subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 5.1s

Alternatives: 8

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: 75.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{elif}\;n \leq 3.4 \cdot 10^{-69}:\\ \;\;\;\;-1 + \frac{n \cdot -2}{f}\\ \mathbf{else}:\\ \;\;\;\;\frac{f + n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -4.8e-11)
   (/ n (- n f))
   (if (<= n 3.4e-69) (+ -1.0 (/ (* n -2.0) f)) (/ (+ f n) n))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -4.8e-11) {
		tmp = n / (n - f);
	} else if (n <= 3.4e-69) {
		tmp = -1.0 + ((n * -2.0) / f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-4.8d-11)) then
        tmp = n / (n - f)
    else if (n <= 3.4d-69) then
        tmp = (-1.0d0) + ((n * (-2.0d0)) / f)
    else
        tmp = (f + n) / n
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -4.8e-11) {
		tmp = n / (n - f);
	} else if (n <= 3.4e-69) {
		tmp = -1.0 + ((n * -2.0) / f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -4.8e-11:
		tmp = n / (n - f)
	elif n <= 3.4e-69:
		tmp = -1.0 + ((n * -2.0) / f)
	else:
		tmp = (f + n) / n
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -4.8e-11)
		tmp = Float64(n / Float64(n - f));
	elseif (n <= 3.4e-69)
		tmp = Float64(-1.0 + Float64(Float64(n * -2.0) / f));
	else
		tmp = Float64(Float64(f + n) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -4.8e-11)
		tmp = n / (n - f);
	elseif (n <= 3.4e-69)
		tmp = -1.0 + ((n * -2.0) / f);
	else
		tmp = (f + n) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -4.8e-11], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 3.4e-69], N[(-1.0 + N[(N[(n * -2.0), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision], N[(N[(f + n), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -4.8000000000000002e-11

   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 85.7%

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

   if -4.8000000000000002e-11 < n < 3.40000000000000008e-69

   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 78.7%

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

   7. Simplified 78.7%

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

   if 3.40000000000000008e-69 < n

   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 76.6%

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

Alternative 3: 75.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{n}{n - f}\\ \mathbf{elif}\;n \leq 7 \cdot 10^{-52}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;\frac{f + n}{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.2e-8)
   (/ n (- n f))
   (if (<= n 7e-52) (/ f (- n f)) (/ (+ f n) n))))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.2e-8) {
		tmp = n / (n - f);
	} else if (n <= 7e-52) {
		tmp = f / (n - f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if (n <= (-1.2d-8)) then
        tmp = n / (n - f)
    else if (n <= 7d-52) then
        tmp = f / (n - f)
    else
        tmp = (f + n) / n
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -1.2e-8) {
		tmp = n / (n - f);
	} else if (n <= 7e-52) {
		tmp = f / (n - f);
	} else {
		tmp = (f + n) / n;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.2e-8:
		tmp = n / (n - f)
	elif n <= 7e-52:
		tmp = f / (n - f)
	else:
		tmp = (f + n) / n
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.2e-8)
		tmp = Float64(n / Float64(n - f));
	elseif (n <= 7e-52)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = Float64(Float64(f + n) / n);
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.2e-8)
		tmp = n / (n - f);
	elseif (n <= 7e-52)
		tmp = f / (n - f);
	else
		tmp = (f + n) / n;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.2e-8], N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision], If[LessEqual[n, 7e-52], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], N[(N[(f + n), $MachinePrecision] / n), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if n < -1.19999999999999999e-8

   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 85.7%

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

   if -1.19999999999999999e-8 < n < 7.0000000000000001e-52

   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 78.2%

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

   if 7.0000000000000001e-52 < n

   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 77.3%

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

Alternative 4: 75.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{n - f}\\ \mathbf{if}\;n \leq -9.5 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{-47}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ n (- n f))))
   (if (<= n -9.5e-14) t_0 (if (<= n 1.2e-47) (/ f (- n f)) t_0))))

C

double code(double f, double n) {
	double t_0 = n / (n - f);
	double tmp;
	if (n <= -9.5e-14) {
		tmp = t_0;
	} else if (n <= 1.2e-47) {
		tmp = f / (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 = n / (n - f)
    if (n <= (-9.5d-14)) then
        tmp = t_0
    else if (n <= 1.2d-47) then
        tmp = f / (n - f)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double t_0 = n / (n - f);
	double tmp;
	if (n <= -9.5e-14) {
		tmp = t_0;
	} else if (n <= 1.2e-47) {
		tmp = f / (n - f);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(f, n):
	t_0 = n / (n - f)
	tmp = 0
	if n <= -9.5e-14:
		tmp = t_0
	elif n <= 1.2e-47:
		tmp = f / (n - f)
	else:
		tmp = t_0
	return tmp

