subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 3.5s

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 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. Step-by-step derivation

   1. clear-num 100.0%

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

   2. associate-/r/ 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-/r/ 100.0%

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

   2. +-commutative 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5.2 \cdot 10^{+22} \lor \neg \left(f \leq 6500000000\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (or (<= f -5.2e+22) (not (<= f 6500000000.0)))
   (+ (* -2.0 (/ n f)) -1.0)
   (+ 1.0 (* 2.0 (/ f n)))))

C

double code(double f, double n) {
	double tmp;
	if ((f <= -5.2e+22) || !(f <= 6500000000.0)) {
		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 <= (-5.2d+22)) .or. (.not. (f <= 6500000000.0d0))) 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 <= -5.2e+22) || !(f <= 6500000000.0)) {
		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 <= -5.2e+22) or not (f <= 6500000000.0):
		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 <= -5.2e+22) || !(f <= 6500000000.0))
		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 <= -5.2e+22) || ~((f <= 6500000000.0)))
		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, -5.2e+22], N[Not[LessEqual[f, 6500000000.0]], $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 < -5.2e22 or 6.5e9 < 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 78.1%

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

   if -5.2e22 < f < 6.5e9

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 76.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5.2 \cdot 10^{+22} \lor \neg \left(f \leq 6500000000\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 3: 74.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 43000000000:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -4.2e+15)
   -1.0
   (if (<= f 43000000000.0) (+ 1.0 (* 2.0 (/ f n))) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -4.2e+15) {
		tmp = -1.0;
	} else if (f <= 43000000000.0) {
		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 <= (-4.2d+15)) then
        tmp = -1.0d0
    else if (f <= 43000000000.0d0) 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 <= -4.2e+15) {
		tmp = -1.0;
	} else if (f <= 43000000000.0) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -4.2e+15:
		tmp = -1.0
	elif f <= 43000000000.0:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -4.2e+15)
		tmp = -1.0;
	elseif (f <= 43000000000.0)
		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 <= -4.2e+15)
		tmp = -1.0;
	elseif (f <= 43000000000.0)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -4.2e+15], -1.0, If[LessEqual[f, 43000000000.0], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.2e15 or 4.3e10 < 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 77.7%

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

   if -4.2e15 < f < 4.3e10

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 76.6%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -4.2 \cdot 10^{+15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 43000000000:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 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 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%

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

Alternative 5: 73.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 9 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -5e+23) -1.0 (if (<= f 9e-34) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -5e+23) {
		tmp = -1.0;
	} else if (f <= 9e-34) {
		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 <= (-5d+23)) then
        tmp = -1.0d0
    else if (f <= 9d-34) 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 <= -5e+23) {
		tmp = -1.0;
	} else if (f <= 9e-34) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -5e+23:
		tmp = -1.0
	elif f <= 9e-34:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -5e+23)
		tmp = -1.0;
	elseif (f <= 9e-34)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -5e+23)
		tmp = -1.0;
	elseif (f <= 9e-34)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -5e+23], -1.0, If[LessEqual[f, 9e-34], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -4.9999999999999999e23 or 9.00000000000000085e-34 < 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 76.8%

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

   if -4.9999999999999999e23 < f < 9.00000000000000085e-34

   1. Initial program 99.9%

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

      1. neg-mul-1 99.9%

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

      2. remove-double-neg 99.9%

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

      3. unsub-neg 99.9%

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

      4. distribute-neg-in 99.9%

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

      5. neg-mul-1 99.9%

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

      6. times-frac 99.9%

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

      7. metadata-eval 99.9%

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

      8. +-commutative 99.9%

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

      9. +-commutative 99.9%

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

      10. sub-neg 99.9%

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

      11. *-lft-identity 99.9%

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

      12. +-commutative 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in f around 0 76.0%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+23}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 9 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 48.9% 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 48.9%

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

5. Final simplification 48.9%

   \[\leadsto -1 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))