subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 4.9s

Alternatives: 9

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{-\left(f + n\right)}{f - n} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (- (+ f n)) (- f n)))

C

double code(double f, double n) {
	return -(f + n) / (f - n);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -(f + n) / (f - n)
end function

Java

public static double code(double f, double n) {
	return -(f + n) / (f - n);
}

Python

def code(f, n):
	return -(f + n) / (f - n)

Julia

function code(f, n)
	return Float64(Float64(-Float64(f + n)) / Float64(f - n))
end

MATLAB

function tmp = code(f, n)
	tmp = -(f + n) / (f - n);
end

Wolfram

code[f_, n_] := N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{f + n}{n - f} \end{array} \]

FPCore

(FPCore (f n) :precision binary64 (/ (+ f n) (- n f)))

C

double code(double f, double n) {
	return (f + n) / (n - f);
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = (f + n) / (n - f)
end function

Java

public static double code(double f, double n) {
	return (f + n) / (n - f);
}

Python

def code(f, n):
	return (f + n) / (n - f)

Julia

function code(f, n)
	return Float64(Float64(f + n) / Float64(n - f))
end

MATLAB

function tmp = code(f, n)
	tmp = (f + n) / (n - f);
end

Wolfram

code[f_, n_] := N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

   8. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

   10. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -7.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{n \cdot -2}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -7.5e-56)
   (/ f (- n f))
   (if (<= f 4.6e+14) (+ 1.0 (* 2.0 (/ f n))) (+ -1.0 (/ (* n -2.0) f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 4.6e+14) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0 + ((n * -2.0) / f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-7.5d-56)) then
        tmp = f / (n - f)
    else if (f <= 4.6d+14) then
        tmp = 1.0d0 + (2.0d0 * (f / n))
    else
        tmp = (-1.0d0) + ((n * (-2.0d0)) / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -7.5e-56) {
		tmp = f / (n - f);
	} else if (f <= 4.6e+14) {
		tmp = 1.0 + (2.0 * (f / n));
	} else {
		tmp = -1.0 + ((n * -2.0) / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -7.5e-56:
		tmp = f / (n - f)
	elif f <= 4.6e+14:
		tmp = 1.0 + (2.0 * (f / n))
	else:
		tmp = -1.0 + ((n * -2.0) / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -7.5e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 4.6e+14)
		tmp = Float64(1.0 + Float64(2.0 * Float64(f / n)));
	else
		tmp = Float64(-1.0 + Float64(Float64(n * -2.0) / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -7.5e-56)
		tmp = f / (n - f);
	elseif (f <= 4.6e+14)
		tmp = 1.0 + (2.0 * (f / n));
	else
		tmp = -1.0 + ((n * -2.0) / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -7.5e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 4.6e+14], N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(n * -2.0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -7.50000000000000041e-56

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.7%

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

   if -7.50000000000000041e-56 < f < 4.6e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

         \[\leadsto 1 + 2 \cdot \left(\frac{1}{n} \cdot \color{blue}{f}\right) \]

      3. associate-*l* N/A

         \[\leadsto 1 + \left(2 \cdot \frac{1}{n}\right) \cdot \color{blue}{f} \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\left(2 \cdot \frac{1}{n}\right) \cdot f\right)}\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \color{blue}{\left(\frac{1}{n} \cdot f\right)}\right)\right) \]

      6. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \frac{1 \cdot f}{\color{blue}{n}}\right)\right) \]

      7. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(2 \cdot \frac{f}{n}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \color{blue}{\left(\frac{f}{n}\right)}\right)\right) \]

      9. /-lowering-/.f64 80.5%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(2, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right)\right) \]

   7. Simplified 80.5%

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

   if 4.6e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot n - n}{f}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(-1 \cdot n - n\right), \color{blue}{f}\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(-1 \cdot n - 1 \cdot n\right), f\right)\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(n \cdot \left(-1 - 1\right)\right), f\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(n \cdot -2\right), f\right)\right) \]

      13. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, -2\right), f\right)\right) \]

