subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 6

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, 0.9× 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. inv-pow 100.0%

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

6. Applied egg-rr 100.0%

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

   1. unpow-1 100.0%

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

   2. +-commutative 100.0%

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

8. Applied egg-rr 100.0%

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

9. Final simplification 100.0%

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

Alternative 2: 71.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \frac{n}{f} + -1\\ \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -9.5 \cdot 10^{-5} \lor \neg \left(f \leq 1.02 \cdot 10^{-16}\right) \land \left(f \leq 1.1 \cdot 10^{+37} \lor \neg \left(f \leq 1.35 \cdot 10^{+131}\right)\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ (* -2.0 (/ n f)) -1.0)))
   (if (<= f -2.7e+158)
     t_0
     (if (<= f -4.3e+102)
       1.0
       (if (or (<= f -9.5e-5)
               (and (not (<= f 1.02e-16))
                    (or (<= f 1.1e+37) (not (<= f 1.35e+131)))))
         t_0
         (+ 1.0 (* 2.0 (/ f n))))))))

C

double code(double f, double n) {
	double t_0 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (f <= -2.7e+158) {
		tmp = t_0;
	} else if (f <= -4.3e+102) {
		tmp = 1.0;
	} else if ((f <= -9.5e-5) || (!(f <= 1.02e-16) && ((f <= 1.1e+37) || !(f <= 1.35e+131)))) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	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) :: tmp
    t_0 = ((-2.0d0) * (n / f)) + (-1.0d0)
    if (f <= (-2.7d+158)) then
        tmp = t_0
    else if (f <= (-4.3d+102)) then
        tmp = 1.0d0
    else if ((f <= (-9.5d-5)) .or. (.not. (f <= 1.02d-16)) .and. (f <= 1.1d+37) .or. (.not. (f <= 1.35d+131))) then
        tmp = t_0
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = (-2.0 * (n / f)) + -1.0;
	double tmp;
	if (f <= -2.7e+158) {
		tmp = t_0;
	} else if (f <= -4.3e+102) {
		tmp = 1.0;
	} else if ((f <= -9.5e-5) || (!(f <= 1.02e-16) && ((f <= 1.1e+37) || !(f <= 1.35e+131)))) {
		tmp = t_0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = (-2.0 * (n / f)) + -1.0
	tmp = 0
	if f <= -2.7e+158:
		tmp = t_0
	elif f <= -4.3e+102:
		tmp = 1.0
	elif (f <= -9.5e-5) or (not (f <= 1.02e-16) and ((f <= 1.1e+37) or not (f <= 1.35e+131))):
		tmp = t_0
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	t_0 = Float64(Float64(-2.0 * Float64(n / f)) + -1.0)
	tmp = 0.0
	if (f <= -2.7e+158)
		tmp = t_0;
	elseif (f <= -4.3e+102)
		tmp = 1.0;
	elseif ((f <= -9.5e-5) || (!(f <= 1.02e-16) && ((f <= 1.1e+37) || !(f <= 1.35e+131))))
		tmp = t_0;
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = (-2.0 * (n / f)) + -1.0;
	tmp = 0.0;
	if (f <= -2.7e+158)
		tmp = t_0;
	elseif (f <= -4.3e+102)
		tmp = 1.0;
	elseif ((f <= -9.5e-5) || (~((f <= 1.02e-16)) && ((f <= 1.1e+37) || ~((f <= 1.35e+131)))))
		tmp = t_0;
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[f, -2.7e+158], t$95$0, If[LessEqual[f, -4.3e+102], 1.0, If[Or[LessEqual[f, -9.5e-5], And[N[Not[LessEqual[f, 1.02e-16]], $MachinePrecision], Or[LessEqual[f, 1.1e+37], N[Not[LessEqual[f, 1.35e+131]], $MachinePrecision]]]], t$95$0, N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if f < -2.69999999999999979e158 or -4.3000000000000001e102 < f < -9.5000000000000005e-5 or 1.0200000000000001e-16 < f < 1.1e37 or 1.35000000000000002e131 < 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 83.2%

