subtraction fraction

Average Error: 0.0 → 0.0

Time: 16.0s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

def code(f, n):
	return (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(n - f))
end

MATLAB

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

↓

function tmp = code(f, n)
	tmp = (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), $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	\[ \frac{-\left(f + n\right)}{f - n} \]
   rational_best-simplify-14 [=>] 0.0	\[ \frac{\color{blue}{0 - \left(f + n\right)}}{f - n} \]
   rational_best-simplify-52 [<=] 0.0	\[ \frac{\color{blue}{\left(0 - n\right) - f}}{f - n} \]
   rational_best-simplify-14 [<=] 0.0	\[ \frac{\color{blue}{\left(-n\right)} - f}{f - n} \]
   rational_best-simplify-12 [<=] 0.0	\[ \frac{\color{blue}{\frac{n}{-1}} - f}{f - n} \]
   rational_best-simplify-8 [<=] 0.0	\[ \frac{\frac{n}{-1} - f}{f - \color{blue}{\frac{n}{1}}} \]
   metadata-eval [<=] 0.0	\[ \frac{\frac{n}{-1} - f}{f - \frac{n}{\color{blue}{-1 \cdot -1}}} \]
   rational_best-simplify-53 [<=] 0.0	\[ \frac{\frac{n}{-1} - f}{f - \color{blue}{\frac{\frac{n}{-1}}{-1}}} \]
   rational_best-simplify-12 [=>] 0.0	\[ \frac{\frac{n}{-1} - f}{f - \color{blue}{\left(-\frac{n}{-1}\right)}} \]
   rational_best-simplify-77 [=>] 0.0	\[ \color{blue}{\frac{f - \frac{n}{-1}}{\left(-\frac{n}{-1}\right) - f}} \]
   rational_best-simplify-8 [<=] 0.0	\[ \frac{\color{blue}{\frac{f}{1}} - \frac{n}{-1}}{\left(-\frac{n}{-1}\right) - f} \]
   metadata-eval [<=] 0.0	\[ \frac{\frac{f}{\color{blue}{-1 \cdot -1}} - \frac{n}{-1}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-53 [<=] 0.0	\[ \frac{\color{blue}{\frac{\frac{f}{-1}}{-1}} - \frac{n}{-1}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-12 [=>] 0.0	\[ \frac{\color{blue}{\left(-\frac{f}{-1}\right)} - \frac{n}{-1}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-9 [<=] 0.0	\[ \frac{\left(-\frac{f}{-1}\right) - \color{blue}{\left(\frac{n}{-1} - 0\right)}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-51 [<=] 0.0	\[ \frac{\color{blue}{0 - \left(\frac{n}{-1} - \left(-\frac{f}{-1}\right)\right)}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-12 [=>] 0.0	\[ \frac{0 - \left(\color{blue}{\left(-n\right)} - \left(-\frac{f}{-1}\right)\right)}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-11 [=>] 0.0	\[ \frac{0 - \left(\color{blue}{n \cdot -1} - \left(-\frac{f}{-1}\right)\right)}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-59 [<=] 0.0	\[ \frac{0 - \color{blue}{\left(\frac{f}{-1} + n \cdot -1\right)}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-12 [=>] 0.0	\[ \frac{0 - \left(\color{blue}{\left(-f\right)} + n \cdot -1\right)}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-14 [=>] 0.0	\[ \frac{0 - \left(\color{blue}{\left(0 - f\right)} + n \cdot -1\right)}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-11 [<=] 0.0	\[ \frac{0 - \left(\left(0 - f\right) + \color{blue}{\left(-n\right)}\right)}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-57 [<=] 0.0	\[ \frac{0 - \color{blue}{\left(0 - \left(f + n\right)\right)}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-51 [=>] 0.0	\[ \frac{\color{blue}{\left(f + n\right) - \left(0 - 0\right)}}{\left(-\frac{n}{-1}\right) - f} \]
   metadata-eval [=>] 0.0	\[ \frac{\left(f + n\right) - \color{blue}{0}}{\left(-\frac{n}{-1}\right) - f} \]
   rational_best-simplify-9 [=>] 0.0	\[ \frac{\color{blue}{f + n}}{\left(-\frac{n}{-1}\right) - f} \]

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	16.8
Cost	712

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -14000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.35 \cdot 10^{+48}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	16.4
Cost	712

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -102000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 4.2 \cdot 10^{+44}:\\ \;\;\;\;-2 \cdot \frac{n}{f} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.0
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 3.6 \cdot 10^{+40}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	32.1
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023099 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))