ab-angle->ABCF A

Percentage Accurate: 79.4% → 79.2%

Time: 16.2s

Alternatives: 14

Speedup: N/A×

• 35.4% 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: 79.4% 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: 79.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \sin \left(\frac{1}{\frac{1}{angle}} \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + b \cdot b \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ 1.0 (/ 1.0 angle)) (* PI 0.005555555555555556)))) 2.0)
  (* b b)))

C

double code(double a, double b, double angle) {
	return pow((a * sin(((1.0 / (1.0 / angle)) * (((double) M_PI) * 0.005555555555555556)))), 2.0) + (b * b);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((1.0 / (1.0 / angle)) * (Math.PI * 0.005555555555555556)))), 2.0) + (b * b);
}

Python

def code(a, b, angle):
	return math.pow((a * math.sin(((1.0 / (1.0 / angle)) * (math.pi * 0.005555555555555556)))), 2.0) + (b * b)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((1.0 / (1.0 / angle)) * (pi * 0.005555555555555556)))) ^ 2.0) + (b * b);
end

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in angle around 0

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

5. Simplified 75.6%

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

   1. clear-num N/A

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

   2. associate-*l/ N/A

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

   3. div-inv N/A

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

   4. *-commutative N/A

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

   5. times-frac N/A

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

   6. div-inv N/A

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

   7. metadata-eval N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 N/A

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

   11. /-lowering-/.f64 N/A

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

   12. /-lowering-/.f64 75.6

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

7. Applied egg-rr 75.6%

   \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \frac{1}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
8. Step-by-step derivation

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 75.6

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

9. Applied egg-rr 75.6%

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

10. Final simplification 75.6%

    \[\leadsto {\left(a \cdot \sin \left(\frac{1}{\frac{1}{angle}} \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} + b \cdot b \]
11. Add Preprocessing

Alternative 2: 79.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in angle around 0

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

5. Simplified 75.6%

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

   1. *-rgt-identity N/A

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

   2. pow2 N/A

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

   3. *-lowering-*.f64 75.6

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

7. Applied egg-rr 75.6%

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

8. Final simplification 75.6%

   \[\leadsto b \cdot b + {\left(a \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
9. Add Preprocessing

Alternative 3: 76.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.003:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 0.5\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.003)
   (+ (* b b) (pow (* a (* angle (* PI 0.005555555555555556))) 2.0))
   (fma
    a
    (* a (* (- 1.0 (cos (* 0.011111111111111112 (* PI angle)))) 0.5))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.003) {
		tmp = (b * b) + pow((a * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma(a, (a * ((1.0 - cos((0.011111111111111112 * (((double) M_PI) * angle)))) * 0.5)), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.003)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(a, Float64(a * Float64(Float64(1.0 - cos(Float64(0.011111111111111112 * Float64(pi * angle)))) * 0.5)), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.003], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(a * N[(a * N[(N[(1.0 - N[Cos[N[(0.011111111111111112 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.0030000000000000001

   1. Initial program 83.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 83.3%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 83.4

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

   7. Applied egg-rr 83.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \frac{1}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 83.4

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

   9. Applied egg-rr 83.4%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 80.6

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

   12. Simplified 80.6%

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

   if 0.0030000000000000001 < angle

   1. Initial program 54.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 55.4%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. pow2 N/A

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

      5. sin-mult N/A

         \[\leadsto {\left(a \cdot \sqrt{\color{blue}{\frac{\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) - \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) + \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}{2}}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      6. sqrt-div N/A

         \[\leadsto {\left(a \cdot \color{blue}{\frac{\sqrt{\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) - \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) + \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}}{\sqrt{2}}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      7. /-lowering-/.f64 N/A

         \[\leadsto {\left(a \cdot \color{blue}{\frac{\sqrt{\cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) - \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right) - \cos \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right) + \frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)}}{\sqrt{2}}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

