Average Error: 15.3 → 0.3
Time: 11.1s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[\frac{\sin b \cdot r}{\mathsf{fma}\left(\sin a, -\sin b, \cos a \cdot \cos b\right)} \]
(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

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.3

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

  2. Applied egg-rr0.3

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

  3. Taylor expanded in r around 0 0.3

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

  4. Taylor expanded in b around inf 0.3

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

  5. Simplified0.3

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

  6. Taylor expanded in a around inf 0.3

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

  7. Simplified0.3

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

  8. Final simplification0.3

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

Reproduce

herbie shell --seed 2022150 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))