Average Error: 15.1 → 0.4
Time: 10.6s
Precision: binary64
\[\frac{r \cdot \sin b}{\cos \left(a + b\right)} \]
↓
\[r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sqrt[3]{\sqrt[3]{{\left({\left(\sin b \cdot \sin a\right)}^{3}\right)}^{3}}}\right)} \]
(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)
    (- (cbrt (cbrt (pow (pow (* (sin b) (sin a)) 3.0) 3.0))))))))
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), -cbrt(cbrt(pow(pow((sin(b) * sin(a)), 3.0), 3.0)))));
}

function code(r, a, b)
	return Float64(Float64(r * sin(b)) / cos(Float64(a + b)))
end
↓
function code(r, a, b)
	return Float64(r * Float64(sin(b) / fma(cos(a), cos(b), Float64(-cbrt(cbrt(((Float64(sin(b) * sin(a)) ^ 3.0) ^ 3.0)))))))
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[(r * N[(N[Sin[b], $MachinePrecision] / N[(N[Cos[a], $MachinePrecision] * N[Cos[b], $MachinePrecision] + (-N[Power[N[Power[N[Power[N[Power[N[(N[Sin[b], $MachinePrecision] * N[Sin[a], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{r \cdot \sin b}{\cos \left(a + b\right)}
↓
r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sqrt[3]{\sqrt[3]{{\left({\left(\sin b \cdot \sin a\right)}^{3}\right)}^{3}}}\right)}

Error

Derivation

  1. Initial program 15.1

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

  2. Simplified15.0

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

  3. Applied egg-rr0.3

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

  4. Applied egg-rr0.4

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\color{blue}{\sqrt[3]{{\left(\sin b \cdot \sin a\right)}^{3}}}\right)} \]

  5. Applied egg-rr0.4

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sqrt[3]{\color{blue}{\sqrt[3]{{\left({\left(\sin b \cdot \sin a\right)}^{3}\right)}^{3}}}}\right)} \]

  6. Final simplification0.4

    \[\leadsto r \cdot \frac{\sin b}{\mathsf{fma}\left(\cos a, \cos b, -\sqrt[3]{\sqrt[3]{{\left({\left(\sin b \cdot \sin a\right)}^{3}\right)}^{3}}}\right)} \]

Reproduce

herbie shell --seed 2022210 
(FPCore (r a b)
  :name "rsin A"
  :precision binary64
  (/ (* r (sin b)) (cos (+ a b))))