Average Error: 20.5 → 20.6
Time: 19.4s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0\right)\right)\right)}^{2} \]
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a 1.0) 2.0)
  (pow (* b (sin (fma (* PI angle) 0.005555555555555556 0.0))) 2.0)))
double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow((a * 1.0), 2.0) + pow((b * sin(fma((((double) M_PI) * angle), 0.005555555555555556, 0.0))), 2.0);
}

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end
↓
function code(a, b, angle)
	return Float64((Float64(a * 1.0) ^ 2.0) + (Float64(b * sin(fma(Float64(pi * angle), 0.005555555555555556, 0.0))) ^ 2.0))
end

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, angle_] := N[(N[Power[N[(a * 1.0), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556 + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2}
↓
{\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0\right)\right)\right)}^{2}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Derivation

  1. Initial program 20.5

    \[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

  2. Taylor expanded in angle around 0 20.6

    \[\leadsto {\left(a \cdot \color{blue}{1}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

  3. Applied egg-rr20.7

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\frac{\pi \cdot angle}{180}\right)}\right)}^{2} \]

  4. Applied egg-rr20.6

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0\right)\right)}\right)}^{2} \]

  5. Final simplification20.6

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{fma}\left(\pi \cdot angle, 0.005555555555555556, 0\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022131 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))