subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

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, 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. 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. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

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

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

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

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

   5. +-lowering-+.f64 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 75.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + 2 \cdot \frac{f}{n}\\ \mathbf{if}\;n \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.6 \cdot 10^{+18}:\\ \;\;\;\;-1 + \frac{n}{f} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* 2.0 (/ f n)))))
   (if (<= n -4.8e+21) t_0 (if (<= n 4.6e+18) (+ -1.0 (* (/ n f) -2.0)) t_0))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -4.8e+21) {
		tmp = t_0;
	} else if (n <= 4.6e+18) {
		tmp = -1.0 + ((n / f) * -2.0);
	} 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 + (2.0d0 * (f / n))
    if (n <= (-4.8d+21)) then
        tmp = t_0
    else if (n <= 4.6d+18) then
        tmp = (-1.0d0) + ((n / f) * (-2.0d0))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (2.0 * (f / n));
	double tmp;
	if (n <= -4.8e+21) {
		tmp = t_0;
	} else if (n <= 4.6e+18) {
		tmp = -1.0 + ((n / f) * -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (2.0 * (f / n))
	tmp = 0
	if n <= -4.8e+21:
		tmp = t_0
	elif n <= 4.6e+18:
		tmp = -1.0 + ((n / f) * -2.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(2.0 * Float64(f / n)))
	tmp = 0.0
	if (n <= -4.8e+21)
		tmp = t_0;
	elseif (n <= 4.6e+18)
		tmp = Float64(-1.0 + Float64(Float64(n / f) * -2.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (2.0 * (f / n));
	tmp = 0.0;
	if (n <= -4.8e+21)
		tmp = t_0;
	elseif (n <= 4.6e+18)
		tmp = -1.0 + ((n / f) * -2.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(2.0 * N[(f / n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -4.8e+21], t$95$0, If[LessEqual[n, 4.6e+18], N[(-1.0 + N[(N[(n / f), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -4.8e21 or 4.6e18 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 85.0%

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

   7. Simplified 85.0%

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

   if -4.8e21 < n < 4.6e18

   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. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

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

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

      14. /-lowering-/.f64 76.6%

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

   7. Simplified 76.6%

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

Alternative 3: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+23}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 2.45 \cdot 10^{+16}:\\ \;\;\;\;-1 + \frac{n}{f} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.8e+23)
   (+ 1.0 (/ f n))
   (if (<= n 2.45e+16) (+ -1.0 (* (/ n f) -2.0)) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.8e+23) {
		tmp = 1.0 + (f / n);
	} else if (n <= 2.45e+16) {
		tmp = -1.0 + ((n / f) * -2.0);
	} else {
		tmp = n / (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 (n <= (-2.8d+23)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 2.45d+16) then
        tmp = (-1.0d0) + ((n / f) * (-2.0d0))
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -2.8e+23) {
		tmp = 1.0 + (f / n);
	} else if (n <= 2.45e+16) {
		tmp = -1.0 + ((n / f) * -2.0);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.8e+23:
		tmp = 1.0 + (f / n)
	elif n <= 2.45e+16:
		tmp = -1.0 + ((n / f) * -2.0)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.8e+23)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 2.45e+16)
		tmp = Float64(-1.0 + Float64(Float64(n / f) * -2.0));
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.8e+23)
		tmp = 1.0 + (f / n);
	elseif (n <= 2.45e+16)
		tmp = -1.0 + ((n / f) * -2.0);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.8e+23], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.45e+16], N[(-1.0 + N[(N[(n / f), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8e23

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 85.2%

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

   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 85.2%

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

   10. Simplified 85.2%

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

   if -2.8e23 < n < 2.45e16

   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. *-lft-identity N/A

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

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

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

      11. metadata-eval N/A

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

      12. associate-*l/ N/A

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

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

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

      14. /-lowering-/.f64 76.9%

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

   7. Simplified 76.9%

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

   if 2.45e16 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 82.2%

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

Alternative 4: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.8 \cdot 10^{+20}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{+16}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n}{n - f}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.8e+20)
   (+ 1.0 (/ f n))
   (if (<= n 2.7e+16) (- -1.0 (/ n f)) (/ n (- n f)))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.8e+20) {
		tmp = 1.0 + (f / n);
	} else if (n <= 2.7e+16) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = n / (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 (n <= (-2.8d+20)) then
        tmp = 1.0d0 + (f / n)
    else if (n <= 2.7d+16) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = n / (n - f)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -2.8e+20) {
		tmp = 1.0 + (f / n);
	} else if (n <= 2.7e+16) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = n / (n - f);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.8e+20:
		tmp = 1.0 + (f / n)
	elif n <= 2.7e+16:
		tmp = -1.0 - (n / f)
	else:
		tmp = n / (n - f)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.8e+20)
		tmp = Float64(1.0 + Float64(f / n));
	elseif (n <= 2.7e+16)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = Float64(n / Float64(n - f));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.8e+20)
		tmp = 1.0 + (f / n);
	elseif (n <= 2.7e+16)
		tmp = -1.0 - (n / f);
	else
		tmp = n / (n - f);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.8e+20], N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 2.7e+16], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -2.8e20

