subtraction fraction

Average Error: 0.0 → 0.0

Time: 4.0s

Precision: binary64

Cost: 704

Math

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

↓

\[\frac{n}{n - f} + \frac{f}{n - f} \]

FPCore

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

↓

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

C

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

↓

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 = -(f + n) / (f - n)
end function

↓

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 -(f + n) / (f - n);
}

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 0.0

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

2. Simplified 0.0

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   Proof
   (/.f64 (+.f64 f n) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 (+.f64 f n) 1)) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (+.f64 f n) (Rewrite<= metadata-eval (/.f64 -1 -1))) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 f n) -1) -1)) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (+.f64 f n))) -1) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (+.f64 f n))) -1) (-.f64 n f)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r*_binary64 (/.f64 (neg.f64 (+.f64 f n)) (*.f64 -1 (-.f64 n f)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 n f)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 n f)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 n) f))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 n)) f)): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (Rewrite<= +-commutative_binary64 (+.f64 f (neg.f64 n)))): 0 points increase in error, 0 points decrease in error
   (/.f64 (neg.f64 (+.f64 f n)) (Rewrite<= sub-neg_binary64 (-.f64 f n))): 0 points increase in error, 0 points decrease in error

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.0

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

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	17.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -4.5632091554398244 \cdot 10^{+82}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.8784326835701253 \cdot 10^{+56}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	15.7
Cost	712

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -12489669798063233000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 0.0021275774150482834:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

\[\frac{n + f}{n - f} \]

Alternative 4
Error	16.4
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -12489669798063233000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 0.0021275774150482834:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	32.1
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2022308 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))