Average Error: 14.8 → 0.3
Time: 15.5s
Precision: binary64
Cost: 39040
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\sin b \cdot \frac{r}{\mathsf{fma}\left(-\sin a, \sin b, \cos a \cdot \cos b\right)} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (* (sin b) (/ r (fma (- (sin a)) (sin b) (* (cos a) (cos 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(-sin(a), sin(b), (cos(a) * cos(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(Float64(-sin(a)), sin(b), Float64(cos(a) * cos(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[Sin[a], $MachinePrecision]) * N[Sin[b], $MachinePrecision] + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 14.8

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

  2. Applied egg-rr14.9

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

  3. Applied egg-rr0.4

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

  4. Taylor expanded in r around 0 0.4

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

  5. Simplified0.4

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

  6. Applied egg-rr0.3

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

  7. Final simplification0.3

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

Alternatives

Alternative 1
Error0.3
Cost32704
\[\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 2
Error0.4
Cost32704
\[\sin b \cdot \frac{r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error15.7
Cost13384
\[\begin{array}{l} t_0 := \sin b \cdot \frac{r}{\cos a}\\ \mathbf{if}\;a \leq -142667178061.19138:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 4.337008163156099 \cdot 10^{-31}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error15.7
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -142667178061.19138:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 4.337008163156099 \cdot 10^{-31}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 5
Error15.7
Cost13384
\[\begin{array}{l} \mathbf{if}\;a \leq -142667178061.19138:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos a}\\ \mathbf{elif}\;a \leq 4.337008163156099 \cdot 10^{-31}:\\ \;\;\;\;\sin b \cdot \frac{r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;r \cdot \frac{\sin b}{\cos a}\\ \end{array} \]
Alternative 6
Error14.8
Cost13248
\[\frac{\sin b}{\frac{\cos \left(b + a\right)}{r}} \]
Alternative 7
Error14.9
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(b - a\right)} \]
Alternative 8
Error28.4
Cost13120
\[\sin b \cdot \frac{r}{\cos a} \]
Alternative 9
Error28.3
Cost7112
\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -122.22465916837005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2740453461538332:\\ \;\;\;\;b \cdot \left(r \cdot \frac{1}{\cos a}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error28.3
Cost6984
\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -122.22465916837005:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3.2740453461538332:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error39.0
Cost6592
\[\sin b \cdot r \]
Alternative 12
Error41.9
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022217 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))