rsin A (should all be same)

Average Error: 14.3 → 0.3

Time: 15.6s

Precision: binary64

Cost: 39040

Math

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

↓

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

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(Float64(r * sin(b)) / cos(Float64(a + b)))
end

↓

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

Derivation

1. Initial program 14.3

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

2. Applied egg-rr 0.3

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

3. Simplified 0.3

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

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

4. Final simplification 0.3

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

Alternatives

Alternative 1
Error	0.3
Cost	32704

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

Alternative 2
Error	0.4
Cost	32512

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

Alternative 3
Error	14.4
Cost	13248

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

Alternative 4
Error	14.3
Cost	6985

\[\begin{array}{l} \mathbf{if}\;b \leq -9.2 \cdot 10^{-5} \lor \neg \left(b \leq 2 \cdot 10^{-9}\right):\\ \;\;\;\;r \cdot \tan b\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]

Alternative 5
Error	38.7
Cost	6592

\[r \cdot \sin b \]

Alternative 6
Error	24.6
Cost	6592

\[r \cdot \tan b \]

Alternative 7
Error	41.5
Cost	192

\[r \cdot b \]

Reproduce

herbie shell --seed 2023040 
(FPCore (r a b)
  :name "rsin A (should all be same)"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))