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

   if -2.69999999999999979e158 < f < -4.3000000000000001e102

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 100.0%

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

   if -9.5000000000000005e-5 < f < 1.0200000000000001e-16 or 1.1e37 < f < 1.35000000000000002e131

   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 80.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;f \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -9.5 \cdot 10^{-5} \lor \neg \left(f \leq 1.02 \cdot 10^{-16}\right) \land \left(f \leq 1.1 \cdot 10^{+37} \lor \neg \left(f \leq 1.35 \cdot 10^{+131}\right)\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.75 \cdot 10^{-16} \lor \neg \left(f \leq 1.95 \cdot 10^{+36}\right) \land f \leq 1.35 \cdot 10^{+131}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.7e+158)
   -1.0
   (if (<= f -4.3e+102)
     1.0
     (if (<= f -7.5e-10)
       -1.0
       (if (or (<= f 1.75e-16) (and (not (<= f 1.95e+36)) (<= f 1.35e+131)))
         (+ 1.0 (* 2.0 (/ f n)))
         -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.7e+158) {
		tmp = -1.0;
	} else if (f <= -4.3e+102) {
		tmp = 1.0;
	} else if (f <= -7.5e-10) {
		tmp = -1.0;
	} else if ((f <= 1.75e-16) || (!(f <= 1.95e+36) && (f <= 1.35e+131))) {
		tmp = 1.0 + (2.0 * (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 <= (-2.7d+158)) then
        tmp = -1.0d0
    else if (f <= (-4.3d+102)) then
        tmp = 1.0d0
    else if (f <= (-7.5d-10)) then
        tmp = -1.0d0
    else if ((f <= 1.75d-16) .or. (.not. (f <= 1.95d+36)) .and. (f <= 1.35d+131)) then
        tmp = 1.0d0 + (2.0d0 * (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 <= -2.7e+158) {
		tmp = -1.0;
	} else if (f <= -4.3e+102) {
		tmp = 1.0;
	} else if (f <= -7.5e-10) {
		tmp = -1.0;
	} else if ((f <= 1.75e-16) || (!(f <= 1.95e+36) && (f <= 1.35e+131))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.7e+158:
		tmp = -1.0
	elif f <= -4.3e+102:
		tmp = 1.0
	elif f <= -7.5e-10:
		tmp = -1.0
	elif (f <= 1.75e-16) or (not (f <= 1.95e+36) and (f <= 1.35e+131)):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.7e+158)
		tmp = -1.0;
	elseif (f <= -4.3e+102)
		tmp = 1.0;
	elseif (f <= -7.5e-10)
		tmp = -1.0;
	elseif ((f <= 1.75e-16) || (!(f <= 1.95e+36) && (f <= 1.35e+131)))
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.7e+158)
		tmp = -1.0;
	elseif (f <= -4.3e+102)
		tmp = 1.0;
	elseif (f <= -7.5e-10)
		tmp = -1.0;
	elseif ((f <= 1.75e-16) || (~((f <= 1.95e+36)) && (f <= 1.35e+131)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.7e+158], -1.0, If[LessEqual[f, -4.3e+102], 1.0, If[LessEqual[f, -7.5e-10], -1.0, If[Or[LessEqual[f, 1.75e-16], And[N[Not[LessEqual[f, 1.95e+36]], $MachinePrecision], LessEqual[f, 1.35e+131]]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if f < -2.69999999999999979e158 or -4.3000000000000001e102 < f < -7.49999999999999995e-10 or 1.75000000000000009e-16 < f < 1.9500000000000001e36 or 1.35000000000000002e131 < 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 81.6%

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

   if -2.69999999999999979e158 < f < -4.3000000000000001e102

   1. Initial program 100.0%

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

      1. distribute-frac-neg 100.0%

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

      2. distribute-neg-frac2 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. sub-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0 100.0%

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

   if -7.49999999999999995e-10 < f < 1.75000000000000009e-16 or 1.9500000000000001e36 < f < 1.35000000000000002e131

