ab-angle->ABCF A

Percentage Accurate: 78.9% → 78.8%

Time: 29.9s

Alternatives: 8

Speedup: N/A×

• 35.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t_0\right)}^{2} + {\left(b \cdot \cos t_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Initial Program: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t_0\right)}^{2} + {\left(b \cdot \cos t_0\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI)))
   (+ (pow (* a (sin t_0)) 2.0) (pow (* b (cos t_0)) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	return pow((a * sin(t_0)), 2.0) + pow((b * cos(t_0)), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (angle / 180.0) * Math.PI;
	return Math.pow((a * Math.sin(t_0)), 2.0) + Math.pow((b * Math.cos(t_0)), 2.0);
}

Python

def code(a, b, angle):
	t_0 = (angle / 180.0) * math.pi
	return math.pow((a * math.sin(t_0)), 2.0) + math.pow((b * math.cos(t_0)), 2.0)

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	return Float64((Float64(a * sin(t_0)) ^ 2.0) + (Float64(b * cos(t_0)) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = (angle / 180.0) * pi;
	tmp = ((a * sin(t_0)) ^ 2.0) + ((b * cos(t_0)) ^ 2.0);
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[(a * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[t$95$0], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {b}^{2} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (expm1 (log1p (* angle (* PI 0.005555555555555556)))))) 2.0)
  (pow b 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	return pow((a * sin(expm1(log1p((angle * (((double) M_PI) * 0.005555555555555556)))))), 2.0) + pow(b, 2.0);
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(Math.expm1(Math.log1p((angle * (Math.PI * 0.005555555555555556)))))), 2.0) + Math.pow(b, 2.0);
}

Python

angle = abs(angle)
def code(a, b, angle):
	return math.pow((a * math.sin(math.expm1(math.log1p((angle * (math.pi * 0.005555555555555556)))))), 2.0) + math.pow(b, 2.0)

Julia

angle = abs(angle)
function code(a, b, angle)
	return Float64((Float64(a * sin(expm1(log1p(Float64(angle * Float64(pi * 0.005555555555555556)))))) ^ 2.0) + (b ^ 2.0))
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(Exp[N[Log[1 + N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. associate-*l/ 79.9%

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

   2. associate-*r/ 80.0%

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

   3. associate-*l/ 79.8%

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

   4. associate-*r/ 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in angle around 0 80.3%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
5. Step-by-step derivation

   1. div-inv 80.3%

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

   2. metadata-eval 80.3%

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

   3. expm1-log1p-u 67.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 67.8%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Final simplification 67.8%

   \[\leadsto {\left(a \cdot \sin \left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} + {b}^{2} \]

Alternative 2: 78.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ {b}^{2} + {\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (+ (pow b 2.0) (pow (* a (sin (* 0.005555555555555556 (* angle PI)))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	return pow(b, 2.0) + pow((a * sin((0.005555555555555556 * (angle * ((double) M_PI))))), 2.0);
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return Math.pow(b, 2.0) + Math.pow((a * Math.sin((0.005555555555555556 * (angle * Math.PI)))), 2.0);
}

Python

angle = abs(angle)
def code(a, b, angle):
	return math.pow(b, 2.0) + math.pow((a * math.sin((0.005555555555555556 * (angle * math.pi)))), 2.0)

Julia

angle = abs(angle)
function code(a, b, angle)
	return Float64((b ^ 2.0) + (Float64(a * sin(Float64(0.005555555555555556 * Float64(angle * pi)))) ^ 2.0))
end

MATLAB

angle = abs(angle)
function tmp = code(a, b, angle)
	tmp = (b ^ 2.0) + ((a * sin((0.005555555555555556 * (angle * pi)))) ^ 2.0);
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(N[Power[b, 2.0], $MachinePrecision] + N[Power[N[(a * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. associate-*l/ 79.9%

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

   2. associate-*r/ 80.0%

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

   3. associate-*l/ 79.8%

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

   4. associate-*r/ 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in angle around 0 80.3%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
5. Step-by-step derivation

   1. associate-*r/ 80.2%

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

   2. expm1-log1p-u 80.2%

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

   3. associate-*r/ 80.3%

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

   4. div-inv 80.3%

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

   5. metadata-eval 80.3%

      \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 80.3%

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Taylor expanded in a around 0 80.3%

