subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 6.9s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (f n) :precision binary64 (/ (/ 1.0 (- n f)) (/ 1.0 (+ 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)) / (1.0 / (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)) / (1.0d0 / (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)) / (1.0 / (n + f));
}

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

code[f_, n_] := N[(N[(1.0 / N[(n - f), $MachinePrecision]), $MachinePrecision] / N[(1.0 / 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 100.0%

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

   [Start] 99.7	\[ \frac{1}{n - f} \cdot n + \frac{1}{n - f} \cdot f \]
   distribute-lft-out [=>] 99.7	\[ \color{blue}{\frac{1}{n - f} \cdot \left(n + f\right)} \]
   associate-*l/ [=>] 100.0	\[ \color{blue}{\frac{1 \cdot \left(n + f\right)}{n - f}} \]
   *-un-lft-identity [<=] 100.0	\[ \frac{\color{blue}{n + f}}{n - f} \]
   flip-- [=>] 52.1	\[ \frac{n + f}{\color{blue}{\frac{n \cdot n - f \cdot f}{n + f}}} \]
   div-inv [=>] 51.9	\[ \frac{n + f}{\color{blue}{\left(n \cdot n - f \cdot f\right) \cdot \frac{1}{n + f}}} \]
   associate-/r* [=>] 51.9	\[ \color{blue}{\frac{\frac{n + f}{n \cdot n - f \cdot f}}{\frac{1}{n + f}}} \]
   *-un-lft-identity [=>] 51.9	\[ \frac{\frac{\color{blue}{1 \cdot \left(n + f\right)}}{n \cdot n - f \cdot f}}{\frac{1}{n + f}} \]
   associate-/l* [=>] 52.1	\[ \frac{\color{blue}{\frac{1}{\frac{n \cdot n - f \cdot f}{n + f}}}}{\frac{1}{n + f}} \]
   flip-- [<=] 100.0	\[ \frac{\frac{1}{\color{blue}{n - f}}}{\frac{1}{n + f}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	73.0%
Cost	978

\[\begin{array}{l} \mathbf{if}\;n \leq -2.05 \cdot 10^{+82} \lor \neg \left(n \leq -4 \cdot 10^{-26}\right) \land \left(n \leq -1.8 \cdot 10^{-95} \lor \neg \left(n \leq 1.95 \cdot 10^{-16}\right)\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	74.4%
Cost	978

\[\begin{array}{l} \mathbf{if}\;n \leq -4.8 \cdot 10^{+48} \lor \neg \left(n \leq -1.7 \cdot 10^{-26} \lor \neg \left(n \leq -2.1 \cdot 10^{-95}\right) \land n \leq 7 \cdot 10^{-16}\right):\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]

Alternative 3
Accuracy	73.6%
Cost	592

\[\begin{array}{l} \mathbf{if}\;n \leq -4 \cdot 10^{+47}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.42 \cdot 10^{-61}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -2.1 \cdot 10^{-95}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 2.2 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Accuracy	100.0%
Cost	448

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

Alternative 5
Accuracy	49.1%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023136 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))