subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.0s

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. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 2: 75.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{+64} \lor \neg \left(f \leq 5.4 \cdot 10^{-10}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -1.05e+64) (not (<= f 5.4e-10)))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -1.05e+64) || !(f <= 5.4e-10)) {
		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 <= (-1.05d+64)) .or. (.not. (f <= 5.4d-10))) 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 <= -1.05e+64) || !(f <= 5.4e-10)) {
		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 <= -1.05e+64) or not (f <= 5.4e-10):
		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 <= -1.05e+64) || !(f <= 5.4e-10))
		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 <= -1.05e+64) || ~((f <= 5.4e-10)))
		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, -1.05e+64], N[Not[LessEqual[f, 5.4e-10]], $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 < -1.05e64 or 5.4e-10 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in n around 0 82.7%

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

   if -1.05e64 < f < 5.4e-10

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 84.9%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{+64} \lor \neg \left(f \leq 5.4 \cdot 10^{-10}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 3: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 6.8 \cdot 10^{+42}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.05e+63) -1.0 (if (<= f 6.8e+42) (+ 1.0 (* 2.0 (/ f n))) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.05e+63) {
		tmp = -1.0;
	} else if (f <= 6.8e+42) {
		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 <= (-1.05d+63)) then
        tmp = -1.0d0
    else if (f <= 6.8d+42) 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 <= -1.05e+63) {
		tmp = -1.0;
	} else if (f <= 6.8e+42) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.05e+63:
		tmp = -1.0
	elif f <= 6.8e+42:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.05e+63)
		tmp = -1.0;
	elseif (f <= 6.8e+42)
		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 <= -1.05e+63)
		tmp = -1.0;
	elseif (f <= 6.8e+42)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.05e+63], -1.0, If[LessEqual[f, 6.8e+42], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.0500000000000001e63 or 6.7999999999999995e42 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 84.3%

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

   if -1.0500000000000001e63 < f < 6.7999999999999995e42

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 82.8%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -1.05 \cdot 10^{+63}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 6.8 \cdot 10^{+42}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5.7 \cdot 10^{+63}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -5.7e+63) -1.0 (if (<= f 5e-16) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -5.7e+63) {
		tmp = -1.0;
	} else if (f <= 5e-16) {
		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 <= (-5.7d+63)) then
        tmp = -1.0d0
    else if (f <= 5d-16) 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 <= -5.7e+63) {
		tmp = -1.0;
	} else if (f <= 5e-16) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -5.7e+63:
		tmp = -1.0
	elif f <= 5e-16:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -5.7e+63)
		tmp = -1.0;
	elseif (f <= 5e-16)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -5.7e+63)
		tmp = -1.0;
	elseif (f <= 5e-16)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -5.7e+63], -1.0, If[LessEqual[f, 5e-16], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -5.7000000000000002e63 or 5.0000000000000004e-16 < f

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around inf 81.3%

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

   if -5.7000000000000002e63 < f < 5.0000000000000004e-16

   1. Initial program 100.0%

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

      1. neg-mul-1 100.0%

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

      2. remove-double-neg 100.0%

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

      3. unsub-neg 100.0%

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

      4. distribute-neg-in 100.0%

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

      5. neg-mul-1 100.0%

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

      6. times-frac 100.0%

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

      7. metadata-eval 100.0%

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

      8. +-commutative 100.0%

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

      9. +-commutative 100.0%

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

      10. sub-neg 100.0%

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

      11. *-lft-identity 100.0%

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

      12. +-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in f around 0 84.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5.7 \cdot 10^{+63}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 5 \cdot 10^{-16}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 49.0% 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. neg-mul-1 100.0%

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

   2. remove-double-neg 100.0%

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

   3. unsub-neg 100.0%

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

   4. distribute-neg-in 100.0%

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

   5. neg-mul-1 100.0%

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

   6. times-frac 100.0%

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

   7. metadata-eval 100.0%

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

   8. +-commutative 100.0%

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

   9. +-commutative 100.0%

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

   10. sub-neg 100.0%

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

   11. *-lft-identity 100.0%

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

   12. +-commutative 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in f around inf 43.4%

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

5. Final simplification 43.4%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023335 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))