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

8. Final simplification 80.3%

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

Alternative 3: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 2.35 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 2.35e-33)
   (* b b)
   (+
    (pow b 2.0)
    (* 3.08641975308642e-5 (* PI (* (* angle (* a PI)) (* a angle)))))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.35e-33) {
		tmp = b * b;
	} else {
		tmp = pow(b, 2.0) + (3.08641975308642e-5 * (((double) M_PI) * ((angle * (a * ((double) M_PI))) * (a * angle))));
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 2.35e-33) {
		tmp = b * b;
	} else {
		tmp = Math.pow(b, 2.0) + (3.08641975308642e-5 * (Math.PI * ((angle * (a * Math.PI)) * (a * angle))));
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if a <= 2.35e-33:
		tmp = b * b
	else:
		tmp = math.pow(b, 2.0) + (3.08641975308642e-5 * (math.pi * ((angle * (a * math.pi)) * (a * angle))))
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 2.35e-33)
		tmp = Float64(b * b);
	else
		tmp = Float64((b ^ 2.0) + Float64(3.08641975308642e-5 * Float64(pi * Float64(Float64(angle * Float64(a * pi)) * Float64(a * angle)))));
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 2.35e-33)
		tmp = b * b;
	else
		tmp = (b ^ 2.0) + (3.08641975308642e-5 * (pi * ((angle * (a * pi)) * (a * angle))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 2.35e-33], N[(b * b), $MachinePrecision], N[(N[Power[b, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[(Pi * N[(N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision] * N[(a * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 2.3500000000000001e-33

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 77.7%

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

      4. associate-*r/ 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 78.2%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 78.2%

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

      2. metadata-eval 78.2%

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

      3. expm1-log1p-u 66.4%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 66.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 63.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 63.9%

       \[\leadsto \color{blue}{{b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 63.9%

       \[\leadsto \color{blue}{b \cdot b} \]

   if 2.3500000000000001e-33 < a

   1. Initial program 85.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 85.5%

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

      3. associate-*l/ 85.5%

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

      4. associate-*r/ 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in angle around 0 85.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 85.6%

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

      2. metadata-eval 85.6%

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

      3. expm1-log1p-u 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 72.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 83.5%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 83.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
   10. Step-by-step derivation

       1. unpow2 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\pi \cdot \left(angle \cdot a\right)\right) \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

       2. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\pi \cdot \left(angle \cdot a\right)\right) \cdot \color{blue}{\left(\left(angle \cdot a\right) \cdot \pi\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

       3. associate-*r* 83.6%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(\pi \cdot \left(angle \cdot a\right)\right) \cdot \left(angle \cdot a\right)\right) \cdot \pi\right)} + {\left(b \cdot 1\right)}^{2} \]

       4. *-commutative 83.6%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\pi \cdot \color{blue}{\left(a \cdot angle\right)}\right) \cdot \left(angle \cdot a\right)\right) \cdot \pi\right) + {\left(b \cdot 1\right)}^{2} \]

       5. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(\left(\pi \cdot a\right) \cdot angle\right)} \cdot \left(angle \cdot a\right)\right) \cdot \pi\right) + {\left(b \cdot 1\right)}^{2} \]

       6. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \color{blue}{\left(a \cdot angle\right)}\right) \cdot \pi\right) + {\left(b \cdot 1\right)}^{2} \]

   11. Applied egg-rr 83.5%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \pi\right)} + {\left(b \cdot 1\right)}^{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 2.35 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \left(\left(angle \cdot \left(a \cdot \pi\right)\right) \cdot \left(a \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.6e-33)
   (* b b)
   (fma 3.08641975308642e-5 (pow (* a (* angle PI)) 2.0) (* b b))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.6e-33) {
		tmp = b * b;
	} else {
		tmp = fma(3.08641975308642e-5, pow((a * (angle * ((double) M_PI))), 2.0), (b * b));
	}
	return tmp;
}

