subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 6.3s

Alternatives: 8

Speedup: 1.0×

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, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{\frac{f - n}{f + n}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(f, n):
	return -1.0 / ((f - n) / (f + n))

Julia

function code(f, n)
	return Float64(-1.0 / Float64(Float64(f - n) / Float64(f + n)))
end

MATLAB

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

Wolfram

code[f_, n_] := N[(-1.0 / N[(N[(f - n), $MachinePrecision] / N[(f + n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. difference-of-squares N/A

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 N/A

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

   12. lower-/.f64 99.9

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

   13. lift-+.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 99.9

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

4. Applied rewrites 99.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. frac-2neg N/A

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

   5. frac-2neg N/A

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

   6. distribute-frac-neg N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-+.f64 N/A

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

   11. remove-double-neg N/A

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

   12. *-inverses N/A

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

   13. metadata-eval N/A

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

   14. lift-/.f64 N/A

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

   15. distribute-frac-neg N/A

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

   16. distribute-frac-neg2 N/A

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

   17. frac-2neg N/A

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

   18. lift-/.f64 N/A

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

6. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\frac{-1}{\frac{f - n}{f + n}}} \]
7. Add Preprocessing

Alternative 2: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{n}{f}, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{n}, f, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (- (+ f n)) (- f n)) -0.5)
   (fma (/ n f) -2.0 -1.0)
   (fma (/ 2.0 n) f 1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = fma((n / f), -2.0, -1.0);
	} else {
		tmp = fma((2.0 / n), f, 1.0);
	}
	return tmp;
}

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = fma(Float64(n / f), -2.0, -1.0);
	else
		tmp = fma(Float64(2.0 / n), f, 1.0);
	end
	return tmp
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(n / f), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision], N[(N[(2.0 / n), $MachinePrecision] * f + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

   5. Applied rewrites 1.6%

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

   6. Taylor expanded in f around -inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

      3. *-lft-identity N/A

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

      4. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

      6. mul-1-neg N/A

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

      7. mul-1-neg N/A

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

      8. distribute-neg-in N/A

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

      9. count-2-rev N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. distribute-lft-neg-in N/A

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

      16. mul-1-neg N/A

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

      17. rgt-mult-inverse N/A

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

      18. fp-cancel-sub-sign N/A

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

      19. mul-1-neg N/A

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

   8. Applied rewrites 97.0%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2}}{n}, f, 1\right) \]

      8. lower-/.f64 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{n}}, f, 1\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{n}, f, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{n}, f, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (- (+ f n)) (- f n)) -0.5)
   (/ (- f) (- f n))
   (fma (/ 2.0 n) f 1.0)))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -f / (f - n);
	} else {
		tmp = fma((2.0 / n), f, 1.0);
	}
	return tmp;
}

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = fma(Float64(2.0 / n), f, 1.0);
	end
	return tmp
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / n), $MachinePrecision] * f + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

      1. +-commutative N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{2}}{n}, f, 1\right) \]

      8. lower-/.f64 99.7

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{n}}, f, 1\right) \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{n}, f, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;\frac{-n}{f - n}\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (- (+ f n)) (- f n)) -0.5) (/ (- f) (- f n)) (/ (- n) (- f n))))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -f / (f - n);
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((-(f + n) / (f - n)) <= (-0.5d0)) then
        tmp = -f / (f - n)
    else
        tmp = -n / (f - n)
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -f / (f - n);
	} else {
		tmp = -n / (f - n);
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (-(f + n) / (f - n)) <= -0.5:
		tmp = -f / (f - n)
	else:
		tmp = -n / (f - n)
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = Float64(Float64(-n) / Float64(f - n));
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((-(f + n) / (f - n)) <= -0.5)
		tmp = -f / (f - n);
	else
		tmp = -n / (f - n);
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], N[((-n) / N[(f - n), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -0.5:\\ \;\;\;\;\frac{-f}{f - n}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (- (+ f n)) (- f n)) -0.5) (/ (- f) (- f n)) 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -f / (f - n);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((-(f + n) / (f - n)) <= (-0.5d0)) then
        tmp = -f / (f - n)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -0.5) {
		tmp = -f / (f - n);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (-(f + n) / (f - n)) <= -0.5:
		tmp = -f / (f - n)
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -0.5)
		tmp = Float64(Float64(-f) / Float64(f - n));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((-(f + n) / (f - n)) <= -0.5)
		tmp = -f / (f - n);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -0.5], N[((-f) / N[(f - n), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -0.5

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -0.5 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

   5. Applied rewrites 99.1%

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

Alternative 6: 98.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{-\left(f + n\right)}{f - n} \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (f n)
 :precision binary64
 (if (<= (/ (- (+ f n)) (- f n)) -5e-310) -1.0 1.0))

C

double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(f, n)
    real(8), intent (in) :: f
    real(8), intent (in) :: n
    real(8) :: tmp
    if ((-(f + n) / (f - n)) <= (-5d-310)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double f, double n) {
	double tmp;
	if ((-(f + n) / (f - n)) <= -5e-310) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(f, n):
	tmp = 0
	if (-(f + n) / (f - n)) <= -5e-310:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(f, n)
	tmp = 0.0
	if (Float64(Float64(-Float64(f + n)) / Float64(f - n)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(f, n)
	tmp = 0.0;
	if ((-(f + n) / (f - n)) <= -5e-310)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[f_, n_] := If[LessEqual[N[((-N[(f + n), $MachinePrecision]) / N[(f - n), $MachinePrecision]), $MachinePrecision], -5e-310], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n)) < -4.999999999999985e-310

   1. Initial program 99.9%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around inf

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

   5. Applied rewrites 95.3%

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

   if -4.999999999999985e-310 < (/.f64 (neg.f64 (+.f64 f n)) (-.f64 f n))

   1. Initial program 100.0%

      \[\frac{-\left(f + n\right)}{f - n} \]
   2. Add Preprocessing

   3. Taylor expanded in f around 0

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

   5. Applied rewrites 99.1%

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

Alternative 7: 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]

Derivation

1. Initial program 99.9%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 8: 49.1% accurate, 20.0× 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 99.9%

   \[\frac{-\left(f + n\right)}{f - n} \]
2. Add Preprocessing

3. Taylor expanded in f around inf

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

5. Applied rewrites 48.8%

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

Reproduce

herbie shell --seed 2024337 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))