subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 7.3s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

Python

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

↓

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

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(f / Float64(n - f)))
end

MATLAB

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

↓

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

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

4. Applied egg-rr 99.9%

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

5. Simplified 100.0%

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

   [Start] 99.9	\[ e^{\mathsf{log1p}\left(\frac{n}{n - f}\right)} - \left(1 - \frac{f}{n - f}\right) \]
   associate--r- [=>] 99.9	\[ \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{n}{n - f}\right)} - 1\right) + \frac{f}{n - f}} \]
   expm1-def [=>] 99.9	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{n}{n - f}\right)\right)} + \frac{f}{n - f} \]
   expm1-log1p [=>] 100.0	\[ \color{blue}{\frac{n}{n - f}} + \frac{f}{n - f} \]

6. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	72.8%
Cost	976

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -5.6 \cdot 10^{+30}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-78}:\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{n + f}{-f}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	73.1%
Cost	976

\[\begin{array}{l} t_0 := 2 \cdot \frac{f}{n} + 1\\ t_1 := -2 \cdot \frac{n}{f} + -1\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.36 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -1.75 \cdot 10^{-78}:\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{elif}\;n \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	72.6%
Cost	912

\[\begin{array}{l} t_0 := \frac{f}{n} + 1\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -1.16 \cdot 10^{+31}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \mathbf{elif}\;n \leq -1.35 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 1.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{n + f}{-f}\\ \mathbf{else}:\\ \;\;\;\;\frac{n + f}{n}\\ \end{array} \]

Alternative 4
Accuracy	72.3%
Cost	850

\[\begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+113} \lor \neg \left(n \leq -1.1 \cdot 10^{+31}\right) \land \left(n \leq -1.22 \cdot 10^{-85} \lor \neg \left(n \leq 6.5 \cdot 10^{-6}\right)\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5
Accuracy	72.5%
Cost	850

\[\begin{array}{l} \mathbf{if}\;n \leq -1.9 \cdot 10^{+113} \lor \neg \left(n \leq -3.9 \cdot 10^{+30}\right) \land \left(n \leq -5.8 \cdot 10^{-86} \lor \neg \left(n \leq 9.5 \cdot 10^{-7}\right)\right):\\ \;\;\;\;\frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1 - \frac{n}{f}\\ \end{array} \]

Alternative 6
Accuracy	72.6%
Cost	848

\[\begin{array}{l} t_0 := \frac{f}{n} + 1\\ t_1 := -1 - \frac{n}{f}\\ \mathbf{if}\;n \leq -1.9 \cdot 10^{+113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;n \leq -9 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;n \leq 8.5 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{n + f}{n}\\ \end{array} \]

Alternative 7
Accuracy	72.0%
Cost	592

\[\begin{array}{l} \mathbf{if}\;n \leq -5 \cdot 10^{+113}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq -1 \cdot 10^{+31}:\\ \;\;\;\;-1\\ \mathbf{elif}\;n \leq -1 \cdot 10^{-78}:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Accuracy	100.0%
Cost	448

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

Alternative 9
Accuracy	49.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023122 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))