subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. distribute-neg-in N/A

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

   5. +-commutative N/A

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

   6. sub-neg N/A

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

   7. div-sub N/A

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

   8. lower--.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 100.0

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

4. Applied rewrites 100.0%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lower-/.f64 N/A

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

   6. lift--.f64 N/A

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

   7. sub-neg N/A

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

   8. lift-neg.f64 N/A

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

   9. distribute-neg-in N/A

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

   10. lift-neg.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. remove-double-neg N/A

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

   13. +-commutative N/A

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

   14. lift-neg.f64 N/A

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

   15. sub-neg N/A

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

   16. lift--.f64 100.0

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

6. Applied rewrites 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 98.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[f_, n_] := If[LessEqual[N[(N[(n + f), $MachinePrecision] / N[(n - f), $MachinePrecision]), $MachinePrecision], -0.5], N[(n * N[(-2.0 / f), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(N[(f + f), $MachinePrecision] / 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 100.0%

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

   3. Taylor expanded in f around inf

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. distribute-neg-frac2 N/A

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

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

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

      5. metadata-eval N/A

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

      6. distribute-rgt1-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. associate-*l/ N/A

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

      10. mul-1-neg N/A

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

      11. distribute-neg-frac2 N/A

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

      12. metadata-eval N/A

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

      13. associate-*r/ N/A

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

      14. *-commutative N/A

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

      15. metadata-eval N/A

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

      16. lower-fma.f64 N/A

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

      17. associate-*r/ N/A

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

      18. metadata-eval N/A

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

      19. distribute-neg-frac N/A

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

      20. metadata-eval N/A

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

      21. lower-/.f64 99.1

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

   5. Applied rewrites 99.1%

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

   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. associate-*r/ N/A

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

      2. count-2 N/A

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

      3. remove-double-neg N/A

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

      4. distribute-neg-out N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. distribute-neg-frac N/A

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

      8. mul-1-neg N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r/ N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. mul-1-neg N/A

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

      14. distribute-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. lower-/.f64 N/A

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

      17. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 98.9%

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

Reproduce

herbie shell --seed 2024228 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))