   7. Simplified 81.6%

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

Alternative 3: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -3.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 2.3 \cdot 10^{+14}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{n \cdot -2}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -3.7e-56)
   (/ f (- n f))
   (if (<= f 2.3e+14) (/ n (- n f)) (+ -1.0 (/ (* n -2.0) f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -3.7e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.3e+14) {
		tmp = n / (n - f);
	} else {
		tmp = -1.0 + ((n * -2.0) / f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-3.7d-56)) then
        tmp = f / (n - f)
    else if (f <= 2.3d+14) then
        tmp = n / (n - f)
    else
        tmp = (-1.0d0) + ((n * (-2.0d0)) / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -3.7e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.3e+14) {
		tmp = n / (n - f);
	} else {
		tmp = -1.0 + ((n * -2.0) / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -3.7e-56:
		tmp = f / (n - f)
	elif f <= 2.3e+14:
		tmp = n / (n - f)
	else:
		tmp = -1.0 + ((n * -2.0) / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -3.7e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 2.3e+14)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = Float64(-1.0 + Float64(Float64(n * -2.0) / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -3.7e-56)
		tmp = f / (n - f);
	elseif (f <= 2.3e+14)
		tmp = n / (n - f);
	else
		tmp = -1.0 + ((n * -2.0) / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -3.7e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 2.3e+14], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(n * -2.0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -3.7000000000000002e-56

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.7%

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

   if -3.7000000000000002e-56 < f < 2.3e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{n}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.5%

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

   if 2.3e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. associate-*r/ N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. metadata-eval N/A

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

      7. +-commutative N/A

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

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left(\frac{-1 \cdot n - n}{f}\right)}\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(-1 \cdot n - n\right), \color{blue}{f}\right)\right) \]

      10. *-lft-identity N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(-1 \cdot n - 1 \cdot n\right), f\right)\right) \]

      11. distribute-rgt-out-- N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(n \cdot \left(-1 - 1\right)\right), f\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(n \cdot -2\right), f\right)\right) \]

      13. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(n, -2\right), f\right)\right) \]

   7. Simplified 81.6%

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

Alternative 4: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.55 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 2.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.55e-56)
   (/ f (- n f))
   (if (<= f 2.1e+14) (/ n (- n f)) (- -1.0 (/ n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.55e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.1e+14) {
		tmp = n / (n - f);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-1.55d-56)) then
        tmp = f / (n - f)
    else if (f <= 2.1d+14) then
        tmp = n / (n - f)
    else
        tmp = (-1.0d0) - (n / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -1.55e-56) {
		tmp = f / (n - f);
	} else if (f <= 2.1e+14) {
		tmp = n / (n - f);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.55e-56:
		tmp = f / (n - f)
	elif f <= 2.1e+14:
		tmp = n / (n - f)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.55e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 2.1e+14)
		tmp = Float64(n / Float64(n - f));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.55e-56)
		tmp = f / (n - f);
	elseif (f <= 2.1e+14)
		tmp = n / (n - f);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.55e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 2.1e+14], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -1.54999999999999994e-56

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.7%

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

   if -1.54999999999999994e-56 < f < 2.1e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{n}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.5%

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

   if 2.1e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 81.1%

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

   8. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{n}{f}\right)}\right) \]

      7. /-lowering-/.f64 81.2%

         \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(n, \color{blue}{f}\right)\right) \]

   10. Simplified 81.2%

       \[\leadsto \color{blue}{-1 - \frac{n}{f}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{-56}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{elif}\;f \leq 5.4 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -6.6e-56)
   (/ f (- n f))
   (if (<= f 5.4e+14) (+ 1.0 (/ f n)) (- -1.0 (/ n f)))))