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 85.2%

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

   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 85.2%

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

   10. Simplified 85.2%

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

   if -2.8e20 < n < 2.7e16

   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 76.5%

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

   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 76.5%

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

   10. Simplified 76.5%

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

   if 2.7e16 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 82.2%

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

Alternative 5: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{f}{n}\\ \mathbf{if}\;n \leq -3 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 3.9 \cdot 10^{+18}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ f n))))
   (if (<= n -3e+23) t_0 (if (<= n 3.9e+18) (- -1.0 (/ n f)) t_0))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (f / n);
	double tmp;
	if (n <= -3e+23) {
		tmp = t_0;
	} else if (n <= 3.9e+18) {
		tmp = -1.0 - (n / f);
	} 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 + (f / n)
    if (n <= (-3d+23)) then
        tmp = t_0
    else if (n <= 3.9d+18) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (f / n);
	double tmp;
	if (n <= -3e+23) {
		tmp = t_0;
	} else if (n <= 3.9e+18) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (f / n)
	tmp = 0
	if n <= -3e+23:
		tmp = t_0
	elif n <= 3.9e+18:
		tmp = -1.0 - (n / f)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(f / n))
	tmp = 0.0
	if (n <= -3e+23)
		tmp = t_0;
	elseif (n <= 3.9e+18)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (f / n);
	tmp = 0.0;
	if (n <= -3e+23)
		tmp = t_0;
	elseif (n <= 3.9e+18)
		tmp = -1.0 - (n / f);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -3e+23], t$95$0, If[LessEqual[n, 3.9e+18], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -3.0000000000000001e23 or 3.9e18 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 84.2%

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

   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 84.2%

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

   10. Simplified 84.2%

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

   if -3.0000000000000001e23 < n < 3.9e18

   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 76.1%

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

   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 76.2%

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

   10. Simplified 76.2%

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

Alternative 6: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{f}{n}\\ \mathbf{if}\;n \leq -1.16 \cdot 10^{+23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ f n))))
   (if (<= n -1.16e+23) t_0 (if (<= n 4.2e+19) -1.0 t_0))))

C

double code(double f, double n) {
	double t_0 = 1.0 + (f / n);
	double tmp;
	if (n <= -1.16e+23) {
		tmp = t_0;
	} else if (n <= 4.2e+19) {
		tmp = -1.0;
	} 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 + (f / n)
    if (n <= (-1.16d+23)) then
        tmp = t_0
    else if (n <= 4.2d+19) then
        tmp = -1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = 1.0 + (f / n);
	double tmp;
	if (n <= -1.16e+23) {
		tmp = t_0;
	} else if (n <= 4.2e+19) {
		tmp = -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = 1.0 + (f / n)
	tmp = 0
	if n <= -1.16e+23:
		tmp = t_0
	elif n <= 4.2e+19:
		tmp = -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(1.0 + Float64(f / n))
	tmp = 0.0
	if (n <= -1.16e+23)
		tmp = t_0;
	elseif (n <= 4.2e+19)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = 1.0 + (f / n);
	tmp = 0.0;
	if (n <= -1.16e+23)
		tmp = t_0;
	elseif (n <= 4.2e+19)
		tmp = -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(1.0 + N[(f / n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -1.16e+23], t$95$0, If[LessEqual[n, 4.2e+19], -1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -1.16e23 or 4.2e19 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   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 84.2%

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

   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 84.2%

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

   10. Simplified 84.2%

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

   if -1.16e23 < n < 4.2e19

   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 75.5%

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

Alternative 7: 74.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.95 \cdot 10^{+23}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.95 \cdot 10^{+17}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.95e+23) 1.0 (if (<= n 1.95e+17) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.95e+23) {
		tmp = 1.0;
	} else if (n <= 1.95e+17) {
		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 (n <= (-1.95d+23)) then
        tmp = 1.0d0
    else if (n <= 1.95d+17) 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 (n <= -1.95e+23) {
		tmp = 1.0;
	} else if (n <= 1.95e+17) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.95e+23:
		tmp = 1.0
	elif n <= 1.95e+17:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.95e+23)
		tmp = 1.0;
	elseif (n <= 1.95e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.95e+23)
		tmp = 1.0;
	elseif (n <= 1.95e+17)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.95e+23], 1.0, If[LessEqual[n, 1.95e+17], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.95e23 or 1.95e17 < n

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in f around 0

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

   7. Simplified 83.0%

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

   if -1.95e23 < n < 1.95e17

   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 75.8%

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

Alternative 8: 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. 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. Final simplification 100.0%

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

Alternative 9: 49.4% 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 50.4%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))