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.6e-33)
		tmp = Float64(b * b);
	else
		tmp = fma(3.08641975308642e-5, (Float64(a * Float64(angle * pi)) ^ 2.0), Float64(b * b));
	end
	return tmp
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.6e-33], N[(b * b), $MachinePrecision], N[(3.08641975308642e-5 * N[Power[N[(a * N[(angle * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.59999999999999988e-33

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 77.7%

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

      4. associate-*r/ 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 78.2%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 78.2%

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

      2. metadata-eval 78.2%

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

      3. expm1-log1p-u 66.4%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 66.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 63.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 63.9%

       \[\leadsto \color{blue}{{b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 63.9%

       \[\leadsto \color{blue}{b \cdot b} \]

   if 1.59999999999999988e-33 < a

   1. Initial program 85.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 85.5%

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

      3. associate-*l/ 85.5%

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

      4. associate-*r/ 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in angle around 0 85.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 85.6%

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

      2. metadata-eval 85.6%

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

      3. expm1-log1p-u 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 72.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 83.5%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 83.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 72.2%

       \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right) + {b}^{2}} \]
   11. Step-by-step derivation

       1. fma-def 72.2%

          \[\leadsto \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right), {b}^{2}\right)} \]

       2. associate-*r* 72.2%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left({a}^{2} \cdot {angle}^{2}\right) \cdot {\pi}^{2}}, {b}^{2}\right) \]

       3. unpow2 72.2%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(\color{blue}{\left(a \cdot a\right)} \cdot {angle}^{2}\right) \cdot {\pi}^{2}, {b}^{2}\right) \]

       4. unpow2 72.2%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(\left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot angle\right)}\right) \cdot {\pi}^{2}, {b}^{2}\right) \]

       5. swap-sqr 83.5%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right)} \cdot {\pi}^{2}, {b}^{2}\right) \]

       6. unpow2 83.5%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(\left(a \cdot angle\right) \cdot \left(a \cdot angle\right)\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}, {b}^{2}\right) \]

       7. swap-sqr 83.5%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right) \cdot \left(\left(a \cdot angle\right) \cdot \pi\right)}, {b}^{2}\right) \]

       8. *-commutative 83.5%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)} \cdot \left(\left(a \cdot angle\right) \cdot \pi\right), {b}^{2}\right) \]

       9. associate-*l* 83.5%

          \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{\left(\left(\pi \cdot a\right) \cdot angle\right)} \cdot \left(\left(a \cdot angle\right) \cdot \pi\right), {b}^{2}\right) \]

       10. *-commutative 83.5%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}, {b}^{2}\right) \]

       11. associate-*l* 83.4%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \left(\left(\pi \cdot a\right) \cdot angle\right) \cdot \color{blue}{\left(\left(\pi \cdot a\right) \cdot angle\right)}, {b}^{2}\right) \]

       12. unpow2 83.4%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, \color{blue}{{\left(\left(\pi \cdot a\right) \cdot angle\right)}^{2}}, {b}^{2}\right) \]

       13. associate-*l* 83.5%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2}, {b}^{2}\right) \]

       14. *-commutative 83.5%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2}, {b}^{2}\right) \]

       15. associate-*r* 83.4%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2}, {b}^{2}\right) \]

       16. unpow2 83.4%

           \[\leadsto \mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}, \color{blue}{b \cdot b}\right) \]

   12. Simplified 83.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(3.08641975308642 \cdot 10^{-5}, {\left(a \cdot \left(angle \cdot \pi\right)\right)}^{2}, b \cdot b\right)\\ \end{array} \]

Alternative 5: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.05e-33)
   (* b b)
   (fma b b (* 3.08641975308642e-5 (pow (* angle (* a PI)) 2.0)))))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.05e-33) {
		tmp = b * b;
	} else {
		tmp = fma(b, b, (3.08641975308642e-5 * pow((angle * (a * ((double) M_PI))), 2.0)));
	}
	return tmp;
}

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.05e-33)
		tmp = Float64(b * b);
	else
		tmp = fma(b, b, Float64(3.08641975308642e-5 * (Float64(angle * Float64(a * pi)) ^ 2.0)));
	end
	return tmp
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.05e-33], N[(b * b), $MachinePrecision], N[(b * b + N[(3.08641975308642e-5 * N[Power[N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.05e-33