   7. Applied egg-rr 55.3%

      \[\leadsto {\left(a \cdot \color{blue}{\frac{\sqrt{1 - \cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right)}}{\sqrt{2}}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. unpow-prod-down N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      2. pow2 N/A

         \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(a \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2}\right)} + {\left(b \cdot 1\right)}^{2} \]

      4. *-rgt-identity N/A

         \[\leadsto a \cdot \left(a \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2}\right) + {\color{blue}{b}}^{2} \]

      5. pow2 N/A

         \[\leadsto a \cdot \left(a \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2}\right) + \color{blue}{b \cdot b} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot {\left(\frac{\sqrt{1 - \cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{90} \cdot angle\right)\right)}}{\sqrt{2}}\right)}^{2}, b \cdot b\right)} \]

   9. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right) \cdot 0.5\right), b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.003:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, a \cdot \left(\left(1 - \cos \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right) \cdot 0.5\right), b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.00044:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a \cdot a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.00044)
   (+ (* b b) (pow (* a (* angle (* PI 0.005555555555555556))) 2.0))
   (fma
    (fma (cos (* PI (* angle 0.011111111111111112))) -0.5 0.5)
    (* a a)
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.00044) {
		tmp = (b * b) + pow((a * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma(fma(cos((((double) M_PI) * (angle * 0.011111111111111112))), -0.5, 0.5), (a * a), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.00044)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(fma(cos(Float64(pi * Float64(angle * 0.011111111111111112))), -0.5, 0.5), Float64(a * a), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.00044], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[(a * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 4.40000000000000016e-4

   1. Initial program 83.3%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 83.4%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 83.5

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

   7. Applied egg-rr 83.5%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \frac{1}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 83.5

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

   9. Applied egg-rr 83.5%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 80.7

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

   12. Simplified 80.7%

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

   if 4.40000000000000016e-4 < angle

   1. Initial program 54.8%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 55.6%

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

      1. pow-prod-down N/A

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

      2. pow2 N/A

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

      3. sqr-sin-a N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{angle}{180} \cdot \mathsf{PI}\left(\right)\right)\right)\right)} + {\left(b \cdot 1\right)}^{2} \]

      4. div-inv N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{180}\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      5. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \color{blue}{\frac{1}{180}}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) + {\left(b \cdot 1\right)}^{2} \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot {a}^{2}} + {\left(b \cdot 1\right)}^{2} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(angle \cdot \frac{1}{180}\right) \cdot \mathsf{PI}\left(\right)\right)\right), {a}^{2}, {\left(b \cdot 1\right)}^{2}\right)} \]

   7. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right), a \cdot a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.00044:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a \cdot a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;angle \leq 0.003:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= angle 0.003)
   (+ (* b b) (pow (* a (* angle (* PI 0.005555555555555556))) 2.0))
   (fma
    (* a (fma (cos (* PI (* angle 0.011111111111111112))) -0.5 0.5))
    a
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (angle <= 0.003) {
		tmp = (b * b) + pow((a * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	} else {
		tmp = fma((a * fma(cos((((double) M_PI) * (angle * 0.011111111111111112))), -0.5, 0.5)), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (angle <= 0.003)
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	else
		tmp = fma(Float64(a * fma(cos(Float64(pi * Float64(angle * 0.011111111111111112))), -0.5, 0.5)), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[angle, 0.003], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a * N[(N[Cos[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if angle < 0.0030000000000000001

   1. Initial program 83.4%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 83.3%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 83.4

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

   7. Applied egg-rr 83.4%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \frac{1}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 83.4