C

double code(double f, double n) {
	double tmp;
	if (f <= -6.6e-56) {
		tmp = f / (n - f);
	} else if (f <= 5.4e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-6.6d-56)) then
        tmp = f / (n - f)
    else if (f <= 5.4d+14) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = (-1.0d0) - (n / f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -6.6e-56) {
		tmp = f / (n - f);
	} else if (f <= 5.4e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0 - (n / f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -6.6e-56:
		tmp = f / (n - f)
	elif f <= 5.4e+14:
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0 - (n / f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -6.6e-56)
		tmp = Float64(f / Float64(n - f));
	elseif (f <= 5.4e+14)
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = Float64(-1.0 - Float64(n / f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -6.6e-56)
		tmp = f / (n - f);
	elseif (f <= 5.4e+14)
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0 - (n / f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -6.6e-56], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[f, 5.4e+14], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if f < -6.59999999999999967e-56

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.7%

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

   if -6.59999999999999967e-56 < f < 5.4e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \color{blue}{n}\right) \]
   6. Step-by-step derivation

   7. Simplified 79.9%

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

   8. Taylor expanded in f around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{f}{n}\right)}\right) \]

      2. /-lowering-/.f64 79.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right) \]

   10. Simplified 79.9%

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

   if 5.4e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 81.1%

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

   8. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{n}{f}\right)}\right) \]

      7. /-lowering-/.f64 81.2%

         \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(n, \color{blue}{f}\right)\right) \]

   10. Simplified 81.2%

       \[\leadsto \color{blue}{-1 - \frac{n}{f}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 - \frac{n}{f}\\ \mathbf{if}\;f \leq -2.05 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;f \leq 1.35 \cdot 10^{+16}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (- -1.0 (/ n f))))
   (if (<= f -2.05e-15) t_0 (if (<= f 1.35e+16) (+ 1.0 (/ f n)) t_0))))

C

double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -2.05e-15) {
		tmp = t_0;
	} else if (f <= 1.35e+16) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (-1.0d0) - (n / f)
    if (f <= (-2.05d-15)) then
        tmp = t_0
    else if (f <= 1.35d+16) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = -1.0 - (n / f);
	double tmp;
	if (f <= -2.05e-15) {
		tmp = t_0;
	} else if (f <= 1.35e+16) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = -1.0 - (n / f)
	tmp = 0
	if f <= -2.05e-15:
		tmp = t_0
	elif f <= 1.35e+16:
		tmp = 1.0 + (f / n)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(-1.0 - Float64(n / f))
	tmp = 0.0
	if (f <= -2.05e-15)
		tmp = t_0;
	elseif (f <= 1.35e+16)
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = -1.0 - (n / f);
	tmp = 0.0;
	if (f <= -2.05e-15)
		tmp = t_0;
	elseif (f <= 1.35e+16)
		tmp = 1.0 + (f / n);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[f, -2.05e-15], t$95$0, If[LessEqual[f, 1.35e+16], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if f < -2.05000000000000018e-15 or 1.35e16 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{f}, \mathsf{\_.f64}\left(n, f\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 82.5%

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

   8. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(-1, \color{blue}{\left(\frac{n}{f}\right)}\right) \]

      7. /-lowering-/.f64 82.2%

         \[\leadsto \mathsf{\_.f64}\left(-1, \mathsf{/.f64}\left(n, \color{blue}{f}\right)\right) \]

   10. Simplified 82.2%

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

   if -2.05000000000000018e-15 < f < 1.35e16

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \color{blue}{n}\right) \]
   6. Step-by-step derivation

   7. Simplified 77.9%

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

   8. Taylor expanded in f around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{f}{n}\right)}\right) \]

      2. /-lowering-/.f64 77.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right) \]