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 77.7%

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

      4. associate-*r/ 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 78.2%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 78.2%

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

      2. metadata-eval 78.2%

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

      3. expm1-log1p-u 66.4%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 66.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 63.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 63.9%

       \[\leadsto \color{blue}{{b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 63.9%

       \[\leadsto \color{blue}{b \cdot b} \]

   if 1.05e-33 < a

   1. Initial program 85.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 85.5%

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

      3. associate-*l/ 85.5%

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

      4. associate-*r/ 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in angle around 0 85.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 85.6%

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

      2. metadata-eval 85.6%

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

      3. expm1-log1p-u 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 72.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 83.5%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 83.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
   10. Step-by-step derivation

       1. +-commutative 83.5%

          \[\leadsto \color{blue}{{\left(b \cdot 1\right)}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} \]

       2. *-rgt-identity 83.5%

          \[\leadsto {\color{blue}{b}}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \]

       3. pow2 83.5%

          \[\leadsto \color{blue}{b \cdot b} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \]

       4. fma-def 83.5%

          \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}\right)} \]

       5. *-commutative 83.5%

          \[\leadsto \mathsf{fma}\left(b, b, \color{blue}{{\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}}\right) \]

       6. *-commutative 83.5%

          \[\leadsto \mathsf{fma}\left(b, b, {\left(\pi \cdot \color{blue}{\left(a \cdot angle\right)}\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

       7. associate-*r* 83.5%

          \[\leadsto \mathsf{fma}\left(b, b, {\color{blue}{\left(\left(\pi \cdot a\right) \cdot angle\right)}}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right) \]

   11. Applied egg-rr 83.5%

       \[\leadsto \color{blue}{\mathsf{fma}\left(b, b, {\left(\left(\pi \cdot a\right) \cdot angle\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.05 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, b, 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2}\right)\\ \end{array} \]

Alternative 6: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.35e-33)
   (* b b)
   (pow (hypot b (* 0.005555555555555556 (* angle (* a PI)))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.35e-33) {
		tmp = b * b;
	} else {
		tmp = pow(hypot(b, (0.005555555555555556 * (angle * (a * ((double) M_PI))))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.35e-33) {
		tmp = b * b;
	} else {
		tmp = Math.pow(Math.hypot(b, (0.005555555555555556 * (angle * (a * Math.PI)))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if a <= 1.35e-33:
		tmp = b * b
	else:
		tmp = math.pow(math.hypot(b, (0.005555555555555556 * (angle * (a * math.pi)))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.35e-33)
		tmp = Float64(b * b);
	else
		tmp = hypot(b, Float64(0.005555555555555556 * Float64(angle * Float64(a * pi)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.35e-33)
		tmp = b * b;
	else
		tmp = hypot(b, (0.005555555555555556 * (angle * (a * pi)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1.35e-33], N[(b * b), $MachinePrecision], N[Power[N[Sqrt[b ^ 2 + N[(0.005555555555555556 * N[(angle * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.35e-33

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 77.7%

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

      4. associate-*r/ 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 78.2%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 78.2%

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

      2. metadata-eval 78.2%

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

      3. expm1-log1p-u 66.4%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 66.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 63.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 63.9%

       \[\leadsto \color{blue}{{b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 63.9%

       \[\leadsto \color{blue}{b \cdot b} \]

   if 1.35e-33 < a

   1. Initial program 85.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 85.5%

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

      3. associate-*l/ 85.5%

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

      4. associate-*r/ 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in angle around 0 85.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 85.6%

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

      2. metadata-eval 85.6%

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

      3. expm1-log1p-u 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 72.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 83.5%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 83.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 83.4%

          \[\leadsto \color{blue}{\sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}} \cdot \sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}}} \]

       2. pow2 83.4%

          \[\leadsto \color{blue}{{\left(\sqrt{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}}\right)}^{2}} \]

