Average Error: 15.5 → 0.3
Time: 9.4s
Precision: binary64
\[r \cdot \frac{\sin b}{\cos \left(a + b\right)} \]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, \sin b \cdot \left(-\sin a\right)\right) + \mathsf{fma}\left(-\sin b, \sin a, \sin b \cdot \sin a\right)} \]
(FPCore (r a b) :precision binary64 (* r (/ (sin b) (cos (+ a b)))))
↓
(FPCore (r a b)
 :precision binary64
 (*
  r
  (/
   (sin b)
   (+
    (fma (cos a) (cos b) (* (sin b) (- (sin a))))
    (fma (- (sin b)) (sin 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(a), cos(b), (sin(b) * -sin(a))) + fma(-sin(b), sin(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) / Float64(fma(cos(a), cos(b), Float64(sin(b) * Float64(-sin(a)))) + fma(Float64(-sin(b)), sin(a), Float64(sin(b) * 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[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] + N[((-N[Sin[b], $MachinePrecision]) * N[Sin[a], $MachinePrecision] + N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

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

Error

Bits error versus r

Bits error versus a

Bits error versus b

Derivation

  1. Initial program 15.5

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

  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 a around inf 0.3

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2022162 
(FPCore (r a b)
  :name "rsin B"
  :precision binary64
  (* r (/ (sin b) (cos (+ a b)))))