Average Error: 15.6 → 0.3
Time: 14.8s
Precision: binary64
Cost: 39040
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b \cdot r}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \]
(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(r * Float64(sin(b) / cos(Float64(a + b))))
end
↓
function code(r, a, b)
	return Float64(Float64(sin(b) * r) / fma(sin(a), Float64(-sin(b)), Float64(cos(a) * cos(b))))
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[(N[(N[Sin[b], $MachinePrecision] * r), $MachinePrecision] / N[(N[Sin[a], $MachinePrecision] * (-N[Sin[b], $MachinePrecision]) + N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Derivation

  1. Initial program 15.6

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

  2. Applied egg-rr0.3

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

  3. Taylor expanded in r around 0 0.3

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

  4. Applied egg-rr0.3

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

  5. Final simplification0.3

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

Alternatives

Alternative 1
Error0.4
Cost32704
\[\frac{r}{\frac{\cos a \cdot \cos b - \sin b \cdot \sin a}{\sin b}} \]
Alternative 2
Error0.3
Cost32704
\[\frac{\sin b \cdot r}{\cos a \cdot \cos b - \sin b \cdot \sin a} \]
Alternative 3
Error15.5
Cost13512
\[\begin{array}{l} t_0 := \frac{r}{\frac{\cos a}{b} - \sin a}\\ \mathbf{if}\;a \leq -124.45381542025724:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 3.300500232441911 \cdot 10^{-33}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 4
Error16.0
Cost13384
\[\begin{array}{l} t_0 := \frac{\sin b \cdot r}{\cos b}\\ \mathbf{if}\;b \leq -176672074.5232325:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 1.1492680206531809 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 5
Error16.0
Cost13384
\[\begin{array}{l} \mathbf{if}\;b \leq -176672074.5232325:\\ \;\;\;\;\frac{r}{\frac{\cos b}{\sin b}}\\ \mathbf{elif}\;b \leq 1.1492680206531809 \cdot 10^{-11}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin b \cdot r}{\cos b}\\ \end{array} \]
Alternative 6
Error16.4
Cost13384
\[\begin{array}{l} t_0 := \sin b \cdot r\\ t_1 := \frac{t_0}{\cos a}\\ \mathbf{if}\;a \leq -124.45381542025724:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 8.659823856620301 \cdot 10^{-39}:\\ \;\;\;\;\frac{t_0}{\cos b}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error15.7
Cost13376
\[\left(\sin b \cdot r\right) \cdot \frac{1}{\cos \left(a - b\right)} \]
Alternative 8
Error15.6
Cost13248
\[r \cdot \frac{\sin b}{\cos \left(a - b\right)} \]
Alternative 9
Error29.5
Cost6984
\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -176672074.5232325:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.405554412989323 \cdot 10^{-6}:\\ \;\;\;\;b \cdot \frac{r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 10
Error29.4
Cost6984
\[\begin{array}{l} t_0 := \sin b \cdot r\\ \mathbf{if}\;b \leq -176672074.5232325:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.405554412989323 \cdot 10^{-6}:\\ \;\;\;\;\frac{b \cdot r}{\cos a}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error40.0
Cost6592
\[\sin b \cdot r \]
Alternative 12
Error42.3
Cost576
\[\frac{r}{b \cdot -0.3333333333333333 + \frac{1}{b}} \]
Alternative 13
Error42.9
Cost192
\[b \cdot r \]

Error

Reproduce

herbie shell --seed 2022228 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))