Average Error: 15.0 → 0.4
Time: 12.1s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[\begin{array}{l} t_0 := \sin b \cdot \sin a\\ r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, t_0\right) + \mathsf{fma}\left(t_0, -1, \sin b \cdot \left(-\sin a\right)\right)} \end{array} \]
(FPCore (r a b) :precision binary64 (/ (* r (sin b)) (cos (+ a b))))
↓
(FPCore (r a b)
 :precision binary64
 (let* ((t_0 (* (sin b) (sin a))))
   (*
    r
    (/
     (sin b)
     (+ (fma (cos a) (cos b) t_0) (fma t_0 -1.0 (* (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) {
	double t_0 = sin(b) * sin(a);
	return r * (sin(b) / (fma(cos(a), cos(b), t_0) + fma(t_0, -1.0, (sin(b) * -sin(a)))));
}

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	t_0 = Float64(sin(b) * sin(a))
	return Float64(r * Float64(sin(b) / Float64(fma(cos(a), cos(b), t_0) + fma(t_0, -1.0, Float64(sin(b) * Float64(-sin(a)))))))
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_] := Block[{t$95$0 = N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision]}, N[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + t$95$0), $MachinePrecision] + N[(t$95$0 * -1.0 + N[(N[Sin[b], $MachinePrecision] * (-N[Sin[a], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
\begin{array}{l}
t_0 := \sin b \cdot \sin a\\
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, t_0\right) + \mathsf{fma}\left(t_0, -1, \sin b \cdot \left(-\sin a\right)\right)}
\end{array}

Error

Derivation

  1. Initial program 15.0

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

  2. Simplified15.0

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

  3. Applied egg-rr0.3

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

  4. Applied egg-rr0.4

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

  5. Final simplification0.4

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

Reproduce

herbie shell --seed 2022209 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))