ab-angle->ABCF C

Percentage Accurate: 79.7% → 79.7%

Time: 1.0min

Alternatives: 12

Speedup: 1.5×

• 37.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt{\pi}}\\ {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\left(t_0 \cdot t_0\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (cbrt (sqrt PI))))
   (+
    (pow
     (*
      a
      (cos
       (* (pow (cbrt PI) 2.0) (* (* t_0 t_0) (* angle 0.005555555555555556)))))
     2.0)
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = cbrt(sqrt(((double) M_PI)));
	return pow((a * cos((pow(cbrt(((double) M_PI)), 2.0) * ((t_0 * t_0) * (angle * 0.005555555555555556))))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = Math.cbrt(Math.sqrt(Math.PI));
	return Math.pow((a * Math.cos((Math.pow(Math.cbrt(Math.PI), 2.0) * ((t_0 * t_0) * (angle * 0.005555555555555556))))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = cbrt(sqrt(pi))
	return Float64((Float64(a * cos(Float64((cbrt(pi) ^ 2.0) * Float64(Float64(t_0 * t_0) * Float64(angle * 0.005555555555555556))))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[Power[N[Sqrt[Pi], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[(a * N[Cos[N[(N[Power[N[Power[Pi, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[(t$95$0 * t$95$0), $MachinePrecision] * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.3%

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

   1. associate-*r/ 82.4%

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

   2. clear-num 82.4%

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

4. Applied egg-rr 82.4%

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

   1. clear-num 82.4%

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

   2. associate-*r/ 82.3%

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

   3. add-cube-cbrt 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \sqrt[3]{\pi}\right)} \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   4. associate-*l* 82.5%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\left(\sqrt[3]{\pi} \cdot \sqrt[3]{\pi}\right) \cdot \left(\sqrt[3]{\pi} \cdot \frac{angle}{180}\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. pow2 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\sqrt[3]{\pi} \cdot \frac{angle}{180}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. div-inv 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. metadata-eval 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 82.5%

   \[\leadsto {\left(a \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\sqrt[3]{\pi} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
7. Step-by-step derivation

   1. pow1/3 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{{\pi}^{0.3333333333333333}} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. add-sqr-sqrt 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left({\color{blue}{\left(\sqrt{\pi} \cdot \sqrt{\pi}\right)}}^{0.3333333333333333} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   3. unpow-prod-down 82.3%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Applied egg-rr 82.3%

   \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\left({\left(\sqrt{\pi}\right)}^{0.3333333333333333} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right)} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
9. Step-by-step derivation

   1. unpow1/3 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\left(\color{blue}{\sqrt[3]{\sqrt{\pi}}} \cdot {\left(\sqrt{\pi}\right)}^{0.3333333333333333}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   2. unpow1/3 82.5%

      \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \color{blue}{\sqrt[3]{\sqrt{\pi}}}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

10. Simplified 82.5%

    \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\color{blue}{\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right)} \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

11. Final simplification 82.5%

    \[\leadsto {\left(a \cdot \cos \left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\left(\sqrt[3]{\sqrt{\pi}} \cdot \sqrt[3]{\sqrt{\pi}}\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
12. Add Preprocessing

Alternative 2: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\\ {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \left(\cos t_0 \cdot \cos -1 - \sin t_0 \cdot \sin -1\right)\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (fma PI (* angle 0.005555555555555556) 1.0)))
   (+
    (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)
    (pow (* a (- (* (cos t_0) (cos -1.0)) (* (sin t_0) (sin -1.0)))) 2.0))))

C

double code(double a, double b, double angle) {
	double t_0 = fma(((double) M_PI), (angle * 0.005555555555555556), 1.0);
	return pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((a * ((cos(t_0) * cos(-1.0)) - (sin(t_0) * sin(-1.0)))), 2.0);
}

Julia

function code(a, b, angle)
	t_0 = fma(pi, Float64(angle * 0.005555555555555556), 1.0)
	return Float64((Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(a * Float64(Float64(cos(t_0) * cos(-1.0)) - Float64(sin(t_0) * sin(-1.0)))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision] + 1.0), $MachinePrecision]}, N[(N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[(N[(N[Cos[t$95$0], $MachinePrecision] * N[Cos[-1.0], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 82.3%

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

   1. associate-*r/ 82.4%

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

   2. clear-num 82.4%

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

4. Applied egg-rr 82.4%

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

   1. clear-num 82.4%

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

   2. associate-*r/ 82.3%

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

   3. div-inv 82.3%

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

   4. metadata-eval 82.3%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   5. expm1-log1p-u 65.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   6. expm1-udef 65.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} - 1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. sub-neg 65.9%

      \[\leadsto {\left(a \cdot \cos \color{blue}{\left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} + \left(-1\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   8. metadata-eval 65.9%

      \[\leadsto {\left(a \cdot \cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)} + \color{blue}{-1}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   9. cos-sum 65.9%