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

   9. Applied egg-rr 83.4%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 80.6

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

   12. Simplified 80.6%

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

   if 0.0030000000000000001 < angle

   1. Initial program 54.1%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 55.4%

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

   7. Applied egg-rr 55.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\cos \left(\pi \cdot \left(0.011111111111111112 \cdot angle\right)\right), -0.5, 0.5\right) \cdot a, a, b \cdot b\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;angle \leq 0.003:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a \cdot \mathsf{fma}\left(\cos \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right), -0.5, 0.5\right), a, b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.32e+20)
   (* b b)
   (+ (* b b) (pow (* a (* angle (* PI 0.005555555555555556))) 2.0))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.32e+20) {
		tmp = b * b;
	} else {
		tmp = (b * b) + pow((a * (angle * (((double) M_PI) * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.32e+20) {
		tmp = b * b;
	} else {
		tmp = (b * b) + Math.pow((a * (angle * (Math.PI * 0.005555555555555556))), 2.0);
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.32e+20:
		tmp = b * b
	else:
		tmp = (b * b) + math.pow((a * (angle * (math.pi * 0.005555555555555556))), 2.0)
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.32e+20)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(b * b) + (Float64(a * Float64(angle * Float64(pi * 0.005555555555555556))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.32e+20)
		tmp = b * b;
	else
		tmp = (b * b) + ((a * (angle * (pi * 0.005555555555555556))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.32e+20], N[(b * b), $MachinePrecision], N[(N[(b * b), $MachinePrecision] + N[Power[N[(a * N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.32e20

   1. Initial program 70.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.6

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

   5. Simplified 57.6%

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

   if 1.32e20 < a

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.9%

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

      1. clear-num N/A

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

      2. associate-*l/ N/A

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

      3. div-inv N/A

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

      4. *-commutative N/A

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

      5. times-frac N/A

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

      6. div-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lowering-*.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. PI-lowering-PI.f64 N/A

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

      11. /-lowering-/.f64 N/A

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

      12. /-lowering-/.f64 87.1

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

   7. Applied egg-rr 87.1%

      \[\leadsto {\left(a \cdot \sin \color{blue}{\left(\left(\pi \cdot 0.005555555555555556\right) \cdot \frac{1}{\frac{1}{angle}}\right)}\right)}^{2} + {\left(b \cdot 1\right)}^{2} \]
   8. Step-by-step derivation

      1. *-rgt-identity N/A

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

      2. pow2 N/A

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

      3. *-lowering-*.f64 87.1

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

   9. Applied egg-rr 87.1%

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

   10. Taylor expanded in angle around 0

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

       1. *-commutative N/A

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

       2. associate-*r* N/A

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

       3. *-commutative N/A

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

       4. *-lowering-*.f64 N/A

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

       5. *-lowering-*.f64 N/A

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

       6. PI-lowering-PI.f64 84.1

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

   12. Simplified 84.1%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.32 \cdot 10^{+20}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;b \cdot b + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.3% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 8 \cdot 10^{+18}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\right), a, b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 8e+18)
   (* b b)
   (fma
    (* (* a angle) (* (* angle 3.08641975308642e-5) (* PI PI)))
    a
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 8e+18) {
		tmp = b * b;
	} else {
		tmp = fma(((a * angle) * ((angle * 3.08641975308642e-5) * (((double) M_PI) * ((double) M_PI)))), a, (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 8e+18)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(Float64(a * angle) * Float64(Float64(angle * 3.08641975308642e-5) * Float64(pi * pi))), a, Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 8e+18], N[(b * b), $MachinePrecision], N[(N[(N[(a * angle), $MachinePrecision] * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 8e18

   1. Initial program 70.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.6

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

   5. Simplified 57.6%

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

   if 8e18 < a

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} + b \cdot b \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot a} + b \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot \left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), a, b \cdot b\right)} \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{fma}\left(a \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)}, a, b \cdot b\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, a, b \cdot b\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)}, a, b \cdot b\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(a \cdot angle\right)} \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), a, b \cdot b\right) \]

      8. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, a, b \cdot b\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}, a, b \cdot b\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\color{blue}{\left(angle \cdot \frac{1}{32400}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right), a, b \cdot b\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right), a, b \cdot b\right) \]

      12. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right), a, b \cdot b\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\left(a \cdot angle\right) \cdot \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right), a, b \cdot b\right) \]

