rsin A (should all be same)

Average Error: 15.0 → 0.4

Time: 13.9s

Precision: binary64

Cost: 19648

Math

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

↓

\[\frac{r}{\frac{\cos a}{\tan b} - \sin a} \]

FPCore

(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))

↓

(FPCore (r a b) :precision binary64 (/ r (- (/ (cos a) (tan b)) (sin a))))

C

double code(double r, double a, double b) {
	return (r * sin(b)) / cos((a + b));
}

↓

double code(double r, double a, double b) {
	return r / ((cos(a) / tan(b)) - sin(a));
}

Fortran

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = (r * sin(b)) / cos((a + b))
end function

↓

real(8) function code(r, a, b)
    real(8), intent (in) :: r
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = r / ((cos(a) / tan(b)) - sin(a))
end function

Java

public static double code(double r, double a, double b) {
	return (r * Math.sin(b)) / Math.cos((a + b));
}

↓

public static double code(double r, double a, double b) {
	return r / ((Math.cos(a) / Math.tan(b)) - Math.sin(a));
}

Python

def code(r, a, b):
	return (r * math.sin(b)) / math.cos((a + b))

↓

def code(r, a, b):
	return r / ((math.cos(a) / math.tan(b)) - math.sin(a))

Julia

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end

↓

function code(r, a, b)
	return Float64(r / Float64(Float64(cos(a) / tan(b)) - sin(a)))
end

MATLAB

function tmp = code(r, a, b)
	tmp = (r * sin(b)) / cos((a + b));
end

↓

function tmp = code(r, a, b)
	tmp = r / ((cos(a) / tan(b)) - sin(a));
end

Wolfram

code[r_, a_, b_] := N[(N[(r * N[Sin[b], $MachinePrecision]), $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[r_, a_, b_] := N[(r / N[(N[(N[Cos[a], $MachinePrecision] / N[Tan[b], $MachinePrecision]), $MachinePrecision] - N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0

   \[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]

2. Simplified 15.1

   \[\leadsto \color{blue}{\frac{r}{\frac{\cos \left(b + a\right)}{\sin b}}} \]
   Proof

   [Start] 15.0	\[ \frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
   associate-/l* [=>] 15.1	\[ \color{blue}{\frac{r}{\frac{\cos \left(a + b\right)}{\sin b}}} \]
   +-commutative [=>] 15.1	\[ \frac{r}{\frac{\cos \color{blue}{\left(b + a\right)}}{\sin b}} \]

3. Applied egg-rr 0.4

   \[\leadsto \frac{r}{\color{blue}{\frac{1}{\sin b} \cdot \left(\cos b \cdot \cos a\right) + \frac{1}{\sin b} \cdot \left(\left(-\sin b\right) \cdot \sin a\right)}} \]

4. Simplified 0.4

   \[\leadsto \frac{r}{\color{blue}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)}} \]
   Proof

   [Start] 0.4	\[ \frac{r}{\frac{1}{\sin b} \cdot \left(\cos b \cdot \cos a\right) + \frac{1}{\sin b} \cdot \left(\left(-\sin b\right) \cdot \sin a\right)} \]
   distribute-lft-neg-in [<=] 0.4	\[ \frac{r}{\frac{1}{\sin b} \cdot \left(\cos b \cdot \cos a\right) + \frac{1}{\sin b} \cdot \color{blue}{\left(-\sin b \cdot \sin a\right)}} \]
   associate-*l/ [=>] 0.4	\[ \frac{r}{\color{blue}{\frac{1 \cdot \left(\cos b \cdot \cos a\right)}{\sin b}} + \frac{1}{\sin b} \cdot \left(-\sin b \cdot \sin a\right)} \]
   *-lft-identity [=>] 0.4	\[ \frac{r}{\frac{\color{blue}{\cos b \cdot \cos a}}{\sin b} + \frac{1}{\sin b} \cdot \left(-\sin b \cdot \sin a\right)} \]
   associate-*l/ [<=] 0.4	\[ \frac{r}{\color{blue}{\frac{\cos b}{\sin b} \cdot \cos a} + \frac{1}{\sin b} \cdot \left(-\sin b \cdot \sin a\right)} \]
   fma-def [=>] 0.4	\[ \frac{r}{\color{blue}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \frac{1}{\sin b} \cdot \left(-\sin b \cdot \sin a\right)\right)}} \]
   associate-*l/ [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \color{blue}{\frac{1 \cdot \left(-\sin b \cdot \sin a\right)}{\sin b}}\right)} \]
   *-lft-identity [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \frac{\color{blue}{-\sin b \cdot \sin a}}{\sin b}\right)} \]
   distribute-frac-neg [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \color{blue}{-\frac{\sin b \cdot \sin a}{\sin b}}\right)} \]
   associate-*l/ [<=] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\color{blue}{\frac{\sin b}{\sin b} \cdot \sin a}\right)} \]
   distribute-lft-neg-in [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \color{blue}{\left(-\frac{\sin b}{\sin b}\right) \cdot \sin a}\right)} \]
   *-inverses [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \left(-\color{blue}{1}\right) \cdot \sin a\right)} \]
   metadata-eval [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \color{blue}{-1} \cdot \sin a\right)} \]
   mul-1-neg [=>] 0.4	\[ \frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, \color{blue}{-\sin a}\right)} \]

5. Applied egg-rr 0.4

   \[\leadsto \frac{r}{\color{blue}{0 - \left(\sin a - \frac{\cos a}{\tan b}\right)}} \]

6. Simplified 0.4

   \[\leadsto \frac{r}{\color{blue}{\frac{\cos a}{\tan b} - \sin a}} \]
   Proof

   [Start] 0.4	\[ \frac{r}{0 - \left(\sin a - \frac{\cos a}{\tan b}\right)} \]
   associate--r- [=>] 0.4	\[ \frac{r}{\color{blue}{\left(0 - \sin a\right) + \frac{\cos a}{\tan b}}} \]
   +-commutative [=>] 0.4	\[ \frac{r}{\color{blue}{\frac{\cos a}{\tan b} + \left(0 - \sin a\right)}} \]
   sub0-neg [=>] 0.4	\[ \frac{r}{\frac{\cos a}{\tan b} + \color{blue}{\left(-\sin a\right)}} \]
   sub-neg [<=] 0.4	\[ \frac{r}{\color{blue}{\frac{\cos a}{\tan b} - \sin a}} \]

7. Final simplification 0.4

   \[\leadsto \frac{r}{\frac{\cos a}{\tan b} - \sin a} \]

Alternatives

Alternative 1
Error	15.0
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -0.18 \lor \neg \left(a \leq 0.0016\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \end{array} \]

Alternative 2
Error	15.0
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.18:\\ \;\;\;\;\frac{r}{\frac{\cos a}{\sin b}}\\ \mathbf{elif}\;a \leq 0.025:\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 3
Error	15.1
Cost	13248

\[\frac{r}{\cos \left(a + b\right)} \cdot \sin b \]

Alternative 4
Error	16.1
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{-8} \lor \neg \left(b \leq 0.0024\right):\\ \;\;\;\;\frac{r}{\frac{1}{\tan b} - a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 5
Error	30.7
Cost	6852

\[\begin{array}{l} \mathbf{if}\;b \leq 0.56:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \sin b\\ \end{array} \]

Alternative 6
Error	32.1
Cost	6720

\[r \cdot \frac{b}{\cos a} \]

Alternative 7
Error	32.2
Cost	6720

\[b \cdot \frac{r}{\cos a} \]

Alternative 8
Error	41.9
Cost	576

\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]

Alternative 9
Error	42.5
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023045 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))