rsin A (should all be same)

Average Error: 23.54% → 0.49%

Time: 15.7s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

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

Julia

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

↓

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

Derivation

1. Initial program 23.54

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

2. Applied egg-rr 0.52

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

3. Simplified 0.51

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

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

4. Taylor expanded in r around 0 0.52

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

5. Simplified 0.5

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

   [Start] 0.52	\[ \frac{\sin b \cdot r}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \]
   associate-/l* [=>] 0.61	\[ \color{blue}{\frac{\sin b}{\frac{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b}{r}}} \]
   associate-/r/ [=>] 0.5	\[ \color{blue}{\frac{\sin b}{-1 \cdot \left(\sin a \cdot \sin b\right) + \cos a \cdot \cos b} \cdot r} \]
   +-commutative [=>] 0.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b + -1 \cdot \left(\sin a \cdot \sin b\right)}} \cdot r \]
   *-commutative [<=] 0.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a} + -1 \cdot \left(\sin a \cdot \sin b\right)} \cdot r \]
   *-commutative [<=] 0.5	\[ \frac{\sin b}{\cos b \cdot \cos a + -1 \cdot \color{blue}{\left(\sin b \cdot \sin a\right)}} \cdot r \]
   mul-1-neg [=>] 0.5	\[ \frac{\sin b}{\cos b \cdot \cos a + \color{blue}{\left(-\sin b \cdot \sin a\right)}} \cdot r \]
   unsub-neg [=>] 0.5	\[ \frac{\sin b}{\color{blue}{\cos b \cdot \cos a - \sin b \cdot \sin a}} \cdot r \]
   *-commutative [=>] 0.5	\[ \frac{\sin b}{\color{blue}{\cos a \cdot \cos b} - \sin b \cdot \sin a} \cdot r \]

6. Applied egg-rr 0.49

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

7. Final simplification 0.49

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

Alternatives

Alternative 1
Error	0.5%
Cost	32704

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

Alternative 2
Error	0.57%
Cost	32512

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

Alternative 3
Error	23.7%
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095 \lor \neg \left(b \leq 2.9 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 4
Error	23.69%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 5
Error	23.7%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\frac{\sin b}{\frac{\cos b}{r}}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 6
Error	23.7%
Cost	13384

\[\begin{array}{l} \mathbf{if}\;b \leq -0.0095:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{elif}\;b \leq 2.9 \cdot 10^{-8}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \end{array} \]

Alternative 7
Error	23.52%
Cost	13248

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

Alternative 8
Error	23.66%
Cost	13248

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

Alternative 9
Error	44.97%
Cost	13120

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

Alternative 10
Error	44.74%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;b \leq -35 \lor \neg \left(b \leq 3.3\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;\frac{b \cdot r}{\cos \left(b + a\right)}\\ \end{array} \]

Alternative 11
Error	44.74%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.7 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 12
Error	44.76%
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1.75 \lor \neg \left(b \leq 4.8\right):\\ \;\;\;\;\sin b \cdot r\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 13
Error	61.21%
Cost	6592

\[\sin b \cdot r \]

Alternative 14
Error	65.5%
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023121 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))