      14. *-lowering-*.f64 80.2

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

   10. Applied egg-rr 80.2%

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

Alternative 8: 64.8% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.02 \cdot 10^{+19}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot \left(\pi \cdot 3.08641975308642 \cdot 10^{-5}\right), a \cdot \left(a \cdot \left(angle \cdot angle\right)\right), b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.02e+19)
   (* b b)
   (fma
    (* PI (* PI 3.08641975308642e-5))
    (* a (* a (* angle angle)))
    (* b b))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.02e+19) {
		tmp = b * b;
	} else {
		tmp = fma((((double) M_PI) * (((double) M_PI) * 3.08641975308642e-5)), (a * (a * (angle * angle))), (b * b));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.02e+19)
		tmp = Float64(b * b);
	else
		tmp = fma(Float64(pi * Float64(pi * 3.08641975308642e-5)), Float64(a * Float64(a * Float64(angle * angle))), Float64(b * b));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.02e+19], N[(b * b), $MachinePrecision], N[(N[(Pi * N[(Pi * 3.08641975308642e-5), $MachinePrecision]), $MachinePrecision] * N[(a * N[(a * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.02e19

   1. Initial program 70.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 57.6

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

   5. Simplified 57.6%

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

   if 1.02e19 < a

   1. Initial program 86.9%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 86.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 57.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} + b \cdot b \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot angle\right)\right)} + b \cdot b \]

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right)} \]

      4. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)}, \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}, \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right)}, \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \left(\frac{1}{32400} \cdot \mathsf{PI}\left(\right)\right), \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}, \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right)}, \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \frac{1}{32400}\right), \left(a \cdot a\right) \cdot \left(angle \cdot angle\right), b \cdot b\right) \]

      11. associate-*l* N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), \color{blue}{a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)}, b \cdot b\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), \color{blue}{a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)}, b \cdot b\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), a \cdot \color{blue}{\left(a \cdot \left(angle \cdot angle\right)\right)}, b \cdot b\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \frac{1}{32400}\right), a \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right), b \cdot b\right) \]

      15. *-lowering-*.f64 73.9

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

   10. Applied egg-rr 73.9%

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

Alternative 9: 70.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(angle, \left(a \cdot a\right) \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.8e+174)
   (fma angle (* (* a a) (* (* angle 3.08641975308642e-5) (* PI PI))) (* b b))
   (* a (* (* a angle) (* 3.08641975308642e-5 (* angle (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.8e+174) {
		tmp = fma(angle, ((a * a) * ((angle * 3.08641975308642e-5) * (((double) M_PI) * ((double) M_PI)))), (b * b));
	} else {
		tmp = a * ((a * angle) * (3.08641975308642e-5 * (angle * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.8e+174)
		tmp = fma(angle, Float64(Float64(a * a) * Float64(Float64(angle * 3.08641975308642e-5) * Float64(pi * pi))), Float64(b * b));
	else
		tmp = Float64(a * Float64(Float64(a * angle) * Float64(3.08641975308642e-5 * Float64(angle * Float64(pi * pi)))));
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.8e+174], N[(angle * N[(N[(a * a), $MachinePrecision] * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision], N[(a * N[(N[(a * angle), $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.8000000000000001e174

   1. Initial program 71.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 71.9%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 56.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a \cdot a, \left(angle \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(angle \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(a \cdot a\right)} + b \cdot b \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot \left(a \cdot a\right) + b \cdot b \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{angle \cdot \left(\left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(a \cdot a\right)\right)} + b \cdot b \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(angle, \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(a \cdot a\right), b \cdot b\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle, \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right) \cdot \left(a \cdot a\right)}, b \cdot b\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(angle, \color{blue}{\left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(a \cdot a\right), b \cdot b\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle, \color{blue}{\left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \left(a \cdot a\right), b \cdot b\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle, \left(\color{blue}{\left(angle \cdot \frac{1}{32400}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right), b \cdot b\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(angle, \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \left(a \cdot a\right), b \cdot b\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(angle, \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(a \cdot a\right), b \cdot b\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{fma}\left(angle, \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(a \cdot a\right), b \cdot b\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(angle, \left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\left(a \cdot a\right)}, b \cdot b\right) \]