Julia

function code(f, n)
	t_0 = Float64(n / Float64(n - f))
	tmp = 0.0
	if (n <= -9.5e-14)
		tmp = t_0;
	elseif (n <= 1.2e-47)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	t_0 = n / (n - f);
	tmp = 0.0;
	if (n <= -9.5e-14)
		tmp = t_0;
	elseif (n <= 1.2e-47)
		tmp = f / (n - f);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := Block[{t$95$0 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[n, -9.5e-14], t$95$0, If[LessEqual[n, 1.2e-47], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if n < -9.4999999999999999e-14 or 1.2e-47 < n

   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 81.1%

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

   if -9.4999999999999999e-14 < n < 1.2e-47

   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 78.2%

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

Alternative 5: 74.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -2.5 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 10^{-50}:\\ \;\;\;\;\frac{f}{n - f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -2.5e+14) 1.0 (if (<= n 1e-50) (/ f (- n f)) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -2.5e+14) {
		tmp = 1.0;
	} else if (n <= 1e-50) {
		tmp = f / (n - f);
	} 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 <= (-2.5d+14)) then
        tmp = 1.0d0
    else if (n <= 1d-50) then
        tmp = f / (n - f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -2.5e+14) {
		tmp = 1.0;
	} else if (n <= 1e-50) {
		tmp = f / (n - f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -2.5e+14:
		tmp = 1.0
	elif n <= 1e-50:
		tmp = f / (n - f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -2.5e+14)
		tmp = 1.0;
	elseif (n <= 1e-50)
		tmp = Float64(f / Float64(n - f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -2.5e+14)
		tmp = 1.0;
	elseif (n <= 1e-50)
		tmp = f / (n - f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -2.5e+14], 1.0, If[LessEqual[n, 1e-50], N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -2.5e14 or 1.00000000000000001e-50 < n

   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 81.0%

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

   if -2.5e14 < n < 1.00000000000000001e-50

   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 77.7%

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

Alternative 6: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.7 \cdot 10^{-74}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -3.7e-10) 1.0 (if (<= n 2.7e-74) (- -1.0 (/ n f)) 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -3.7e-10) {
		tmp = 1.0;
	} else if (n <= 2.7e-74) {
		tmp = -1.0 - (n / f);
	} 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 <= (-3.7d-10)) then
        tmp = 1.0d0
    else if (n <= 2.7d-74) then
        tmp = (-1.0d0) - (n / f)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if (n <= -3.7e-10) {
		tmp = 1.0;
	} else if (n <= 2.7e-74) {
		tmp = -1.0 - (n / f);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -3.7e-10:
		tmp = 1.0
	elif n <= 2.7e-74:
		tmp = -1.0 - (n / f)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -3.7e-10)
		tmp = 1.0;
	elseif (n <= 2.7e-74)
		tmp = Float64(-1.0 - Float64(n / f));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -3.7e-10)
		tmp = 1.0;
	elseif (n <= 2.7e-74)
		tmp = -1.0 - (n / f);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -3.7e-10], 1.0, If[LessEqual[n, 2.7e-74], N[(-1.0 - N[(n / f), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -3.70000000000000015e-10 or 2.70000000000000018e-74 < n

   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 80.1%

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

   if -3.70000000000000015e-10 < n < 2.70000000000000018e-74

   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 78.6%

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

   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 78.5%

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

   10. Simplified 78.5%

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

Alternative 7: 73.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;n \leq -1.45 \cdot 10^{+14}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3 \cdot 10^{-69}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= n -1.45e+14) 1.0 (if (<= n 3e-69) -1.0 1.0)))

C

double code(double f, double n) {
	double tmp;
	if (n <= -1.45e+14) {
		tmp = 1.0;
	} else if (n <= 3e-69) {
		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 <= (-1.45d+14)) then
        tmp = 1.0d0
    else if (n <= 3d-69) 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 <= -1.45e+14) {
		tmp = 1.0;
	} else if (n <= 3e-69) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if n <= -1.45e+14:
		tmp = 1.0
	elif n <= 3e-69:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (n <= -1.45e+14)
		tmp = 1.0;
	elseif (n <= 3e-69)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if (n <= -1.45e+14)
		tmp = 1.0;
	elseif (n <= 3e-69)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[n, -1.45e+14], 1.0, If[LessEqual[n, 3e-69], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if n < -1.45e14 or 2.99999999999999989e-69 < n

   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 80.5%

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

   if -1.45e14 < n < 2.99999999999999989e-69

   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 77.5%

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

Alternative 8: 50.1% 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 48.3%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))