   10. Simplified 77.9%

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

Alternative 7: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -1.12 \cdot 10^{-29}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2 \cdot 10^{+14}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -1.12e-29) -1.0 (if (<= f 2e+14) (+ 1.0 (/ f n)) -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -1.12e-29) {
		tmp = -1.0;
	} else if (f <= 2e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (f <= (-1.12d-29)) then
        tmp = -1.0d0
    else if (f <= 2d+14) then
        tmp = 1.0d0 + (f / n)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (f <= -1.12e-29) {
		tmp = -1.0;
	} else if (f <= 2e+14) {
		tmp = 1.0 + (f / n);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -1.12e-29:
		tmp = -1.0
	elif f <= 2e+14:
		tmp = 1.0 + (f / n)
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -1.12e-29)
		tmp = -1.0;
	elseif (f <= 2e+14)
		tmp = Float64(1.0 + Float64(f / n));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -1.12e-29)
		tmp = -1.0;
	elseif (f <= 2e+14)
		tmp = 1.0 + (f / n);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -1.12e-29], -1.0, If[LessEqual[f, 2e+14], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -1.11999999999999995e-29 or 2e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Simplified 81.4%

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

   if -1.11999999999999995e-29 < f < 2e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in n around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \color{blue}{n}\right) \]
   6. Step-by-step derivation

   7. Simplified 78.3%

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

   8. Taylor expanded in f around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{f}{n}\right)}\right) \]

      2. /-lowering-/.f64 78.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(f, \color{blue}{n}\right)\right) \]

   10. Simplified 78.3%

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

Alternative 8: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;f \leq -2.9 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.95 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= f -2.9e-15) -1.0 (if (<= f 1.95e+14) 1.0 -1.0)))

C

double code(double f, double n) {
	double tmp;
	if (f <= -2.9e-15) {
		tmp = -1.0;
	} else if (f <= 1.95e+14) {
		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 <= (-2.9d-15)) then
        tmp = -1.0d0
    else if (f <= 1.95d+14) 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 <= -2.9e-15) {
		tmp = -1.0;
	} else if (f <= 1.95e+14) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if f <= -2.9e-15:
		tmp = -1.0
	elif f <= 1.95e+14:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (f <= -2.9e-15)
		tmp = -1.0;
	elseif (f <= 1.95e+14)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (f <= -2.9e-15)
		tmp = -1.0;
	elseif (f <= 1.95e+14)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[f, -2.9e-15], -1.0, If[LessEqual[f, 1.95e+14], 1.0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if f < -2.90000000000000019e-15 or 1.95e14 < f

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around inf

      \[\leadsto \color{blue}{-1} \]
   6. Step-by-step derivation

   7. Simplified 81.9%

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

   if -2.90000000000000019e-15 < f < 1.95e14

   1. Initial program 100.0%

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

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

      2. distribute-neg-frac2 N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

      7. distribute-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

      8. unsub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

      9. remove-double-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

      10. --lowering--.f64 100.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in f around 0

      \[\leadsto \color{blue}{1} \]
   6. Step-by-step derivation

   7. Simplified 77.7%

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

Alternative 9: 49.6% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (f n) :precision binary64 -1.0)

C

double code(double f, double n) {
	return -1.0;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    code = -1.0d0
end function

Java

public static double code(double f, double n) {
	return -1.0;
}

Python

def code(f, n):
	return -1.0

Julia

function code(f, n)
	return -1.0
end

MATLAB

function tmp = code(f, n)
	tmp = -1.0;
end

Wolfram

code[f_, n_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. distribute-frac-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{f + n}{f - n}\right) \]

   2. distribute-neg-frac2 N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(f + n\right), \color{blue}{\left(\mathsf{neg}\left(\left(f - n\right)\right)\right)}\right) \]

   4. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\color{blue}{\left(f - n\right)}\right)\right)\right) \]

   5. sub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(f + \left(\mathsf{neg}\left(n\right)\right)\right)\right)\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(n\right)\right) + f\right)\right)\right)\right) \]

   7. distribute-neg-in N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) + \color{blue}{\left(\mathsf{neg}\left(f\right)\right)}\right)\right) \]

   8. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(n\right)\right)\right)\right) - \color{blue}{f}\right)\right) \]

   9. remove-double-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \left(n - f\right)\right) \]

   10. --lowering--.f64 100.0%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(f, n\right), \mathsf{\_.f64}\left(n, \color{blue}{f}\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in f around inf

   \[\leadsto \color{blue}{-1} \]
6. Step-by-step derivation

7. Simplified 54.9%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))