      13. *-lowering-*.f64 63.9

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

   10. Applied egg-rr 63.9%

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

   if 1.8000000000000001e174 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.6%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 51.0%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       17. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 51.0

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

   11. Simplified 51.0%

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

       1. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot a} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot a \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot a \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(a \cdot angle\right)} \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot a \]

       7. *-commutative N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}\right) \cdot a \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}\right) \cdot a \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}\right) \cdot a \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)\right) \cdot a \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right)\right) \cdot a \]

       12. PI-lowering-PI.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\right) \cdot a \]

       13. PI-lowering-PI.f64 72.0

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

   13. Applied egg-rr 72.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.8 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(angle, \left(a \cdot a\right) \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\right), b \cdot b\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.9% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+181)
   (* b b)
   (* a (* (* a angle) (* 3.08641975308642e-5 (* angle (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+181) {
		tmp = b * b;
	} else {
		tmp = a * ((a * angle) * (3.08641975308642e-5 * (angle * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+181) {
		tmp = b * b;
	} else {
		tmp = a * ((a * angle) * (3.08641975308642e-5 * (angle * (Math.PI * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.5e+181:
		tmp = b * b
	else:
		tmp = a * ((a * angle) * (3.08641975308642e-5 * (angle * (math.pi * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.5e+181)
		tmp = Float64(b * b);
	else
		tmp = Float64(a * Float64(Float64(a * angle) * Float64(3.08641975308642e-5 * Float64(angle * Float64(pi * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.5e+181)
		tmp = b * b;
	else
		tmp = a * ((a * angle) * (3.08641975308642e-5 * (angle * (pi * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.5e+181], N[(b * b), $MachinePrecision], N[(a * N[(N[(a * angle), $MachinePrecision] * N[(3.08641975308642e-5 * N[(angle * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.50000000000000006e181

   1. Initial program 71.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 56.3

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

   5. Simplified 56.3%

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

   if 1.50000000000000006e181 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.6%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 52.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       17. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 52.6

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

   11. Simplified 52.6%

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

       1. associate-*l* N/A

          \[\leadsto \color{blue}{a \cdot \left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot a} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) \cdot a} \]

       4. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot a \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(a \cdot angle\right) \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)} \cdot a \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \left(\color{blue}{\left(a \cdot angle\right)} \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot a \]

       7. *-commutative N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot angle\right)}\right) \cdot a \]

       8. associate-*l* N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}\right) \cdot a \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \left(\left(a \cdot angle\right) \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)}\right) \cdot a \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)}\right)\right) \cdot a \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot angle\right)\right)\right) \cdot a \]

       12. PI-lowering-PI.f64 N/A

           \[\leadsto \left(\left(a \cdot angle\right) \cdot \left(\frac{1}{32400} \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot angle\right)\right)\right) \cdot a \]

       13. PI-lowering-PI.f64 74.0

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

   13. Applied egg-rr 74.0%

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

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;a \cdot \left(\left(a \cdot angle\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.8% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.5e+181)
   (* b b)
   (* (* a (* a (* angle angle))) (* 3.08641975308642e-5 (* PI PI)))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+181) {
		tmp = b * b;
	} else {
		tmp = (a * (a * (angle * angle))) * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.5e+181) {
		tmp = b * b;
	} else {
		tmp = (a * (a * (angle * angle))) * (3.08641975308642e-5 * (Math.PI * Math.PI));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.5e+181:
		tmp = b * b
	else:
		tmp = (a * (a * (angle * angle))) * (3.08641975308642e-5 * (math.pi * math.pi))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.5e+181)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * Float64(a * Float64(angle * angle))) * Float64(3.08641975308642e-5 * Float64(pi * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.5e+181)
		tmp = b * b;
	else
		tmp = (a * (a * (angle * angle))) * (3.08641975308642e-5 * (pi * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.5e+181], N[(b * b), $MachinePrecision], N[(N[(a * N[(a * N[(angle * angle), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.50000000000000006e181

