?

Average Error: 24.06% → 0.52%
Time: 16.3s
Precision: binary64
Cost: 39040

?

\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\sin b \cdot \frac{r}{\mathsf{fma}\left(\cos b, \cos a, \sin a \cdot \left(-\sin b\right)\right)} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (sin b) (/ r (fma (cos b) (cos a) (* (sin a) (- (sin b)))))))
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) * (r / fma(cos(b), cos(a), (sin(a) * -sin(b))));
}

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(r / fma(cos(b), cos(a), Float64(sin(a) * Float64(-sin(b))))))
end

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[(r / N[(N[Cos[b], $MachinePrecision] * N[Cos[a], $MachinePrecision] + N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 24.06

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

  2. Simplified24.13

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

    [Start]24.06

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

    associate-/l* [=>]24.13

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

    +-commutative [=>]24.13

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

  3. Applied egg-rr24.06

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

  4. Applied egg-rr0.52

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

  5. Final simplification0.52

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

Alternatives

Alternative 1
Error0.54%
Cost32704
\[\sin b \cdot \frac{r}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error0.59%
Cost32512
\[\frac{r}{\mathsf{fma}\left(\frac{\cos b}{\sin b}, \cos a, -\sin a\right)} \]
Alternative 3
Error22.97%
Cost19648
\[\sin b \cdot \frac{r}{\cos b \cdot \cos a} \]
Alternative 4
Error24.36%
Cost13385
\[\begin{array}{l} \mathbf{if}\;b \leq -4.8 \cdot 10^{-6} \lor \neg \left(b \leq 3.2 \cdot 10^{-8}\right):\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 5
Error24.06%
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 6
Error46.75%
Cost7113
\[\begin{array}{l} \mathbf{if}\;b \leq -0.98 \lor \neg \left(b \leq 0.000106\right):\\ \;\;\;\;\frac{r}{\frac{1}{b} - \sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 7
Error46.94%
Cost6985
\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \lor \neg \left(b \leq 75\right):\\ \;\;\;\;\frac{-r}{\sin a}\\ \mathbf{else}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \end{array} \]
Alternative 8
Error65.08%
Cost576
\[\frac{r}{\frac{1}{b} + b \cdot -0.3333333333333333} \]
Alternative 9
Error65.99%
Cost192
\[r \cdot b \]

Error

Reproduce?

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