subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 3.0s

Alternatives: 5

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. /-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. neg-mul-1 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-/l* 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/r* 100.0%

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

   9. neg-mul-1 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. remove-double-neg 100.0%

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

   14. 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. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3.35 \cdot 10^{+56} \lor \neg \left(f \leq -1.22 \cdot 10^{-7} \lor \neg \left(f \leq -2.7 \cdot 10^{-60}\right) \land f \leq 2.7 \cdot 10^{-17}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -3.35e+56)
         (not (or (<= f -1.22e-7) (and (not (<= f -2.7e-60)) (<= f 2.7e-17)))))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -3.35e+56) || !((f <= -1.22e-7) || (!(f <= -2.7e-60) && (f <= 2.7e-17)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} 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) :: tmp
    if ((f <= (-3.35d+56)) .or. (.not. (f <= (-1.22d-7)) .or. (.not. (f <= (-2.7d-60))) .and. (f <= 2.7d-17))) then
        tmp = ((-2.0d0) * (n / f)) + (-1.0d0)
    else
        tmp = 1.0d0 + (2.0d0 * (f / n))
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((f <= -3.35e+56) || !((f <= -1.22e-7) || (!(f <= -2.7e-60) && (f <= 2.7e-17)))) {
		tmp = (-2.0 * (n / f)) + -1.0;
	} else {
		tmp = 1.0 + (2.0 * (f / n));
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (f <= -3.35e+56) or not ((f <= -1.22e-7) or (not (f <= -2.7e-60) and (f <= 2.7e-17))):
		tmp = (-2.0 * (n / f)) + -1.0
	else:
		tmp = 1.0 + (2.0 * (f / n))
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if ((f <= -3.35e+56) || !((f <= -1.22e-7) || (!(f <= -2.7e-60) && (f <= 2.7e-17))))
		tmp = Float64(Float64(-2.0 * Float64(n / f)) + -1.0);
	else
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((f <= -3.35e+56) || ~(((f <= -1.22e-7) || (~((f <= -2.7e-60)) && (f <= 2.7e-17)))))
		tmp = (-2.0 * (n / f)) + -1.0;
	else
		tmp = 1.0 + (2.0 * (f / n));
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[Or[LessEqual[f, -3.35e+56], N[Not[Or[LessEqual[f, -1.22e-7], And[N[Not[LessEqual[f, -2.7e-60]], $MachinePrecision], LessEqual[f, 2.7e-17]]]], $MachinePrecision]], N[(N[(-2.0 * N[(n / f), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if f < -3.35e56 or -1.2200000000000001e-7 < f < -2.7e-60 or 2.7000000000000001e-17 < f

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. 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 79.0%

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

   if -3.35e56 < f < -1.2200000000000001e-7 or -2.7e-60 < f < 2.7000000000000001e-17

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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 80.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -3.35 \cdot 10^{+56} \lor \neg \left(f \leq -1.22 \cdot 10^{-7} \lor \neg \left(f \leq -2.7 \cdot 10^{-60}\right) \land f \leq 2.7 \cdot 10^{-17}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4 \cdot 10^{+100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -5.5 \cdot 10^{-12} \lor \neg \left(f \leq -1.45 \cdot 10^{-62}\right) \land f \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -4e+100)
   -1.0
   (if (or (<= f -5.5e-12) (and (not (<= f -1.45e-62)) (<= f 3.2e-18)))
     (+ 1.0 (* 2.0 (/ f n)))
     -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -4e+100) {
		tmp = -1.0;
	} else if ((f <= -5.5e-12) || (!(f <= -1.45e-62) && (f <= 3.2e-18))) {
		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 <= (-4d+100)) then
        tmp = -1.0d0
    else if ((f <= (-5.5d-12)) .or. (.not. (f <= (-1.45d-62))) .and. (f <= 3.2d-18)) 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 <= -4e+100) {
		tmp = -1.0;
	} else if ((f <= -5.5e-12) || (!(f <= -1.45e-62) && (f <= 3.2e-18))) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -4e+100:
		tmp = -1.0
	elif (f <= -5.5e-12) or (not (f <= -1.45e-62) and (f <= 3.2e-18)):
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -4e+100)
		tmp = -1.0;
	elseif ((f <= -5.5e-12) || (!(f <= -1.45e-62) && (f <= 3.2e-18)))
		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 <= -4e+100)
		tmp = -1.0;
	elseif ((f <= -5.5e-12) || (~((f <= -1.45e-62)) && (f <= 3.2e-18)))
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -4e+100], -1.0, If[Or[LessEqual[f, -5.5e-12], And[N[Not[LessEqual[f, -1.45e-62]], $MachinePrecision], LessEqual[f, 3.2e-18]]], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.00000000000000006e100 or -5.5000000000000004e-12 < f < -1.44999999999999993e-62 or 3.1999999999999999e-18 < f