   1. Initial program 71.7%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 56.3

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

   5. Simplified 56.3%

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

   if 1.50000000000000006e181 < a

   1. Initial program 99.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 99.6%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 52.6%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       17. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 52.6

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

   11. Simplified 52.6%

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

   12. Taylor expanded in a around 0

       \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{\left(\left({a}^{2} \cdot {angle}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)} \cdot \frac{1}{32400} \]

       3. associate-*l* N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(\color{blue}{\left(a \cdot a\right)} \cdot {angle}^{2}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       6. associate-*l* N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot {angle}^{2}\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot \left(a \cdot {angle}^{2}\right)\right)} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot \color{blue}{\left(a \cdot {angle}^{2}\right)}\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       9. unpow2 N/A

          \[\leadsto \left(a \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot \left(a \cdot \color{blue}{\left(angle \cdot angle\right)}\right)\right) \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot \frac{1}{32400}\right)} \]

       12. unpow2 N/A

           \[\leadsto \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right) \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{32400}\right) \]

       14. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{32400}\right) \]

       15. PI-lowering-PI.f64 60.8

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

   14. Simplified 60.8%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.5 \cdot 10^{+181}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(a \cdot \left(angle \cdot angle\right)\right)\right) \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+115}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(angle \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.25e+115)
   (* b b)
   (* (* a a) (* angle (* (* angle 3.08641975308642e-5) (* PI PI))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e+115) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (angle * ((angle * 3.08641975308642e-5) * (((double) M_PI) * ((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.25e+115) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (angle * ((angle * 3.08641975308642e-5) * (Math.PI * Math.PI)));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.25e+115:
		tmp = b * b
	else:
		tmp = (a * a) * (angle * ((angle * 3.08641975308642e-5) * (math.pi * math.pi)))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.25e+115)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * a) * Float64(angle * Float64(Float64(angle * 3.08641975308642e-5) * Float64(pi * pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.25e+115)
		tmp = b * b;
	else
		tmp = (a * a) * (angle * ((angle * 3.08641975308642e-5) * (pi * pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.25e+115], N[(b * b), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(angle * N[(N[(angle * 3.08641975308642e-5), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.25000000000000002e115

   1. Initial program 70.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 56.9

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

   5. Simplified 56.9%

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

   if 1.25000000000000002e115 < a

   1. Initial program 94.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 94.6%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       17. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 53.3

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

   11. Simplified 53.3%

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

       1. associate-*r* N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(\left(angle \cdot \frac{1}{32400}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

       2. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \frac{1}{32400}\right)\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \frac{1}{32400}\right)\right)}\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(angle \cdot \frac{1}{32400}\right)\right)\right) \]

       5. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \left(angle \cdot \frac{1}{32400}\right)\right)\right) \]

       6. PI-lowering-PI.f64 N/A

          \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot \left(angle \cdot \frac{1}{32400}\right)\right)\right) \]

       7. *-lowering-*.f64 53.4

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

   13. Applied egg-rr 53.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.25 \cdot 10^{+115}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(angle \cdot \left(\left(angle \cdot 3.08641975308642 \cdot 10^{-5}\right) \cdot \left(\pi \cdot \pi\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 1.22 \cdot 10^{+117}:\\ \;\;\;\;b \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(3.08641975308642 \cdot 10^{-5} \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (if (<= a 1.22e+117)
   (* b b)
   (* (* a a) (* angle (* angle (* 3.08641975308642e-5 (* PI PI)))))))

C

double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.22e+117) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (angle * (angle * (3.08641975308642e-5 * (((double) M_PI) * ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double tmp;
	if (a <= 1.22e+117) {
		tmp = b * b;
	} else {
		tmp = (a * a) * (angle * (angle * (3.08641975308642e-5 * (Math.PI * Math.PI))));
	}
	return tmp;
}

