subtraction fraction

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

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 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%

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

   [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 [=>] 99.8	\[ \color{blue}{\frac{n}{n - f}} + f \cdot \frac{1}{n - f} \]
   un-div-inv [=>] 100.0	\[ \frac{n}{n - f} + \color{blue}{\frac{f}{n - f}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;f \leq -3.7 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.9 \cdot 10^{-26}:\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	100.0%
Cost	448

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

Alternative 3
Accuracy	74.4%
Cost	328

\[\begin{array}{l} \mathbf{if}\;f \leq -2.2 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{elif}\;f \leq 4.8 \cdot 10^{-26}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4
Accuracy	49.9%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023160 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))