rsin A (should all be same)

Average Error: 15.3 → 0.4

Time: 15.5s

Precision: binary64

Cost: 39040

Math

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

↓

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

FPCore

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

↓

(FPCore (r a b)
 :precision binary64
 (/ (sin b) (/ (fma (cos a) (cos 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) {
	return sin(b) / (fma(cos(a), cos(b), (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(sin(b) / Float64(fma(cos(a), cos(b), 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[Sin[b], $MachinePrecision] / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / r), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.3

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

2. Applied egg-rr 0.3

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

3. Taylor expanded in r around 0 0.3

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

4. Simplified 0.4

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

   [Start] 0.3	\[ \frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin a \cdot \sin b} \]
   associate-/l* [=>] 0.4	\[ \color{blue}{\frac{\sin b}{\frac{\cos a \cdot \cos b - \sin a \cdot \sin b}{r}}} \]
   *-commutative [<=] 0.4	\[ \frac{\sin b}{\frac{\cos a \cdot \cos b - \color{blue}{\sin b \cdot \sin a}}{r}} \]
   fma-neg [=>] 0.4	\[ \frac{\sin b}{\frac{\color{blue}{\mathsf{fma}\left(\cos a, \cos b, -\sin b \cdot \sin a\right)}}{r}} \]
   distribute-rgt-neg-in [=>] 0.4	\[ \frac{\sin b}{\frac{\mathsf{fma}\left(\cos a, \cos b, \color{blue}{\sin b \cdot \left(-\sin a\right)}\right)}{r}} \]

5. Final simplification 0.4

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

Alternatives

Alternative 1
Error	0.4
Cost	32832

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

Alternative 2
Error	0.4
Cost	32704

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

Alternative 3
Error	0.4
Cost	32704

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

Alternative 4
Error	0.4
Cost	32512

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

Alternative 5
Error	15.8
Cost	13385

\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 8000000000\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]

Alternative 6
Error	15.3
Cost	13248

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

Alternative 7
Error	15.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 53\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{b}{\cos a}\\ \end{array} \]

Alternative 8
Error	15.6
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -1200 \lor \neg \left(b \leq 53\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 9
Error	39.3
Cost	6592

\[\sin b \cdot r \]

Alternative 10
Error	25.6
Cost	6592

\[r \cdot \tan b \]

Alternative 11
Error	42.2
Cost	192

\[b \cdot r \]

Reproduce

herbie shell --seed 2023059 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))