   11. Applied egg-rr 83.4%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot a\right) \cdot angle\right)\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.35 \cdot 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 7: 66.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ \begin{array}{l} \mathbf{if}\;a \leq 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle)
 :precision binary64
 (if (<= a 1e-33)
   (* b b)
   (pow (hypot b (* angle (* 0.005555555555555556 (* a PI)))) 2.0)))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e-33) {
		tmp = b * b;
	} else {
		tmp = pow(hypot(b, (angle * (0.005555555555555556 * (a * ((double) M_PI))))), 2.0);
	}
	return tmp;
}

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1e-33) {
		tmp = b * b;
	} else {
		tmp = Math.pow(Math.hypot(b, (angle * (0.005555555555555556 * (a * Math.PI)))), 2.0);
	}
	return tmp;
}

Python

angle = abs(angle)
def code(a, b, angle):
	tmp = 0
	if a <= 1e-33:
		tmp = b * b
	else:
		tmp = math.pow(math.hypot(b, (angle * (0.005555555555555556 * (a * math.pi)))), 2.0)
	return tmp

Julia

angle = abs(angle)
function code(a, b, angle)
	tmp = 0.0
	if (a <= 1e-33)
		tmp = Float64(b * b);
	else
		tmp = hypot(b, Float64(angle * Float64(0.005555555555555556 * Float64(a * pi)))) ^ 2.0;
	end
	return tmp
end

MATLAB

angle = abs(angle)
function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1e-33)
		tmp = b * b;
	else
		tmp = hypot(b, (angle * (0.005555555555555556 * (a * pi)))) ^ 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := If[LessEqual[a, 1e-33], N[(b * b), $MachinePrecision], N[Power[N[Sqrt[b ^ 2 + N[(angle * N[(0.005555555555555556 * N[(a * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision], 2.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.0000000000000001e-33

   1. Initial program 77.9%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 77.9%

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

      2. associate-*r/ 77.9%

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

      3. associate-*l/ 77.7%

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

      4. associate-*r/ 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in angle around 0 78.2%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 78.2%

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

      2. metadata-eval 78.2%

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

      3. expm1-log1p-u 66.4%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 66.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 63.0%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 63.0%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 72.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 72.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 72.4%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. Taylor expanded in angle around 0 63.9%

       \[\leadsto \color{blue}{{b}^{2}} \]
   11. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \color{blue}{b \cdot b} \]

   12. Simplified 63.9%

       \[\leadsto \color{blue}{b \cdot b} \]

   if 1.0000000000000001e-33 < a

   1. Initial program 85.5%

      \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
   2. Step-by-step derivation

      1. associate-*l/ 85.4%

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

      2. associate-*r/ 85.5%

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

      3. associate-*l/ 85.5%

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

      4. associate-*r/ 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in angle around 0 85.6%

      \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
   5. Step-by-step derivation

      1. div-inv 85.6%

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

      2. metadata-eval 85.6%

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

      3. expm1-log1p-u 71.7%

         \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   6. Applied egg-rr 71.7%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Taylor expanded in angle around 0 72.2%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-commutative 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      2. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      3. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      4. swap-sqr 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

      5. unpow2 72.2%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      6. swap-sqr 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      7. associate-*r* 83.5%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      8. associate-*r* 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

      9. unpow2 83.4%

         \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      10. associate-*r* 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      11. *-commutative 83.4%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      12. associate-*r* 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      13. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

      14. *-commutative 83.5%

          \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   9. Simplified 83.5%

      \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 81.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)\right)} \]

       2. expm1-udef 74.8%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} + {\left(b \cdot 1\right)}^{2}\right)} - 1} \]

   11. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot a\right) \cdot angle\right)\right)\right)}^{2}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 81.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot a\right) \cdot angle\right)\right)\right)}^{2}\right)\right)} \]

       2. expm1-log1p 83.4%

          \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b, 0.005555555555555556 \cdot \left(\left(\pi \cdot a\right) \cdot angle\right)\right)\right)}^{2}} \]

       3. associate-*r* 83.5%

          \[\leadsto {\left(\mathsf{hypot}\left(b, \color{blue}{\left(0.005555555555555556 \cdot \left(\pi \cdot a\right)\right) \cdot angle}\right)\right)}^{2} \]

       4. *-commutative 83.5%

          \[\leadsto {\left(\mathsf{hypot}\left(b, \left(0.005555555555555556 \cdot \color{blue}{\left(a \cdot \pi\right)}\right) \cdot angle\right)\right)}^{2} \]

