?

Average Error: 14.9 → 0.3
Time: 16.0s
Precision: binary64
Cost: 39040

?

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

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

code[r_, a_, b_] := N[(r * N[(N[Sin[b], $MachinePrecision] / N[Cos[N[(a + b), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[r_, a_, b_] := N[(r * 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]), $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 14.9

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

  2. Simplified14.9

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

    [Start]14.9

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

    +-commutative [=>]14.9

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

  3. Applied egg-rr0.3

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

  4. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a - \sin b \cdot \sin a} \]
Alternative 2
Error14.2
Cost19648
\[r \cdot \frac{\sin b}{\cos b \cdot \cos a} \]
Alternative 3
Error15.0
Cost13385
\[\begin{array}{l} \mathbf{if}\;a \leq -0.185 \lor \neg \left(a \leq 0.4\right):\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \end{array} \]
Alternative 4
Error15.0
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.185:\\ \;\;\;\;\frac{\sin b}{\frac{\cos a}{r}}\\ \mathbf{elif}\;a \leq 0.4:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 5
Error15.0
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -0.185:\\ \;\;\;\;\frac{r \cdot \sin b}{\cos a}\\ \mathbf{elif}\;a \leq 0.42:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 6
Error15.0
Cost13376
\[\sin b \cdot \left(r \cdot \frac{1}{\cos \left(b - a\right)}\right) \]
Alternative 7
Error14.9
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b + a\right)} \]
Alternative 8
Error14.9
Cost13248
\[\sin b \cdot \frac{r}{\cos \left(b + a\right)} \]
Alternative 9
Error15.0
Cost13248
\[\frac{r}{\frac{\cos \left(b - a\right)}{\sin b}} \]
Alternative 10
Error15.0
Cost13248
\[\frac{r \cdot \sin b}{\cos \left(b - a\right)} \]
Alternative 11
Error28.7
Cost13120
\[r \cdot \frac{\sin b}{\cos a} \]
Alternative 12
Error30.9
Cost7232
\[\frac{\frac{r}{\cos \left(b + a\right)}}{b \cdot 0.16666666666666666 + \frac{1}{b}} \]
Alternative 13
Error31.5
Cost6848
\[\frac{r \cdot b}{\cos \left(b - a\right)} \]
Alternative 14
Error31.5
Cost6720
\[r \cdot \frac{b}{\cos a} \]
Alternative 15
Error41.8
Cost192
\[r \cdot b \]

Error

Reproduce?

herbie shell --seed 2023046 
(FPCore (r a b)
  :name "rsin B (should all be same)"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))