subtraction fraction

Average Error: 0.0 → 0.0

Time: 7.3s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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 - f) / ((f - n) / ((f + n) / (f - n)))
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 - f) / ((f - n) / ((f + n) / (f - n)));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

code[f_, n_] := N[(N[(n - f), $MachinePrecision] / N[(N[(f - n), $MachinePrecision] / N[(N[(f + n), $MachinePrecision] / N[(f - n), $MachinePrecision]), $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	\[ \frac{-\left(f + n\right)}{f - n} \]
   rational.json-simplify-50 [<=] 0.0	\[ \color{blue}{\frac{f + n}{n - f}} \]

3. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{n - f}{f + n} \cdot \frac{1}{\left(n - f\right) \cdot \frac{\frac{n - f}{f + n}}{f + n}}} \]

4. Simplified 0.0

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

   [Start] 0.2	\[ \frac{n - f}{f + n} \cdot \frac{1}{\left(n - f\right) \cdot \frac{\frac{n - f}{f + n}}{f + n}} \]
   rational.json-simplify-46 [=>] 0.0	\[ \frac{n - f}{f + n} \cdot \color{blue}{\frac{\frac{1}{n - f}}{\frac{\frac{n - f}{f + n}}{f + n}}} \]
   rational.json-simplify-61 [=>] 0.2	\[ \frac{n - f}{f + n} \cdot \color{blue}{\frac{f + n}{\frac{\frac{n - f}{f + n}}{\frac{1}{n - f}}}} \]
   rational.json-simplify-61 [=>] 0.0	\[ \frac{n - f}{f + n} \cdot \frac{f + n}{\color{blue}{\frac{n - f}{\frac{1}{\frac{n - f}{f + n}}}}} \]
   rational.json-simplify-61 [<=] 0.0	\[ \frac{n - f}{f + n} \cdot \frac{f + n}{\frac{n - f}{\color{blue}{\frac{f + n}{\frac{n - f}{1}}}}} \]
   rational.json-simplify-7 [=>] 0.0	\[ \frac{n - f}{f + n} \cdot \frac{f + n}{\frac{n - f}{\frac{f + n}{\color{blue}{n - f}}}} \]

5. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{-\frac{\left(f + n\right) \cdot 2}{\frac{f + n}{n - f}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)}} \]

6. Simplified 0.2

   \[\leadsto \color{blue}{\frac{2 \cdot \left(f - n\right)}{\left(\left(n - f\right) \cdot \frac{2}{f + n}\right) \cdot \left(f - n\right)}} \]
   Proof

   [Start] 0.2	\[ \frac{-\frac{\left(f + n\right) \cdot 2}{\frac{f + n}{n - f}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-10 [=>] 0.2	\[ \frac{\color{blue}{\frac{\frac{\left(f + n\right) \cdot 2}{\frac{f + n}{n - f}}}{-1}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-49 [=>] 0.2	\[ \frac{\frac{\color{blue}{2 \cdot \frac{f + n}{\frac{f + n}{n - f}}}}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-2 [=>] 0.2	\[ \frac{\frac{\color{blue}{\frac{f + n}{\frac{f + n}{n - f}} \cdot 2}}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-61 [=>] 0.2	\[ \frac{\frac{\color{blue}{\frac{n - f}{\frac{f + n}{f + n}}} \cdot 2}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-60 [=>] 0.2	\[ \frac{\frac{\color{blue}{\left(n - f\right)} \cdot 2}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-49 [=>] 0.2	\[ \frac{\color{blue}{2 \cdot \frac{n - f}{-1}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-60 [<=] 0.2	\[ \frac{2 \cdot \frac{\color{blue}{\frac{n - f}{\frac{f + n}{f + n}}}}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-61 [<=] 0.2	\[ \frac{2 \cdot \frac{\color{blue}{\frac{f + n}{\frac{f + n}{n - f}}}}{-1}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-47 [=>] 0.2	\[ \frac{2 \cdot \color{blue}{\frac{f + n}{\frac{f + n}{n - f} \cdot -1}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-2 [=>] 0.2	\[ \frac{2 \cdot \frac{f + n}{\color{blue}{-1 \cdot \frac{f + n}{n - f}}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-49 [<=] 0.2	\[ \frac{2 \cdot \frac{f + n}{\color{blue}{\frac{\left(f + n\right) \cdot -1}{n - f}}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-8 [<=] 0.2	\[ \frac{2 \cdot \frac{f + n}{\frac{\color{blue}{-\left(f + n\right)}}{n - f}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-50 [<=] 0.2	\[ \frac{2 \cdot \frac{f + n}{\color{blue}{\frac{f + n}{f - n}}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-61 [=>] 0.2	\[ \frac{2 \cdot \color{blue}{\frac{f - n}{\frac{f + n}{f + n}}}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]
   rational.json-simplify-60 [=>] 0.2	\[ \frac{2 \cdot \color{blue}{\left(f - n\right)}}{-\left(n - f\right) \cdot \left(\left(n - f\right) \cdot \frac{2}{f + n}\right)} \]

7. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\frac{\frac{f + n}{f - n}}{n - f} \cdot \left(f - n\right)} \]

8. Applied egg-rr 0.0

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

9. Final simplification 0.0

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

Alternatives

Alternative 1
Error	17.1
Cost	976

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -3.7 \cdot 10^{-16}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -5.2 \cdot 10^{-75}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 2.8 \cdot 10^{-25}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	16.1
Cost	712

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -2.7 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 3.45 \cdot 10^{-25}:\\ \;\;\;\;-2 \cdot \frac{n}{f} - 1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.0
Cost	448

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

Alternative 4
Error	16.8
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -2.7 \cdot 10^{+53}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 10^{-24}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	31.7
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023075 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))