subtraction fraction

Average Error: 0.0 → 0.0

Time: 4.0s

Precision: binary64

Cost: 13248

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double f, double n) {
	return log(exp(((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 = log(exp(((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 Math.log(Math.exp(((f + n) / (n - f))));
}

Python

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

↓

def code(f, n):
	return math.log(math.exp(((f + n) / (n - f))))

Julia

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

↓

function code(f, n)
	return log(exp(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 = log(exp(((f + n) / (n - f))));
end

Wolfram

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

↓

code[f_, n_] := N[Log[N[Exp[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

   [Start] 0.0	\[ \frac{-\left(f + n\right)}{f - n} \]
   sub-neg [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]
   +-commutative [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(-n\right) + f}} \]
   neg-sub0 [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]
   associate-+l- [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]
   sub0-neg [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{-\left(n - f\right)}} \]
   neg-mul-1 [=>] 0.0	\[ \frac{-\left(f + n\right)}{\color{blue}{-1 \cdot \left(n - f\right)}} \]
   associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{-\left(f + n\right)}{-1}}{n - f}} \]
   neg-mul-1 [=>] 0.0	\[ \frac{\frac{\color{blue}{-1 \cdot \left(f + n\right)}}{-1}}{n - f} \]
   *-commutative [=>] 0.0	\[ \frac{\frac{\color{blue}{\left(f + n\right) \cdot -1}}{-1}}{n - f} \]
   associate-/l* [=>] 0.0	\[ \frac{\color{blue}{\frac{f + n}{\frac{-1}{-1}}}}{n - f} \]
   metadata-eval [=>] 0.0	\[ \frac{\frac{f + n}{\color{blue}{1}}}{n - f} \]
   /-rgt-identity [=>] 0.0	\[ \frac{\color{blue}{f + n}}{n - f} \]

3. Applied egg-rr 0.0

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

4. Final simplification 0.0

   \[\leadsto \log \left(e^{\frac{f + n}{n - f}}\right) \]

Alternatives

Alternative 1
Error	16.7
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -6 \cdot 10^{+14} \lor \neg \left(n \leq 4.5 \cdot 10^{-97}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	16.4
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -1.1 \cdot 10^{+16} \lor \neg \left(n \leq 4.5 \cdot 10^{-97}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	17.1
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -50000000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 4.5 \cdot 10^{-97}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	32.9
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023073 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))