subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 6.6s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

def code(f, n):
	return (f / (n - 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 / Float64(n - f)) + Float64(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 - 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[(n - f), $MachinePrecision]), $MachinePrecision] + N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{-\left(f + n\right)}{f - n} \]
   sub-neg [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{f + \left(-n\right)}} \]
   +-commutative [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(-n\right) + f}} \]
   neg-sub0 [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{\left(0 - n\right)} + f} \]
   associate-+l- [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{0 - \left(n - f\right)}} \]
   sub0-neg [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{-\left(n - f\right)}} \]
   neg-mul-1 [=>] 100.0	\[ \frac{-\left(f + n\right)}{\color{blue}{-1 \cdot \left(n - f\right)}} \]
   associate-/r* [=>] 100.0	\[ \color{blue}{\frac{\frac{-\left(f + n\right)}{-1}}{n - f}} \]
   neg-mul-1 [=>] 100.0	\[ \frac{\frac{\color{blue}{-1 \cdot \left(f + n\right)}}{-1}}{n - f} \]
   *-commutative [=>] 100.0	\[ \frac{\frac{\color{blue}{\left(f + n\right) \cdot -1}}{-1}}{n - f} \]
   associate-/l* [=>] 100.0	\[ \frac{\color{blue}{\frac{f + n}{\frac{-1}{-1}}}}{n - f} \]
   metadata-eval [=>] 100.0	\[ \frac{\frac{f + n}{\color{blue}{1}}}{n - f} \]
   /-rgt-identity [=>] 100.0	\[ \frac{\color{blue}{f + n}}{n - f} \]

3. Applied egg-rr 99.7%

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

   [Start] 100.0	\[ \frac{f + n}{n - f} \]
   div-inv [=>] 99.7	\[ \color{blue}{\left(f + n\right) \cdot \frac{1}{n - f}} \]
   *-commutative [=>] 99.7	\[ \color{blue}{\frac{1}{n - f} \cdot \left(f + n\right)} \]
   +-commutative [=>] 99.7	\[ \frac{1}{n - f} \cdot \color{blue}{\left(n + f\right)} \]
   distribute-lft-in [=>] 99.7	\[ \color{blue}{\frac{1}{n - f} \cdot n + \frac{1}{n - f} \cdot f} \]

4. Applied egg-rr 98.9%

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

   [Start] 99.7	\[ \frac{1}{n - f} \cdot n + \frac{1}{n - f} \cdot f \]
   +-commutative [=>] 99.7	\[ \color{blue}{\frac{1}{n - f} \cdot f + \frac{1}{n - f} \cdot n} \]
   expm1-log1p-u [=>] 98.6	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{n - f} \cdot f\right)\right)} + \frac{1}{n - f} \cdot n \]
   expm1-udef [=>] 98.6	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{n - f} \cdot f\right)} - 1\right)} + \frac{1}{n - f} \cdot n \]
   associate-+l- [=>] 98.6	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{n - f} \cdot f\right)} - \left(1 - \frac{1}{n - f} \cdot n\right)} \]
   associate-*l/ [=>] 98.7	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot f}{n - f}}\right)} - \left(1 - \frac{1}{n - f} \cdot n\right) \]
   *-un-lft-identity [<=] 98.7	\[ e^{\mathsf{log1p}\left(\frac{\color{blue}{f}}{n - f}\right)} - \left(1 - \frac{1}{n - f} \cdot n\right) \]
   associate-*l/ [=>] 98.9	\[ e^{\mathsf{log1p}\left(\frac{f}{n - f}\right)} - \left(1 - \color{blue}{\frac{1 \cdot n}{n - f}}\right) \]
   *-un-lft-identity [<=] 98.9	\[ e^{\mathsf{log1p}\left(\frac{f}{n - f}\right)} - \left(1 - \frac{\color{blue}{n}}{n - f}\right) \]

5. Simplified 100.0%

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

   [Start] 98.9	\[ e^{\mathsf{log1p}\left(\frac{f}{n - f}\right)} - \left(1 - \frac{n}{n - f}\right) \]
   associate--r- [=>] 98.9	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{f}{n - f}\right)} - 1\right) + \frac{n}{n - f}} \]
   expm1-def [=>] 98.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{f}{n - f}\right)\right)} + \frac{n}{n - f} \]
   expm1-log1p [=>] 100.0	\[ \color{blue}{\frac{f}{n - f}} + \frac{n}{n - f} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.5%
Cost	713

\[\begin{array}{l} \mathbf{if}\;f \leq -2.1 \cdot 10^{+69} \lor \neg \left(f \leq 2600000000000\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 2
Accuracy	74.0%
Cost	712

\[\begin{array}{l} \mathbf{if}\;f \leq -4.6 \cdot 10^{+70}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 55000000000:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	72.9%
Cost	328

\[\begin{array}{l} \mathbf{if}\;f \leq -3.6 \cdot 10^{+85}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 450000000:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	50.1%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023137 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))