subtraction fraction

Average Error: 0.04% → 0.05%

Time: 4.7s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

(FPCore (f n) :precision binary64 (/ 1.0 (/ (- n f) (+ n f))))

C

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

↓

double code(double f, double n) {
	return 1.0 / ((n - 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 = 1.0d0 / ((n - 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 1.0 / ((n - f) / (n + f));
}

Python

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

↓

def code(f, n):
	return 1.0 / ((n - f) / (n + f))

Julia

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

↓

function code(f, n)
	return Float64(1.0 / Float64(Float64(n - f) / Float64(n + f)))
end

MATLAB

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

↓

function tmp = code(f, n)
	tmp = 1.0 / ((n - f) / (n + f));
end

Wolfram

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

↓

code[f_, n_] := N[(1.0 / N[(N[(n - f), $MachinePrecision] / N[(n + f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.04

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

2. Simplified 0.04

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

   [Start] 0.04	\[ \frac{-\left(f + n\right)}{f - n} \]
   sub-neg [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]
   +-commutative [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(-n\right) + f}} \]
   neg-sub0 [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]
   associate-+l- [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]
   sub0-neg [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{-\left(n - f\right)}} \]
   neg-mul-1 [=>] 0.04	\[ \frac{-\left(f + n\right)}{\color{blue}{-1 \cdot \left(n - f\right)}} \]
   associate-/r* [=>] 0.04	\[ \color{blue}{\frac{\frac{-\left(f + n\right)}{-1}}{n - f}} \]
   neg-mul-1 [=>] 0.04	\[ \frac{\frac{\color{blue}{-1 \cdot \left(f + n\right)}}{-1}}{n - f} \]
   *-commutative [=>] 0.04	\[ \frac{\frac{\color{blue}{\left(f + n\right) \cdot -1}}{-1}}{n - f} \]
   associate-/l* [=>] 0.04	\[ \frac{\color{blue}{\frac{f + n}{\frac{-1}{-1}}}}{n - f} \]
   metadata-eval [=>] 0.04	\[ \frac{\frac{f + n}{\color{blue}{1}}}{n - f} \]
   /-rgt-identity [=>] 0.04	\[ \frac{\color{blue}{f + n}}{n - f} \]

3. Applied egg-rr 0.27

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

4. Applied egg-rr 0.05

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

5. Final simplification 0.05

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

Alternatives

Alternative 1
Error	24.38%
Cost	713

\[\begin{array}{l} \mathbf{if}\;f \leq -8.8 \cdot 10^{-17} \lor \neg \left(f \leq 7.8 \cdot 10^{-53}\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 2
Error	24.96%
Cost	712

\[\begin{array}{l} \mathbf{if}\;f \leq -9 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 7.2 \cdot 10^{-53}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Error	0.04%
Cost	448

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

Alternative 4
Error	25.47%
Cost	328

\[\begin{array}{l} \mathbf{if}\;f \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 7.8 \cdot 10^{-53}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Error	49.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023121 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))