      \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \cos -1 - \sin \left(e^{\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right) \cdot \sin -1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

6. Applied egg-rr 82.4%

   \[\leadsto {\left(a \cdot \color{blue}{\left(\cos \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right) \cdot \cos -1 - \sin \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right) \cdot \sin -1\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Final simplification 82.4%

   \[\leadsto {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(a \cdot \left(\cos \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right) \cdot \cos -1 - \sin \left(\mathsf{fma}\left(\pi, angle \cdot 0.005555555555555556, 1\right)\right) \cdot \sin -1\right)\right)}^{2} \]
8. Add Preprocessing

Alternative 3: 79.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (expm1 (log1p (sin (* PI (* angle 0.005555555555555556)))))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((a * cos((((double) M_PI) * (angle / 180.0)))), 2.0) + pow((b * expm1(log1p(sin((((double) M_PI) * (angle * 0.005555555555555556)))))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.cos((Math.PI * (angle / 180.0)))), 2.0) + Math.pow((b * Math.expm1(Math.log1p(Math.sin((Math.PI * (angle * 0.005555555555555556)))))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((a * math.cos((math.pi * (angle / 180.0)))), 2.0) + math.pow((b * math.expm1(math.log1p(math.sin((math.pi * (angle * 0.005555555555555556)))))), 2.0)

Julia

function code(a, b, angle)
	return Float64((Float64(a * cos(Float64(pi * Float64(angle / 180.0)))) ^ 2.0) + (Float64(b * expm1(log1p(sin(Float64(pi * Float64(angle * 0.005555555555555556)))))) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Cos[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. expm1-log1p-u 82.3%

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

   2. div-inv 82.5%

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

   3. metadata-eval 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} \]

4. Applied egg-rr 82.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]

5. Final simplification 82.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)}^{2} \]
6. Add Preprocessing

Alternative 4: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(a \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (sin (* angle (/ PI -180.0)))) 2.0)
  (pow (* a (cos (* -0.005555555555555556 (* PI angle)))) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((b * sin((angle * (((double) M_PI) / -180.0)))), 2.0) + pow((a * cos((-0.005555555555555556 * (((double) M_PI) * angle)))), 2.0);
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow((b * Math.sin((angle * (Math.PI / -180.0)))), 2.0) + Math.pow((a * Math.cos((-0.005555555555555556 * (Math.PI * angle)))), 2.0);
}

Python

def code(a, b, angle):
	return math.pow((b * math.sin((angle * (math.pi / -180.0)))), 2.0) + math.pow((a * math.cos((-0.005555555555555556 * (math.pi * angle)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((b * sin((angle * (pi / -180.0)))) ^ 2.0) + ((a * cos((-0.005555555555555556 * (pi * angle)))) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Simplified 82.4%

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

5. Taylor expanded in angle around inf 82.4%

   \[\leadsto {\left(a \cdot \color{blue}{\cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi}{-180} \cdot angle\right)\right)}^{2} \]

6. Final simplification 82.4%

   \[\leadsto {\left(b \cdot \sin \left(angle \cdot \frac{\pi}{-180}\right)\right)}^{2} + {\left(a \cdot \cos \left(-0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)}^{2} \]
7. Add Preprocessing

Alternative 5: 79.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Simplified 82.4%

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

5. Final simplification 82.4%

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

Alternative 6: 79.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* b (expm1 (log1p (sin (* PI (* angle 0.005555555555555556)))))) 2.0)
  (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(b * N[(Exp[N[Log[1 + N[Sin[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

   1. expm1-log1p-u 82.3%

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

   2. div-inv 82.5%

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

   3. metadata-eval 82.5%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot \color{blue}{0.005555555555555556}\right)\right)\right)\right)\right)}^{2} \]

