Average Error: 20.3 → 20.3
Time: 14.8s
Precision: binary64
\[{\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
↓
\[{a}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(\pi \cdot \frac{1}{\sqrt{180}}\right) \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)\right)}^{2} \]
(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 2.0)
  (pow
   (*
    b
    (log1p
     (expm1 (sin (* (* PI (/ 1.0 (sqrt 180.0))) (/ angle (sqrt 180.0)))))))
   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, 2.0) + pow((b * log1p(expm1(sin(((((double) M_PI) * (1.0 / sqrt(180.0))) * (angle / sqrt(180.0))))))), 2.0);
}

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}
↓
public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + Math.pow((b * Math.log1p(Math.expm1(Math.sin(((Math.PI * (1.0 / Math.sqrt(180.0))) * (angle / Math.sqrt(180.0))))))), 2.0);
}

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.sin((math.pi * (angle / 180.0)))), 2.0)
↓
def code(a, b, angle):
	return math.pow(a, 2.0) + math.pow((b * math.log1p(math.expm1(math.sin(((math.pi * (1.0 / math.sqrt(180.0))) * (angle / math.sqrt(180.0))))))), 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((a ^ 2.0) + (Float64(b * log1p(expm1(sin(Float64(Float64(pi * Float64(1.0 / sqrt(180.0))) * Float64(angle / sqrt(180.0))))))) ^ 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[a, 2.0], $MachinePrecision] + N[Power[N[(b * N[Log[1 + N[(Exp[N[Sin[N[(N[(Pi * N[(1.0 / N[Sqrt[180.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(angle / N[Sqrt[180.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $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}
↓
{a}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(\pi \cdot \frac{1}{\sqrt{180}}\right) \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)\right)}^{2}

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 20.3

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

  2. Taylor expanded in angle around 0 20.3

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

  3. Applied add-sqr-sqrt_binary6420.5

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

  4. Applied *-un-lft-identity_binary6420.5

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

  5. Applied times-frac_binary6420.4

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

  6. Applied associate-*r*_binary6420.3

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

  7. Applied log1p-expm1-u_binary6420.3

    \[\leadsto {\left(a \cdot 1\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(\pi \cdot \frac{1}{\sqrt{180}}\right) \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)}\right)}^{2} \]

  8. Final simplification20.3

    \[\leadsto {a}^{2} + {\left(b \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(\left(\pi \cdot \frac{1}{\sqrt{180}}\right) \cdot \frac{angle}{\sqrt{180}}\right)\right)\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022129 
(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)))