rsin B (should all be same)

Average Error: 14.9 → 0.3

Time: 16.0s

Precision: binary64

Cost: 32704

Math

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

↓

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

FPCore

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

↓

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

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 sin(b) * (r / ((cos(a) * cos(b)) - (sin(a) * sin(b))));
}

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 = sin(b) * (r / ((cos(a) * cos(b)) - (sin(a) * sin(b))))
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 Math.sin(b) * (r / ((Math.cos(a) * Math.cos(b)) - (Math.sin(a) * Math.sin(b))));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Derivation

1. Initial program 14.9

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

2. Simplified 14.9

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

   [Start] 14.9	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 14.9	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.3

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

4. Taylor expanded in r around 0 0.3

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

5. Applied egg-rr 0.3

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

6. Simplified 0.3

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b + \sin a \cdot \left(-\sin b\right)} \]
   +-commutative [<=] 0.3	\[ \frac{\sin b \cdot r}{\color{blue}{\sin a \cdot \left(-\sin b\right) + \cos a \cdot \cos b}} \]
   fma-def [=>] 0.3	\[ \frac{\sin b \cdot r}{\color{blue}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)}} \]

7. Taylor expanded in b around inf 0.3

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

8. Simplified 0.3

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   *-commutative [=>] 0.3	\[ \frac{\color{blue}{r \cdot \sin b}}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{r}{\frac{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}{\sin b}}} \]
   associate-/r/ [=>] 0.3	\[ \color{blue}{\frac{r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \cdot \sin b} \]
   +-commutative [=>] 0.3	\[ \frac{r}{\color{blue}{\cos a \cdot \cos b + -1 \cdot \left(\sin a \cdot \sin b\right)}} \cdot \sin b \]
   *-commutative [<=] 0.3	\[ \frac{r}{\cos a \cdot \cos b + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}} \cdot \sin b \]
   mul-1-neg [=>] 0.3	\[ \frac{r}{\cos a \cdot \cos b + \color{blue}{\left(-\sin b \cdot \sin a\right)}} \cdot \sin b \]
   unsub-neg [=>] 0.3	\[ \frac{r}{\color{blue}{\cos a \cdot \cos b - \sin b \cdot \sin a}} \cdot \sin b \]
   *-commutative [=>] 0.3	\[ \frac{r}{\cos a \cdot \cos b - \color{blue}{\sin a \cdot \sin b}} \cdot \sin b \]

9. Final simplification 0.3

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

Alternatives

Alternative 1
Error	11.6
Cost	45892

\[\begin{array}{l} t_0 := \cos \left(a + b\right)\\ \mathbf{if}\;r \cdot \frac{\sin b}{t_0} \leq 5 \cdot 10^{-298}:\\ \;\;\;\;\sin b \cdot \frac{r}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{r}{\cos a \cdot \frac{\cos b}{\sin b} - \sin a}\right|\\ \end{array} \]

Alternative 2
Error	0.3
Cost	32704

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

Alternative 3
Error	15.3
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 78000000000:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 4
Error	14.9
Cost	13248

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

Alternative 5
Error	14.9
Cost	13248

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

Alternative 6
Error	28.7
Cost	13120

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

Alternative 7
Error	28.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.1 \lor \neg \left(b \leq 200000\right):\\ \;\;\;\;\frac{\sin b}{\frac{1}{r}}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Error	31.5
Cost	6720

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

Alternative 9
Error	41.8
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2022364 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))