   13. Simplified 83.5%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(b, \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right) \cdot angle\right)\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 10^{-33}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(b, angle \cdot \left(0.005555555555555556 \cdot \left(a \cdot \pi\right)\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 8: 56.0% accurate, 205.0× speedup

Math

\[\begin{array}{l} angle = |angle|\\ \\ b \cdot b \end{array} \]

FPCore

NOTE: angle should be positive before calling this function
(FPCore (a b angle) :precision binary64 (* b b))

C

angle = abs(angle);
double code(double a, double b, double angle) {
	return b * b;
}

Fortran

NOTE: angle should be positive before calling this function
real(8) function code(a, b, angle)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = b * b
end function

Java

angle = Math.abs(angle);
public static double code(double a, double b, double angle) {
	return b * b;
}

Python

angle = abs(angle)
def code(a, b, angle):
	return b * b

Julia

angle = abs(angle)
function code(a, b, angle)
	return Float64(b * b)
end

MATLAB

angle = abs(angle)
function tmp = code(a, b, angle)
	tmp = b * b;
end

Wolfram

NOTE: angle should be positive before calling this function
code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 80.0%

   \[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
2. Step-by-step derivation

   1. associate-*l/ 79.9%

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

   2. associate-*r/ 80.0%

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

   3. associate-*l/ 79.8%

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

   4. associate-*r/ 79.9%

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

3. Simplified 79.9%

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

4. Taylor expanded in angle around 0 80.3%

   \[\leadsto {\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{1}\right)}^{2} \]
5. Step-by-step derivation

   1. div-inv 80.3%

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

   2. metadata-eval 80.3%

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

   3. expm1-log1p-u 67.8%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Applied egg-rr 67.8%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

7. Taylor expanded in angle around 0 65.5%

   \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} + {\left(b \cdot 1\right)}^{2} \]
8. Step-by-step derivation

   1. *-commutative 65.5%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left({angle}^{2} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

   2. unpow2 65.5%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\color{blue}{\left(angle \cdot angle\right)} \cdot {\pi}^{2}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

   3. unpow2 65.5%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot angle\right) \cdot \color{blue}{\left(\pi \cdot \pi\right)}\right) \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

   4. swap-sqr 65.5%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right)} \cdot {a}^{2}\right) + {\left(b \cdot 1\right)}^{2} \]

   5. unpow2 65.5%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(\left(angle \cdot \pi\right) \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   6. swap-sqr 75.4%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{\left(\left(\left(angle \cdot \pi\right) \cdot a\right) \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

   7. associate-*r* 75.4%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)} \cdot \left(\left(angle \cdot \pi\right) \cdot a\right)\right) + {\left(b \cdot 1\right)}^{2} \]

   8. associate-*r* 75.4%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \left(\left(angle \cdot \left(\pi \cdot a\right)\right) \cdot \color{blue}{\left(angle \cdot \left(\pi \cdot a\right)\right)}\right) + {\left(b \cdot 1\right)}^{2} \]

   9. unpow2 75.4%

      \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot \color{blue}{{\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

   10. associate-*r* 75.4%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(angle \cdot \pi\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   11. *-commutative 75.4%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(a \cdot \left(angle \cdot \pi\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   12. associate-*r* 75.4%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\left(a \cdot angle\right) \cdot \pi\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   13. *-commutative 75.4%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\color{blue}{\left(\pi \cdot \left(a \cdot angle\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

   14. *-commutative 75.4%

       \[\leadsto 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \color{blue}{\left(angle \cdot a\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

9. Simplified 75.4%

   \[\leadsto \color{blue}{3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

10. Taylor expanded in angle around 0 62.2%

    \[\leadsto \color{blue}{{b}^{2}} \]
11. Step-by-step derivation

    1. unpow2 62.2%

       \[\leadsto \color{blue}{b \cdot b} \]

12. Simplified 62.2%

    \[\leadsto \color{blue}{b \cdot b} \]

13. Final simplification 62.2%

    \[\leadsto b \cdot b \]

Reproduce

herbie shell --seed 2023279 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))