subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 7.0s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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) + -1.0));
}

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) + (-1.0d0)))
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) + -1.0));
}

Python

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

↓

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

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) + -1.0)))
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) + -1.0));
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] + -1.0), $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	\[ \frac{-\left(f + n\right)}{f - n} \]
   neg-mul-1 [=>] 100.0	\[ \frac{\color{blue}{-1 \cdot \left(f + n\right)}}{f - n} \]
   *-commutative [=>] 100.0	\[ \frac{\color{blue}{\left(f + n\right) \cdot -1}}{f - n} \]
   associate-/l* [=>] 100.0	\[ \color{blue}{\frac{f + n}{\frac{f - n}{-1}}} \]
   div-sub [=>] 100.0	\[ \frac{f + n}{\color{blue}{\frac{f}{-1} - \frac{n}{-1}}} \]
   metadata-eval [<=] 100.0	\[ \frac{f + n}{\frac{f}{\color{blue}{\frac{1}{-1}}} - \frac{n}{-1}} \]
   metadata-eval [<=] 100.0	\[ \frac{f + n}{\frac{f}{\frac{\color{blue}{--1}}{-1}} - \frac{n}{-1}} \]
   associate-/l* [<=] 100.0	\[ \frac{f + n}{\color{blue}{\frac{f \cdot -1}{--1}} - \frac{n}{-1}} \]
   *-commutative [<=] 100.0	\[ \frac{f + n}{\frac{\color{blue}{-1 \cdot f}}{--1} - \frac{n}{-1}} \]
   neg-mul-1 [<=] 100.0	\[ \frac{f + n}{\frac{\color{blue}{-f}}{--1} - \frac{n}{-1}} \]
   metadata-eval [<=] 100.0	\[ \frac{f + n}{\frac{-f}{--1} - \frac{n}{\color{blue}{\frac{1}{-1}}}} \]
   metadata-eval [<=] 100.0	\[ \frac{f + n}{\frac{-f}{--1} - \frac{n}{\frac{\color{blue}{--1}}{-1}}} \]
   associate-/l* [<=] 100.0	\[ \frac{f + n}{\frac{-f}{--1} - \color{blue}{\frac{n \cdot -1}{--1}}} \]
   *-commutative [=>] 100.0	\[ \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-1 \cdot n}}{--1}} \]
   neg-mul-1 [<=] 100.0	\[ \frac{f + n}{\frac{-f}{--1} - \frac{\color{blue}{-n}}{--1}} \]
   div-sub [<=] 100.0	\[ \frac{f + n}{\color{blue}{\frac{\left(-f\right) - \left(-n\right)}{--1}}} \]
   unsub-neg [<=] 100.0	\[ \frac{f + n}{\frac{\color{blue}{\left(-f\right) + \left(-\left(-n\right)\right)}}{--1}} \]
   remove-double-neg [=>] 100.0	\[ \frac{f + n}{\frac{\left(-f\right) + \color{blue}{n}}{--1}} \]
   +-commutative [<=] 100.0	\[ \frac{f + n}{\frac{\color{blue}{n + \left(-f\right)}}{--1}} \]
   sub-neg [<=] 100.0	\[ \frac{f + n}{\frac{\color{blue}{n - f}}{--1}} \]
   metadata-eval [=>] 100.0	\[ \frac{f + n}{\frac{n - f}{\color{blue}{1}}} \]
   /-rgt-identity [=>] 100.0	\[ \frac{f + n}{\color{blue}{n - f}} \]

3. Applied egg-rr 99.7%

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

   [Start] 100.0	\[ \frac{f + n}{n - f} \]
   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%

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

   [Start] 99.7	\[ \frac{1}{n - f} \cdot \left(f + n\right) \]
   +-commutative [=>] 99.7	\[ \frac{1}{n - f} \cdot \color{blue}{\left(n + f\right)} \]
   distribute-rgt-in [=>] 99.7	\[ \color{blue}{n \cdot \frac{1}{n - f} + f \cdot \frac{1}{n - f}} \]
   un-div-inv [=>] 99.9	\[ \color{blue}{\frac{n}{n - f}} + f \cdot \frac{1}{n - f} \]
   un-div-inv [=>] 100.0	\[ \frac{n}{n - f} + \color{blue}{\frac{f}{n - f}} \]

5. Applied egg-rr 100.0%

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

   [Start] 100.0	\[ \frac{n}{n - f} + \frac{f}{n - f} \]
   clear-num [=>] 100.0	\[ \frac{n}{n - f} + \color{blue}{\frac{1}{\frac{n - f}{f}}} \]
   inv-pow [=>] 100.0	\[ \frac{n}{n - f} + \color{blue}{{\left(\frac{n - f}{f}\right)}^{-1}} \]

6. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{n}{n - f} + {\left(\frac{n - f}{f}\right)}^{-1} \]
   unpow-1 [=>] 100.0	\[ \frac{n}{n - f} + \color{blue}{\frac{1}{\frac{n - f}{f}}} \]
   div-sub [=>] 100.0	\[ \frac{n}{n - f} + \frac{1}{\color{blue}{\frac{n}{f} - \frac{f}{f}}} \]
   sub-neg [=>] 100.0	\[ \frac{n}{n - f} + \frac{1}{\color{blue}{\frac{n}{f} + \left(-\frac{f}{f}\right)}} \]
   *-inverses [=>] 100.0	\[ \frac{n}{n - f} + \frac{1}{\frac{n}{f} + \left(-\color{blue}{1}\right)} \]
   metadata-eval [=>] 100.0	\[ \frac{n}{n - f} + \frac{1}{\frac{n}{f} + \color{blue}{-1}} \]

7. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.6%
Cost	977

\[\begin{array}{l} \mathbf{if}\;n \leq -2.25 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -6.8 \cdot 10^{-16} \lor \neg \left(n \leq -1.16 \cdot 10^{-57}\right) \land n \leq 4 \cdot 10^{-14}:\\ \;\;\;\;-1 + \frac{n}{f} \cdot -2\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 2
Accuracy	74.1%
Cost	592

\[\begin{array}{l} \mathbf{if}\;n \leq -3.3 \cdot 10^{+28}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1.66 \cdot 10^{-15}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-57}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 9.2 \cdot 10^{-14}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	448

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

Alternative 4
Accuracy	48.7%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023146 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))