subtraction fraction

Average Error: 0.0 → 0.0

Time: 10.9s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[f_, n_] := N[(N[(f - n), $MachinePrecision] / N[(N[(f - n), $MachinePrecision] / N[(N[(f + n), $MachinePrecision] / N[(n - f), $MachinePrecision]), $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 0.0

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{f + n}{n - f}\right)}^{3}}} \]

4. Applied egg-rr 31.5

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

5. Simplified 31.6

   \[\leadsto \color{blue}{\frac{f + n}{\left(n - f\right) \cdot \left(f - n\right)} \cdot \left(f - n\right)} \]
   Proof
   (*.f64 (/.f64 (+.f64 f n) (*.f64 (-.f64 n f) (-.f64 f n))) (-.f64 f n)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (+.f64 f n) (/.f64 (*.f64 (-.f64 n f) (-.f64 f n)) (-.f64 f n)))): 0 points increase in error, 2 points decrease in error

6. Applied egg-rr 0.0

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

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.0
Cost	713

\[\begin{array}{l} \mathbf{if}\;f \leq -6.6 \cdot 10^{+25} \lor \neg \left(f \leq 2.55 \cdot 10^{+32}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 2
Error	16.1
Cost	712

\[\begin{array}{l} \mathbf{if}\;f \leq -1.8 \cdot 10^{+26}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 1.4 \cdot 10^{+25}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \]

Alternative 3
Error	16.3
Cost	585

\[\begin{array}{l} \mathbf{if}\;f \leq -7.2 \cdot 10^{+25} \lor \neg \left(f \leq 5.6 \cdot 10^{+26}\right):\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{f}{n}\\ \end{array} \]

Alternative 4
Error	16.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;f \leq -1.35 \cdot 10^{+26}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 2.85 \cdot 10^{+26}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Error	16.3
Cost	584

\[\begin{array}{l} \mathbf{if}\;f \leq -5.7 \cdot 10^{+25}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;f \leq 9 \cdot 10^{+30}:\\ \;\;\;\;1 + \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{f}{n - f}\\ \end{array} \]

Alternative 6
Error	0.0
Cost	448

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

Alternative 7
Error	16.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;f \leq -5 \cdot 10^{+46}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 60000000000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8
Error	32.3
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2022343 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))