Python

def code(a, b, angle):
	tmp = 0
	if a <= 1.22e+117:
		tmp = b * b
	else:
		tmp = (a * a) * (angle * (angle * (3.08641975308642e-5 * (math.pi * math.pi))))
	return tmp

Julia

function code(a, b, angle)
	tmp = 0.0
	if (a <= 1.22e+117)
		tmp = Float64(b * b);
	else
		tmp = Float64(Float64(a * a) * Float64(angle * Float64(angle * Float64(3.08641975308642e-5 * Float64(pi * pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	tmp = 0.0;
	if (a <= 1.22e+117)
		tmp = b * b;
	else
		tmp = (a * a) * (angle * (angle * (3.08641975308642e-5 * (pi * pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := If[LessEqual[a, 1.22e+117], N[(b * b), $MachinePrecision], N[(N[(a * a), $MachinePrecision] * N[(angle * N[(angle * N[(3.08641975308642e-5 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 1.22000000000000004e117

   1. Initial program 70.8%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 56.9

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

   5. Simplified 56.9%

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

   if 1.22000000000000004e117 < a

   1. Initial program 94.6%

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

   3. Taylor expanded in angle around 0

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

   5. Simplified 94.6%

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

   6. Taylor expanded in angle around 0

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) + {b}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} + {b}^{2} \]

      2. associate-*l* N/A

         \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} + {b}^{2} \]

      3. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + {b}^{2} \]

      4. *-commutative N/A

         \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) + {b}^{2} \]

      5. associate-*r* N/A

         \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} + {b}^{2} \]

      6. metadata-eval N/A

         \[\leadsto {a}^{2} \cdot \left(\left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400}\right)\right)} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) + {b}^{2} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto {a}^{2} \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot {angle}^{2}\right) + {b}^{2} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left({a}^{2}, \left(\mathsf{neg}\left(\frac{-1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot {angle}^{2}, {b}^{2}\right)} \]

   8. Simplified 55.4%

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

   9. Taylor expanded in a around inf

      \[\leadsto \color{blue}{\frac{1}{32400} \cdot \left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\left({a}^{2} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \frac{1}{32400}} \]

       2. associate-*r* N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \frac{1}{32400}\right)} \]

       3. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\frac{1}{32400} \cdot \left({angle}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       4. *-commutative N/A

          \[\leadsto {a}^{2} \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{2} \cdot {angle}^{2}\right)}\right) \]

       5. associate-*r* N/A

          \[\leadsto {a}^{2} \cdot \color{blue}{\left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{{a}^{2} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right)} \]

       7. unpow2 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \color{blue}{\left(a \cdot a\right)} \cdot \left(\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot {angle}^{2}\right) \]

       9. *-commutative N/A

          \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left({angle}^{2} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]

       10. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(\color{blue}{\left(angle \cdot angle\right)} \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \]

       11. associate-*l* N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \color{blue}{\left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right)} \]

       13. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \color{blue}{\left(angle \cdot \left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}\right) \]

       14. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \color{blue}{\left(\frac{1}{32400} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right)\right) \]

       15. unpow2 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \]

       17. PI-lowering-PI.f64 N/A

           \[\leadsto \left(a \cdot a\right) \cdot \left(angle \cdot \left(angle \cdot \left(\frac{1}{32400} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

       18. PI-lowering-PI.f64 53.3

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

   11. Simplified 53.3%

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

Alternative 14: 57.5% accurate, 74.7× speedup

Math

\[\begin{array}{l} \\ b \cdot b \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (* b b))

C

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

Fortran

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

public static double code(double a, double b, double angle) {
	return b * b;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, angle_] := N[(b * b), $MachinePrecision]

Derivation

1. Initial program 75.3%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 52.9

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

5. Simplified 52.9%

   \[\leadsto \color{blue}{b \cdot b} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024197 
(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)))