subtraction fraction

Average Error: 0.0 → 0.0

Time: 5.5s

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

   [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 0.2

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

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	16.4
Cost	978

\[\begin{array}{l} \mathbf{if}\;n \leq -1.05 \cdot 10^{-32} \lor \neg \left(n \leq 2.8 \cdot 10^{+33}\right) \land \left(n \leq 2 \cdot 10^{+73} \lor \neg \left(n \leq 8 \cdot 10^{+98}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Error	16.0
Cost	977

\[\begin{array}{l} t_0 := 1 + 2 \cdot \frac{f}{n}\\ \mathbf{if}\;n \leq -7.5 \cdot 10^{-33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 6 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{elif}\;n \leq 1.36 \cdot 10^{+73} \lor \neg \left(n \leq 1.4 \cdot 10^{+99}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	16.7
Cost	592

\[\begin{array}{l} \mathbf{if}\;n \leq -5.6 \cdot 10^{-34}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 1.02 \cdot 10^{+27}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq 1.2 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.6 \cdot 10^{+99}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	0.0
Cost	576

\[\frac{1}{\frac{n - f}{n + f}} \]

Alternative 5
Error	0.0
Cost	448

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

Alternative 6
Error	31.6
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))