rsin B (should all be same)

Average Error: 24.05% → 0.59%

Time: 18.8s

Precision: binary64

Cost: 77760

Math

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

↓

\[\begin{array}{l} t_0 := -\sin a\\ \frac{\sin b}{\frac{\mathsf{fma}\left(\sin b, t_0, \mathsf{fma}\left(\cos b, \cos a, \mathsf{fma}\left(t_0, \sin b, \sin b \cdot \sin a\right)\right)\right)}{r}} \end{array} \]

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (- (sin a))))
   (/
    (sin b)
    (/
     (fma
      (sin b)
      t_0
      (fma (cos b) (cos a) (fma t_0 (sin b) (* (sin b) (sin a)))))
     r))))

C

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

↓

double code(double r, double a, double b) {
	double t_0 = -sin(a);
	return sin(b) / (fma(sin(b), t_0, fma(cos(b), cos(a), fma(t_0, sin(b), (sin(b) * sin(a))))) / r);
}

Julia

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

↓

function code(r, a, b)
	t_0 = Float64(-sin(a))
	return Float64(sin(b) / Float64(fma(sin(b), t_0, fma(cos(b), cos(a), fma(t_0, sin(b), Float64(sin(b) * sin(a))))) / r))
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_] := Block[{t$95$0 = (-N[Sin[a], $MachinePrecision])}, N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Sin[b], $MachinePrecision] * t$95$0 + N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(t$95$0 * N[Sin[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 24.05

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

2. Simplified 24.13

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

   [Start] 24.05	\[ r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
   *-commutative [=>] 24.05	\[ \color{blue}{\frac{\sin b}{\cos \left(a + b\right)} \cdot r} \]
   associate-/r/ [<=] 24.13	\[ \color{blue}{\frac{\sin b}{\frac{\cos \left(a + b\right)}{r}}} \]
   +-commutative [=>] 24.13	\[ \frac{\sin b}{\frac{\cos \color{blue}{\left(b + a\right)}}{r}} \]

3. Applied egg-rr 0.6

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

4. Applied egg-rr 0.6

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

5. Simplified 0.59

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

   [Start] 0.6	\[ \frac{\sin b}{\frac{\cos b \cdot \cos a + \left(\sin b \cdot \left(-\sin a\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)\right)}{r}} \]
   associate-+r+ [=>] 0.59	\[ \frac{\sin b}{\frac{\color{blue}{\left(\cos b \cdot \cos a + \sin b \cdot \left(-\sin a\right)\right) + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)}}{r}} \]
   +-commutative [<=] 0.59	\[ \frac{\sin b}{\frac{\color{blue}{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right)} + \mathsf{fma}\left(-\sin a, \sin b, \sin b \cdot \sin a\right)}{r}} \]
   fma-udef [=>] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \color{blue}{\left(\left(-\sin a\right) \cdot \sin b + \sin b \cdot \sin a\right)}}{r}} \]
   *-commutative [<=] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\sin b \cdot \left(-\sin a\right)} + \sin b \cdot \sin a\right)}{r}} \]
   neg-mul-1 [=>] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\sin b \cdot \color{blue}{\left(-1 \cdot \sin a\right)} + \sin b \cdot \sin a\right)}{r}} \]
   associate-*r* [=>] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\left(\sin b \cdot -1\right) \cdot \sin a} + \sin b \cdot \sin a\right)}{r}} \]
   *-commutative [<=] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\left(-1 \cdot \sin b\right)} \cdot \sin a + \sin b \cdot \sin a\right)}{r}} \]
   neg-mul-1 [<=] 0.6	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \left(\color{blue}{\left(-\sin b\right)} \cdot \sin a + \sin b \cdot \sin a\right)}{r}} \]
   fma-udef [<=] 0.59	\[ \frac{\sin b}{\frac{\left(\sin b \cdot \left(-\sin a\right) + \cos b \cdot \cos a\right) + \color{blue}{\mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)}}{r}} \]
   associate-+r+ [<=] 0.58	\[ \frac{\sin b}{\frac{\color{blue}{\sin b \cdot \left(-\sin a\right) + \left(\cos b \cdot \cos a + \mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)\right)}}{r}} \]
   fma-def [=>] 0.59	\[ \frac{\sin b}{\frac{\color{blue}{\mathsf{fma}\left(\sin b, -\sin a, \cos b \cdot \cos a + \mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)\right)}}{r}} \]

6. Final simplification 0.59

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

Alternatives

Alternative 1
Error	0.52%
Cost	51840

\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos b, \cos a, -\mathsf{expm1}\left(\mathsf{log1p}\left(\sin b \cdot \sin a\right)\right)\right)} \]

Alternative 2
Error	0.5%
Cost	39040

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

Alternative 3
Error	0.51%
Cost	32704

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

Alternative 4
Error	0.6%
Cost	32704

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

Alternative 5
Error	22.96%
Cost	19648

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

Alternative 6
Error	23.05%
Cost	19648

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

Alternative 7
Error	24.09%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-5} \lor \neg \left(a \leq 0.0012\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \tan b\\ \end{array} \]

Alternative 8
Error	24.09%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.00145:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 9
Error	24.11%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;a \leq -0.00025:\\ \;\;\;\;r \cdot \left(\sin b \cdot \frac{1}{\cos a}\right)\\ \mathbf{elif}\;a \leq 0.001:\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \end{array} \]

Alternative 10
Error	24.05%
Cost	13248

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

Alternative 11
Error	24.06%
Cost	13248

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

Alternative 12
Error	24.28%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-5} \lor \neg \left(b \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 13
Error	39.94%
Cost	6592

\[r \cdot \tan b \]

Alternative 14
Error	65.99%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023115 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))