)) .and. (f <= 280000.0d0)) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -2.5e+31) {
		tmp = -1.0;
	} else if ((f <= 4.2e-88) || (!(f <= 6.8e-46) && (f <= 280000.0))) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.5e+31:
		tmp = -1.0
	elif (f <= 4.2e-88) or (not (f <= 6.8e-46) and (f <= 280000.0)):
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.5e+31)
		tmp = -1.0;
	elseif ((f <= 4.2e-88) || (!(f <= 6.8e-46) && (f <= 280000.0)))
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.5e+31)
		tmp = -1.0;
	elseif ((f <= 4.2e-88) || (~((f <= 6.8e-46)) && (f <= 280000.0)))
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.5e+31], -1.0, If[Or[LessEqual[f, 4.2e-88], And[N[Not[LessEqual[f, 6.8e-46]], $MachinePrecision], LessEqual[f, 280000.0]]], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -2.50000000000000013e31 or 4.1999999999999999e-88 < f < 6.79999999999999992e-46 or 2.8e5 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 78.6%

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

   if -2.50000000000000013e31 < f < 4.1999999999999999e-88 or 6.79999999999999992e-46 < f < 2.8e5

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 79.7%

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

   6. Taylor expanded in n around inf 79.2%

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

      1. +-commutative 79.2%

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

   8. Simplified 79.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.5 \cdot 10^{+31}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88} \lor \neg \left(f \leq 6.8 \cdot 10^{-46}\right) \land f \leq 280000:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ t_1 := \frac{n}{n - f}\\ \mathbf{if}\;f \leq -1.05 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 6.6 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;f \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 12000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))) (t_1 (/ n (- n f))))
   (if (<= f -1.05e+31)
     t_0
     (if (<= f 6.6e-77)
       t_1
       (if (<= f 6.8e-46) (/ f (- n f)) (if (<= f 12000.0) t_1 t_0))))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = n / (n - f);
	double tmp;
	if (f <= -1.05e+31) {
		tmp = t_0;
	} else if (f <= 6.6e-77) {
		tmp = t_1;
	} else if (f <= 6.8e-46) {
		tmp = f / (n - f);
	} else if (f <= 12000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) - (n / f)
    t_1 = n / (n - f)
    if (f <= (-1.05d+31)) then
        tmp = t_0
    else if (f <= 6.6d-77) then
        tmp = t_1
    else if (f <= 6.8d-46) then
        tmp = f / (n - f)
    else if (f <= 12000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = n / (n - f);
	double tmp;
	if (f <= -1.05e+31) {
		tmp = t_0;
	} else if (f <= 6.6e-77) {
		tmp = t_1;
	} else if (f <= 6.8e-46) {
		tmp = f / (n - f);
	} else if (f <= 12000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	t_1 = n / (n - f)
	tmp = 0
	if f <= -1.05e+31:
		tmp = t_0
	elif f <= 6.6e-77:
		tmp = t_1
	elif f <= 6.8e-46:
		tmp = f / (n - f)
	elif f <= 12000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	t_1 = Float64(n / Float64(n - f))
	tmp = 0.0
	if (f <= -1.05e+31)
		tmp = t_0;
	elseif (f <= 6.6e-77)
		tmp = t_1;
	elseif (f <= 6.8e-46)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 12000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	t_1 = n / (n - f);
	tmp = 0.0;
	if (f <= -1.05e+31)
		tmp = t_0;
	elseif (f <= 6.6e-77)
		tmp = t_1;
	elseif (f <= 6.8e-46)
		tmp = f / (n - f);
	elseif (f <= 12000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -1.05e+31], t$95$0, If[LessEqual[f, 6.6e-77], t$95$1, If[LessEqual[f, 6.8e-46], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 12000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if f < -1.04999999999999989e31 or 12000 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 80.3%

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

   6. Taylor expanded in f around inf 80.6%

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

      1. sub-neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. +-commutative 80.6%

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

      4. mul-1-neg 80.6%

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

      5. sub-neg 80.6%

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

   8. Simplified 80.6%

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

   if -1.04999999999999989e31 < f < 6.59999999999999982e-77 or 6.79999999999999992e-46 < f < 12000