   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 81.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -4.3 \cdot 10^{+102}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -7.5 \cdot 10^{-10}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.75 \cdot 10^{-16} \lor \neg \left(f \leq 1.95 \cdot 10^{+36}\right) \land f \leq 1.35 \cdot 10^{+131}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 71.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -7.6 \cdot 10^{+101}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.7e+158)
   -1.0
   (if (<= f -7.6e+101)
     1.0
     (if (<= f -4.1e-5)
       -1.0
       (if (<= f 4e-16)
         1.0
         (if (<= f 2.2e+75) -1.0 (if (<= f 2.55e+122) 1.0 -1.0)))))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.7e+158) {
		tmp = -1.0;
	} else if (f <= -7.6e+101) {
		tmp = 1.0;
	} else if (f <= -4.1e-5) {
		tmp = -1.0;
	} else if (f <= 4e-16) {
		tmp = 1.0;
	} else if (f <= 2.2e+75) {
		tmp = -1.0;
	} else if (f <= 2.55e+122) {
		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 <= (-2.7d+158)) then
        tmp = -1.0d0
    else if (f <= (-7.6d+101)) then
        tmp = 1.0d0
    else if (f <= (-4.1d-5)) then
        tmp = -1.0d0
    else if (f <= 4d-16) then
        tmp = 1.0d0
    else if (f <= 2.2d+75) then
        tmp = -1.0d0
    else if (f <= 2.55d+122) 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 <= -2.7e+158) {
		tmp = -1.0;
	} else if (f <= -7.6e+101) {
		tmp = 1.0;
	} else if (f <= -4.1e-5) {
		tmp = -1.0;
	} else if (f <= 4e-16) {
		tmp = 1.0;
	} else if (f <= 2.2e+75) {
		tmp = -1.0;
	} else if (f <= 2.55e+122) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.7e+158:
		tmp = -1.0
	elif f <= -7.6e+101:
		tmp = 1.0
	elif f <= -4.1e-5:
		tmp = -1.0
	elif f <= 4e-16:
		tmp = 1.0
	elif f <= 2.2e+75:
		tmp = -1.0
	elif f <= 2.55e+122:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.7e+158)
		tmp = -1.0;
	elseif (f <= -7.6e+101)
		tmp = 1.0;
	elseif (f <= -4.1e-5)
		tmp = -1.0;
	elseif (f <= 4e-16)
		tmp = 1.0;
	elseif (f <= 2.2e+75)
		tmp = -1.0;
	elseif (f <= 2.55e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.7e+158)
		tmp = -1.0;
	elseif (f <= -7.6e+101)
		tmp = 1.0;
	elseif (f <= -4.1e-5)
		tmp = -1.0;
	elseif (f <= 4e-16)
		tmp = 1.0;
	elseif (f <= 2.2e+75)
		tmp = -1.0;
	elseif (f <= 2.55e+122)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.7e+158], -1.0, If[LessEqual[f, -7.6e+101], 1.0, If[LessEqual[f, -4.1e-5], -1.0, If[LessEqual[f, 4e-16], 1.0, If[LessEqual[f, 2.2e+75], -1.0, If[LessEqual[f, 2.55e+122], 1.0, -1.0]]]]]]

Derivation

1. Split input into 2 regimes

2. if f < -2.69999999999999979e158 or -7.5999999999999996e101 < f < -4.10000000000000005e-5 or 3.9999999999999999e-16 < f < 2.20000000000000012e75 or 2.55e122 < 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 79.3%

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

   if -2.69999999999999979e158 < f < -7.5999999999999996e101 or -4.10000000000000005e-5 < f < 3.9999999999999999e-16 or 2.20000000000000012e75 < f < 2.55e122

   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 81.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2.7 \cdot 10^{+158}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -7.6 \cdot 10^{+101}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -4.1 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq 2.2 \cdot 10^{+75}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 6: 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 44.0%

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

6. Final simplification 44.0%

   \[\leadsto -1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024045 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))