subtraction fraction

Average Accuracy: 100.0% → 100.0%

Time: 7.7s

Precision: binary64

Cost: 2240

Math

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

↓

\[\begin{array}{l} t_0 := \frac{f}{n - f}\\ t_1 := \frac{n}{n - f}\\ \frac{t_0 \cdot t_0 - t_1 \cdot t_1}{t_0 - t_1} \end{array} \]

FPCore

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

↓

(FPCore (f n)
 :precision binary64
 (let* ((t_0 (/ f (- n f))) (t_1 (/ n (- n f))))
   (/ (- (* t_0 t_0) (* t_1 t_1)) (- t_0 t_1))))

C

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

↓

double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

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
    real(8) :: t_0
    real(8) :: t_1
    t_0 = f / (n - f)
    t_1 = n / (n - f)
    code = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)
end function

Java

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

↓

public static double code(double f, double n) {
	double t_0 = f / (n - f);
	double t_1 = n / (n - f);
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
}

Python

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

↓

def code(f, n):
	t_0 = f / (n - f)
	t_1 = n / (n - f)
	return ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1)

Julia

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

↓

function code(f, n)
	t_0 = Float64(f / Float64(n - f))
	t_1 = Float64(n / Float64(n - f))
	return Float64(Float64(Float64(t_0 * t_0) - Float64(t_1 * t_1)) / Float64(t_0 - t_1))
end

MATLAB

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

↓

function tmp = code(f, n)
	t_0 = f / (n - f);
	t_1 = n / (n - f);
	tmp = ((t_0 * t_0) - (t_1 * t_1)) / (t_0 - t_1);
end

Wolfram

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

↓

code[f_, n_] := Block[{t$95$0 = N[(f / N[(n - f), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(n / N[(n - f), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 - t$95$1), $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 52.2%

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

   [Start] 100.0	\[ \frac{f + n}{n - f} \]
   flip-- [=>] 52.4	\[ \frac{f + n}{\color{blue}{\frac{n \cdot n - f \cdot f}{n + f}}} \]
   +-commutative [<=] 52.4	\[ \frac{f + n}{\frac{n \cdot n - f \cdot f}{\color{blue}{f + n}}} \]
   associate-/r/ [=>] 52.2	\[ \color{blue}{\frac{f + n}{n \cdot n - f \cdot f} \cdot \left(f + n\right)} \]

4. Applied egg-rr 100.0%

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

   [Start] 52.2	\[ \frac{f + n}{n \cdot n - f \cdot f} \cdot \left(f + n\right) \]
   distribute-lft-in [=>] 52.2	\[ \color{blue}{\frac{f + n}{n \cdot n - f \cdot f} \cdot f + \frac{f + n}{n \cdot n - f \cdot f} \cdot n} \]
   flip-+ [=>] 51.1	\[ \color{blue}{\frac{\left(\frac{f + n}{n \cdot n - f \cdot f} \cdot f\right) \cdot \left(\frac{f + n}{n \cdot n - f \cdot f} \cdot f\right) - \left(\frac{f + n}{n \cdot n - f \cdot f} \cdot n\right) \cdot \left(\frac{f + n}{n \cdot n - f \cdot f} \cdot n\right)}{\frac{f + n}{n \cdot n - f \cdot f} \cdot f - \frac{f + n}{n \cdot n - f \cdot f} \cdot n}} \]

5. Final simplification 100.0%

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

Alternatives

Alternative 1
Accuracy	74.6%
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -3.7 \cdot 10^{+16} \lor \neg \left(n \leq 1.4 \cdot 10^{-43}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 2
Accuracy	75.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;n \leq -3.1 \cdot 10^{+15} \lor \neg \left(n \leq 8 \cdot 10^{-43}\right):\\ \;\;\;\;2 \cdot \frac{f}{n} + 1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{n}{f} + -1\\ \end{array} \]

Alternative 3
Accuracy	100.0%
Cost	704

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

Alternative 4
Accuracy	100.0%
Cost	448

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

Alternative 5
Accuracy	74.2%
Cost	328

\[\begin{array}{l} \mathbf{if}\;n \leq -2600000000:\\ \;\;\;\;1\\ \mathbf{elif}\;n \leq 5 \cdot 10^{-46}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	49.8%
Cost	64

\[-1 \]

Reproduce

herbie shell --seed 2023152 
(FPCore (f n)
  :name "subtraction fraction"
  :precision binary64
  (/ (- (+ f n)) (- f n)))