rsin B (should all be same)

Average Error: 24.14% → 0.5%

Time: 16.5s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (* r (/ (sin b) (fma (sin b) (- (sin a)) (* (cos b) (cos 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 * (sin(b) / fma(sin(b), -sin(a), (cos(b) * cos(a))));
}

Julia

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

↓

function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(sin(b), Float64(-sin(a)), Float64(cos(b) * cos(a)))))
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[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision]) + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 24.14

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

2. Simplified 24.14

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

   [Start] 24.14	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   +-commutative [=>] 24.14	\[ r \cdot \frac{\sin b}{\cos \color{blue}{\left(b + a\right)}} \]

3. Applied egg-rr 0.52

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

4. Applied egg-rr 0.52

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

5. Simplified 0.52

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

   [Start] 0.52	\[ r \cdot \frac{\sin b}{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)} \]
   associate-+r+ [=>] 0.51	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\cos b \cdot \cos a + \sin b \cdot \left(-\sin a\right)\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)}} \]
   +-commutative [<=] 0.51	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)} \]
   fma-udef [=>] 0.52	\[ r \cdot \frac{\sin b}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \color{blue}{\left(\left(-\sin a\right) \cdot \sin b + \sin b \cdot \sin a\right)}} \]
   *-commutative [<=] 0.52	\[ r \cdot \frac{\sin b}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\sin b \cdot \left(-\sin a\right)} + \sin b \cdot \sin a\right)} \]
   associate-+r+ [=>] 0.57	\[ r \cdot \frac{\sin b}{\color{blue}{\left(\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \sin b \cdot \left(-\sin a\right)\right) + \sin b \cdot \sin a}} \]

6. Taylor expanded in a around inf 0.5

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

7. Simplified 0.5

   \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)\right)} \]
   Proof

   [Start] 0.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, -1 \cdot \left(\sin a \cdot \sin b\right) + \sin a \cdot \sin b\right)\right)} \]
   +-commutative [=>] 0.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin a \cdot \sin b + -1 \cdot \left(\sin a \cdot \sin b\right)}\right)\right)} \]
   mul-1-neg [=>] 0.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \sin b + \color{blue}{\left(-\sin a \cdot \sin b\right)}\right)\right)} \]
   sub-neg [<=] 0.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{\sin a \cdot \sin b - \sin a \cdot \sin b}\right)\right)} \]
   +-inverses [=>] 0.5	\[ r \cdot \frac{\sin b}{\mathsf{fma}\left(\sin b, -\sin a, \mathsf{fma}\left(\cos b, \cos a, \color{blue}{0}\right)\right)} \]

8. Applied egg-rr 0.5

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

9. Final simplification 0.5

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

Alternatives

Alternative 1
Error	0.52%
Cost	32704

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

Alternative 2
Error	0.53%
Cost	32704

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

Alternative 3
Error	24.57%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-5} \lor \neg \left(a \leq 1500\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]

Alternative 4
Error	24.55%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -2.4 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 1500:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 5
Error	24.14%
Cost	13248

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

Alternative 6
Error	24.14%
Cost	13248

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

Alternative 7
Error	45.89%
Cost	13120

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

Alternative 8
Error	49.28%
Cost	7232

\[\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}} \]

Alternative 9
Error	50.19%
Cost	6720

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

Alternative 10
Error	50.19%
Cost	6720

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

Alternative 11
Error	65.84%
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023090 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))