subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 7.5s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

Julia

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 99.9%

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

2. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{f + n}{n - f}} \]
   Step-by-step derivation

   [Start] 99.9	\[ \frac{-\left(f + n\right)}{f - n} \]
   neg-mul-1 [=>] 99.9	\[ \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]
   *-commutative [=>] 99.9	\[ \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]
   div-sub [=>] 99.9	\[ \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]
   metadata-eval [<=] 99.9	\[ \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]
   metadata-eval [<=] 99.9	\[ \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]
   associate-/l* [<=] 99.9	\[ \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]
   *-commutative [<=] 99.9	\[ \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]
   neg-mul-1 [<=] 99.9	\[ \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]
   metadata-eval [<=] 99.9	\[ \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]
   metadata-eval [<=] 99.9	\[ \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]
   associate-/l* [<=] 99.9	\[ \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]
   *-commutative [=>] 99.9	\[ \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]
   neg-mul-1 [<=] 99.9	\[ \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]
   div-sub [<=] 99.9	\[ \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]
   unsub-neg [<=] 99.9	\[ \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]
   remove-double-neg [=>] 99.9	\[ \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]
   +-commutative [<=] 99.9	\[ \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]
   sub-neg [<=] 99.9	\[ \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]
   metadata-eval [=>] 99.9	\[ \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 99.9	\[ \frac{f + n}{\color{blue}{n - f}} \]

3. Applied egg-rr 99.7%

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

   [Start] 99.9	\[ \frac{f + n}{n - f} \]
   clear-num [=>] 100.0	\[ \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]
   associate-/r/ [=>] 99.7	\[ \color{blue}{\frac{1}{n - f} \cdot \left(f + n\right)} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{n - f}{n + f}}} \]
   Step-by-step derivation

   [Start] 99.7	\[ \frac{1}{n - f} \cdot \left(f + n\right) \]
   associate-*l/ [=>] 99.9	\[ \color{blue}{\frac{1 \cdot \left(f + n\right)}{n - f}} \]
   associate-/l* [=>] 100.0	\[ \color{blue}{\frac{1}{\frac{n - f}{f + n}}} \]
   +-commutative [=>] 100.0	\[ \frac{1}{\frac{n - f}{\color{blue}{n + f}}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	75.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;f \leq -102000000 \lor \neg \left(f \leq 0.002\right):\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \end{array} \]

Alternative 2
Accuracy	74.6%
Cost	712

\[\begin{array}{l} \mathbf{if}\;f \leq -4400000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 11.5:\\ \;\;\;\;1 + 2 \cdot \frac{f}{n}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	73.1%
Cost	328

\[\begin{array}{l} \mathbf{if}\;f \leq -5000000000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 1.18 \cdot 10^{-76}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	50.0%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023159 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))