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. 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 -4.00000000000000006e100 < f < -5.5000000000000004e-12 or -1.44999999999999993e-62 < f < 3.1999999999999999e-18

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4 \cdot 10^{+100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -5.5 \cdot 10^{-12} \lor \neg \left(f \leq -1.45 \cdot 10^{-62}\right) \land f \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+57}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -8.8 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1 \cdot 10^{-59}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 9 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2e+57)
   -1.0
   (if (<= f -8.8e+26)
     1.0
     (if (<= f -1e-59) -1.0 (if (<= f 9e-75) 1.0 -1.0)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2e+57) {
		tmp = -1.0;
	} else if (f <= -8.8e+26) {
		tmp = 1.0;
	} else if (f <= -1e-59) {
		tmp = -1.0;
	} else if (f <= 9e-75) {
		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 <= (-2d+57)) then
        tmp = -1.0d0
    else if (f <= (-8.8d+26)) then
        tmp = 1.0d0
    else if (f <= (-1d-59)) then
        tmp = -1.0d0
    else if (f <= 9d-75) 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 <= -2e+57) {
		tmp = -1.0;
	} else if (f <= -8.8e+26) {
		tmp = 1.0;
	} else if (f <= -1e-59) {
		tmp = -1.0;
	} else if (f <= 9e-75) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2e+57:
		tmp = -1.0
	elif f <= -8.8e+26:
		tmp = 1.0
	elif f <= -1e-59:
		tmp = -1.0
	elif f <= 9e-75:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2e+57)
		tmp = -1.0;
	elseif (f <= -8.8e+26)
		tmp = 1.0;
	elseif (f <= -1e-59)
		tmp = -1.0;
	elseif (f <= 9e-75)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2e+57)
		tmp = -1.0;
	elseif (f <= -8.8e+26)
		tmp = 1.0;
	elseif (f <= -1e-59)
		tmp = -1.0;
	elseif (f <= 9e-75)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2e+57], -1.0, If[LessEqual[f, -8.8e+26], 1.0, If[LessEqual[f, -1e-59], -1.0, If[LessEqual[f, 9e-75], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if f < -2.0000000000000001e57 or -8.80000000000000028e26 < f < -1e-59 or 9.0000000000000006e-75 < f

   1. Initial program 100.0%

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

      1. /-rgt-identity 100.0%

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

      2. metadata-eval 100.0%

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

      3. neg-mul-1 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-/l* 100.0%

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

      6. metadata-eval 100.0%

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

      7. metadata-eval 100.0%

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

      8. associate-/r* 100.0%

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

      9. neg-mul-1 100.0%

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

      10. sub-neg 100.0%

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

      11. +-commutative 100.0%

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

      12. distribute-neg-in 100.0%

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

      13. remove-double-neg 100.0%

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

      14. 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 74.6%

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

   if -2.0000000000000001e57 < f < -8.80000000000000028e26 or -1e-59 < f < 9.0000000000000006e-75

   1. Initial program 99.9%

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

      1. /-rgt-identity 99.9%

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

      2. metadata-eval 99.9%

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

      3. neg-mul-1 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-/l* 99.9%

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

      6. metadata-eval 99.9%

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

      7. metadata-eval 99.9%

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

      8. associate-/r* 99.9%

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

      9. neg-mul-1 99.9%

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

      10. sub-neg 99.9%

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

      11. +-commutative 99.9%

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

      12. distribute-neg-in 99.9%

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

      13. remove-double-neg 99.9%

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

      14. 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 83.1%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -2 \cdot 10^{+57}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq -8.8 \cdot 10^{+26}:\\ \;\;\;\;1\\ \mathbf{elif}\;f \leq -1 \cdot 10^{-59}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 9 \cdot 10^{-75}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.8% 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 100.0%

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

   1. /-rgt-identity 100.0%

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

   2. metadata-eval 100.0%

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

   3. neg-mul-1 100.0%

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

   4. *-commutative 100.0%

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

   5. associate-/l* 100.0%

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

   6. metadata-eval 100.0%

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

   7. metadata-eval 100.0%

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

   8. associate-/r* 100.0%

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

   9. neg-mul-1 100.0%

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

   10. sub-neg 100.0%

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

   11. +-commutative 100.0%

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

   12. distribute-neg-in 100.0%

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

   13. remove-double-neg 100.0%

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

   14. 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 49.0%

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

6. Final simplification 49.0%

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

Reproduce

herbie shell --seed 2024029 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))