4. Applied egg-rr 82.5%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right)}^{2} \]

5. Taylor expanded in angle around 0 82.3%

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

6. Final simplification 82.3%

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

Alternative 7: 79.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

4. Taylor expanded in b around 0 82.2%

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

5. Final simplification 82.2%

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

Alternative 8: 79.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

   1. clear-num 82.2%

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

   2. un-div-inv 82.2%

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

5. Applied egg-rr 82.2%

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

6. Final simplification 82.2%

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

Alternative 9: 74.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

4. Taylor expanded in angle around 0 78.0%

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

   1. *-commutative 78.0%

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

6. Simplified 78.0%

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

   1. unpow2 78.0%

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

   2. associate-*r* 78.0%

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

   3. associate-*r* 78.0%

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

   4. *-commutative 78.0%

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

   5. associate-*l* 78.0%

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

   6. *-commutative 78.0%

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

   7. associate-*l* 78.0%

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

8. Applied egg-rr 78.0%

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

9. Final simplification 78.0%

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

Alternative 10: 74.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

4. Taylor expanded in angle around 0 78.0%

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

   1. *-commutative 78.0%

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

6. Simplified 78.0%

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

   1. unpow2 78.0%

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

   2. *-commutative 78.0%

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

   3. associate-*r* 78.0%

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

   4. associate-*r* 78.0%

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

   5. *-commutative 78.0%

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

   6. associate-*l* 78.0%

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

   7. *-commutative 78.0%

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

   8. associate-*l* 78.1%

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

8. Applied egg-rr 78.1%

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

9. Final simplification 78.1%

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

Alternative 11: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* angle (* PI b)) 2.0))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (3.08641975308642e-5 * pow((angle * (((double) M_PI) * b)), 2.0));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((angle * (Math.PI * b)), 2.0));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((angle * (math.pi * b)), 2.0))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(angle * Float64(pi * b)) ^ 2.0)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((angle * (pi * b)) ^ 2.0));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(angle * N[(Pi * b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

4. Taylor expanded in angle around 0 78.0%

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

   1. *-commutative 78.0%

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

6. Simplified 78.0%

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

7. Taylor expanded in angle around 0 67.3%

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

   1. *-commutative 67.3%

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

   2. *-commutative 67.3%

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

   3. unpow2 67.3%

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

   4. unpow2 67.3%

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

   5. swap-sqr 67.3%

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

   6. unpow2 67.3%

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

   7. swap-sqr 78.0%

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

   8. associate-*r* 78.0%

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

   9. associate-*r* 78.0%

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

   10. unpow2 78.0%

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

   11. associate-*r* 78.0%

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

   12. *-commutative 78.0%

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

9. Simplified 78.0%

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

10. Final simplification 78.0%

    \[\leadsto {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot b\right)\right)}^{2} \]
11. Add Preprocessing

Alternative 12: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot \left(\pi \cdot angle\right)\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+ (pow a 2.0) (* 3.08641975308642e-5 (pow (* b (* PI angle)) 2.0))))

C

double code(double a, double b, double angle) {
	return pow(a, 2.0) + (3.08641975308642e-5 * pow((b * (((double) M_PI) * angle)), 2.0));
}

Java

public static double code(double a, double b, double angle) {
	return Math.pow(a, 2.0) + (3.08641975308642e-5 * Math.pow((b * (Math.PI * angle)), 2.0));
}

Python

def code(a, b, angle):
	return math.pow(a, 2.0) + (3.08641975308642e-5 * math.pow((b * (math.pi * angle)), 2.0))

Julia

function code(a, b, angle)
	return Float64((a ^ 2.0) + Float64(3.08641975308642e-5 * (Float64(b * Float64(pi * angle)) ^ 2.0)))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = (a ^ 2.0) + (3.08641975308642e-5 * ((b * (pi * angle)) ^ 2.0));
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(3.08641975308642e-5 * N[Power[N[(b * N[(Pi * angle), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.3%

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

3. Taylor expanded in angle around 0 82.2%

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

4. Taylor expanded in angle around 0 78.0%

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

   1. *-commutative 78.0%

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

6. Simplified 78.0%

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

7. Taylor expanded in angle around 0 67.3%

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

   1. *-commutative 67.3%

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

   2. *-commutative 67.3%

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

   3. unpow2 67.3%

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

   4. unpow2 67.3%

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

   5. swap-sqr 67.3%

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

   6. unpow2 67.3%

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

   7. swap-sqr 78.0%

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

   8. associate-*r* 78.0%

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

   9. associate-*r* 78.0%

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

   10. unpow2 78.0%

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

   11. associate-*r* 78.0%

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

   12. *-commutative 78.0%

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

9. Simplified 78.0%

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

10. Taylor expanded in angle around 0 67.3%

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

    1. associate-*r* 67.3%

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

    2. unpow2 67.3%

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

    3. unpow2 67.3%

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

    4. swap-sqr 78.0%

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

    5. *-commutative 78.0%

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

    6. unpow2 78.0%

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

    7. swap-sqr 78.0%

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

    8. unpow2 78.0%

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

    9. *-commutative 78.0%

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

    10. *-commutative 78.0%

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

    11. associate-*l* 78.0%

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

12. Simplified 78.0%

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

13. Final simplification 78.0%

    \[\leadsto {a}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(b \cdot \left(\pi \cdot angle\right)\right)}^{2} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024012 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))