   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 78.9%

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

   if 6.59999999999999982e-77 < f < 6.79999999999999992e-46

   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 77.8%

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

Alternative 7: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ t_1 := 1 + \frac{f}{n}\\ \mathbf{if}\;f \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;f \leq 7.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 225000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))) (t_1 (+ 1.0 (/ f n))))
   (if (<= f -2.2e+27)
     t_0
     (if (<= f 4.2e-88)
       t_1
       (if (<= f 7.8e-46) (/ f (- n f)) (if (<= f 225000.0) t_1 t_0))))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = 1.0 + (f / n);
	double tmp;
	if (f <= -2.2e+27) {
		tmp = t_0;
	} else if (f <= 4.2e-88) {
		tmp = t_1;
	} else if (f <= 7.8e-46) {
		tmp = f / (n - f);
	} else if (f <= 225000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-1.0d0) - (n / f)
    t_1 = 1.0d0 + (f / n)
    if (f <= (-2.2d+27)) then
        tmp = t_0
    else if (f <= 4.2d-88) then
        tmp = t_1
    else if (f <= 7.8d-46) then
        tmp = f / (n - f)
    else if (f <= 225000.0d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double t_1 = 1.0 + (f / n);
	double tmp;
	if (f <= -2.2e+27) {
		tmp = t_0;
	} else if (f <= 4.2e-88) {
		tmp = t_1;
	} else if (f <= 7.8e-46) {
		tmp = f / (n - f);
	} else if (f <= 225000.0) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	t_1 = 1.0 + (f / n)
	tmp = 0
	if f <= -2.2e+27:
		tmp = t_0
	elif f <= 4.2e-88:
		tmp = t_1
	elif f <= 7.8e-46:
		tmp = f / (n - f)
	elif f <= 225000.0:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	t_1 = Float64(1.0 + Float64(f / n))
	tmp = 0.0
	if (f <= -2.2e+27)
		tmp = t_0;
	elseif (f <= 4.2e-88)
		tmp = t_1;
	elseif (f <= 7.8e-46)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 225000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	t_1 = 1.0 + (f / n);
	tmp = 0.0;
	if (f <= -2.2e+27)
		tmp = t_0;
	elseif (f <= 4.2e-88)
		tmp = t_1;
	elseif (f <= 7.8e-46)
		tmp = f / (n - f);
	elseif (f <= 225000.0)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -2.2e+27], t$95$0, If[LessEqual[f, 4.2e-88], t$95$1, If[LessEqual[f, 7.8e-46], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 225000.0], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if f < -2.1999999999999999e27 or 225000 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 80.3%

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

   6. Taylor expanded in f around inf 80.6%

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

      1. sub-neg 80.6%

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

      2. metadata-eval 80.6%

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

      3. +-commutative 80.6%

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

      4. mul-1-neg 80.6%

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

      5. sub-neg 80.6%

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

   8. Simplified 80.6%

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

   if -2.1999999999999999e27 < f < 4.1999999999999999e-88 or 7.8000000000000005e-46 < f < 225000

   1. Initial program 99.9%

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

      1. distribute-frac-neg 99.9%

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

      2. distribute-neg-frac2 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0 79.7%

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

   6. Taylor expanded in n around inf 79.2%

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

      1. +-commutative 79.2%

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

   8. Simplified 79.2%

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

   if 4.1999999999999999e-88 < f < 7.8000000000000005e-46

   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 71.2%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.2 \cdot 10^{+27}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 4.2 \cdot 10^{-88}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;f \leq 7.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 225000:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -7 \cdot 10^{+24}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 10^{-88}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq 6.8 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 50000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -7e+24)
   -1.0
   (if (<= f 1e-88)
     1.0
     (if (<= f 6.8e-46) -1.0 (if (<= f 50000.0) 1.0 -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -7e+24) {
		tmp = -1.0;
	} else if (f <= 1e-88) {
		tmp = 1.0;
	} else if (f <= 6.8e-46) {
		tmp = -1.0;
	} else if (f <= 50000.0) {
		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 <= (-7d+24)) then
        tmp = -1.0d0
    else if (f <= 1d-88) then
        tmp = 1.0d0
    else if (f <= 6.8d-46) then
        tmp = -1.0d0
    else if (f <= 50000.0d0) 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 <= -7e+24) {
		tmp = -1.0;
	} else if (f <= 1e-88) {
		tmp = 1.0;
	} else if (f <= 6.8e-46) {
		tmp = -1.0;
	} else if (f <= 50000.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -7e+24:
		tmp = -1.0
	elif f <= 1e-88:
		tmp = 1.0
	elif f <= 6.8e-46:
		tmp = -1.0
	elif f <= 50000.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -7e+24)
		tmp = -1.0;
	elseif (f <= 1e-88)
		tmp = 1.0;
	elseif (f <= 6.8e-46)
		tmp = -1.0;
	elseif (f <= 50000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -7e+24)
		tmp = -1.0;
	elseif (f <= 1e-88)
		tmp = 1.0;
	elseif (f <= 6.8e-46)
		tmp = -1.0;
	elseif (f <= 50000.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -7e+24], -1.0, If[LessEqual[f, 1e-88], 1.0, If[LessEqual[f, 6.8e-46], -1.0, If[LessEqual[f, 50000.0], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if f < -7.0000000000000004e24 or 9.99999999999999934e-89 < f < 6.79999999999999992e-46 or 5e4 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf 78.6%

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

   if -7.0000000000000004e24 < f < 9.99999999999999934e-89 or 6.79999999999999992e-46 < f < 5e4

   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 78.8%

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

Alternative 9: 50.2% 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 47.8%

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

Reproduce

herbie shell --seed 2024111 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))