subtraction fraction

Average Error: 0.0 → 0.0

Time: 10.3s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

MATLAB

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

↓

function tmp = code(f, n)
	tmp = (n / (n - f)) + (1.0 / ((n - f) / 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[(1.0 / N[(N[(n - f), $MachinePrecision] / 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

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

4. Applied egg-rr 0.1

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

5. Applied egg-rr 0.0

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n}{n - f}\right)} - 1\right)} + \frac{1}{\frac{n - f}{f}} \]

6. Simplified 0.0

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

   [Start] 0.0	\[ \left(e^{\mathsf{log1p}\left(\frac{n}{n - f}\right)} - 1\right) + \frac{1}{\frac{n - f}{f}} \]
   expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{n - f}\right)\right)} + \frac{1}{\frac{n - f}{f}} \]
   expm1-log1p [=>] 0.0	\[ \color{blue}{\frac{n}{n - f}} + \frac{1}{\frac{n - f}{f}} \]

7. Final simplification 0.0

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

Alternatives

Alternative 1
Error	15.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;n \leq -1.6 \cdot 10^{+68}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.25 \cdot 10^{+21}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Error	0.0
Cost	704

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

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	16.2
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -2950000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2 \cdot 10^{-13}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	31.8
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023046 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))