ab-angle->ABCF C

Percentage Accurate: 79.8% → 79.8%

Time: 26.3s

Alternatives: 11

Speedup: N/A×

• 36.0% 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: 79.8% 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: 79.8% accurate, N/A× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (sin (* PI (* 0.005555555555555556 angle)))) 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 * sin((((double) M_PI) * (0.005555555555555556 * angle)))), 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.sin((Math.PI * (0.005555555555555556 * angle)))), 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.sin((math.pi * (0.005555555555555556 * angle)))), 2.0)

Julia

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

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * sin((pi * (0.005555555555555556 * angle)))) ^ 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[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

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

   1. lower-*.f64 76.7

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

5. Applied rewrites 76.7%

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

Alternative 2: 79.8% accurate, N/A× speedup

Math

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

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (cos (* PI (/ angle 180.0)))) 2.0)
  (pow (* b (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 * 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.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.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 * sin(Float64(Float64(pi * angle) * 0.005555555555555556))) ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * cos((pi * (angle / 180.0)))) ^ 2.0) + ((b * sin(((pi * 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[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 76.2

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

5. Applied rewrites 76.2%

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

Alternative 3: 79.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. *-commutative N/A

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

   6. lower-/.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 76.2

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

4. Applied rewrites 76.2%

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

6. Applied rewrites 27.8%

   \[\leadsto {\color{blue}{\left(\frac{{\left(\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)}^{3} + {\left(\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)}^{3}}{\mathsf{fma}\left(\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a, \left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a, \left(\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right) \cdot \left(\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right) - \left(\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right) \cdot \left(\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)\right)}\right)}}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

8. Applied rewrites 19.3%

   \[\leadsto {\left(\frac{{\left(\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)}^{3} + {\left(\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)}^{3}}{\mathsf{fma}\left(\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a, \left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a, \color{blue}{\frac{{\left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot a\right)}^{2} \cdot {\left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot a\right)}^{2} - \left(\left(\left(0 \cdot a\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right) \cdot \left(\left(\left(0 \cdot a\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a\right)}{{\left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot a\right)}^{2} + \left(\left(0 \cdot a\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot a}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\frac{\pi \cdot angle}{180}\right)\right)}^{2} \]

9. Taylor expanded in angle around 0

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

11. Applied rewrites 76.1%

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

Alternative 4: 79.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ {\left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* (sin (fma (* 0.005555555555555556 angle) PI (* 0.5 PI))) a) 2.0)
  (pow (* (sin (* (* PI angle) 0.005555555555555556)) b) 2.0)))

C

double code(double a, double b, double angle) {
	return pow((sin(fma((0.005555555555555556 * angle), ((double) M_PI), (0.5 * ((double) M_PI)))) * a), 2.0) + pow((sin(((((double) M_PI) * angle) * 0.005555555555555556)) * b), 2.0);
}

Julia

function code(a, b, angle)
	return Float64((Float64(sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(0.5 * pi))) * a) ^ 2.0) + (Float64(sin(Float64(Float64(pi * angle) * 0.005555555555555556)) * b) ^ 2.0))
end

Wolfram

code[a_, b_, angle_] := N[(N[Power[N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(N[Sin[N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lift-PI.f64 N/A

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in angle around 0

   \[\leadsto {\color{blue}{\left(a \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + angle \cdot \left(\frac{1}{180} \cdot \left(a \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + angle \cdot \left(\frac{-1}{64800} \cdot \left(a \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right) + \frac{-1}{34992000} \cdot \left(a \cdot \left(angle \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

7. Applied rewrites 69.0%

   \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-2.8577960676726107 \cdot 10^{-8} \cdot a, \left(angle \cdot {\pi}^{3}\right) \cdot \cos \left(0.5 \cdot \pi\right), \left(-1.54320987654321 \cdot 10^{-5} \cdot a\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \sin \left(0.5 \cdot \pi\right)\right)\right), angle, \left(\left(a \cdot \pi\right) \cdot \cos \left(0.5 \cdot \pi\right)\right) \cdot 0.005555555555555556\right), angle, \sin \left(0.5 \cdot \pi\right) \cdot a\right)\right)}}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

8. Taylor expanded in a around 0

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

10. Applied rewrites 76.0%

    \[\leadsto \color{blue}{{\left(\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right) \cdot a\right)}^{2} + {\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}} \]
11. Add Preprocessing

Alternative 5: 57.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(t\_2 \cdot a\right)}^{2}}{b \cdot b} + {t\_1}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(t\_1 \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), t\_1, t\_2\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= a 1.6e-163)
     (* (+ (/ (pow (* t_2 a) 2.0) (* b b)) (pow t_1 2.0)) (* b b))
     (if (<= a 1.9e+130)
       (*
        (+
         (/ (pow (* t_1 b) 2.0) (* a a))
         (pow (fma (sin (fma 0.5 PI (/ PI 2.0))) t_1 t_2) 2.0))
        (* a a))
       (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (a <= 1.6e-163) {
		tmp = ((pow((t_2 * a), 2.0) / (b * b)) + pow(t_1, 2.0)) * (b * b);
	} else if (a <= 1.9e+130) {
		tmp = ((pow((t_1 * b), 2.0) / (a * a)) + pow(fma(sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))), t_1, t_2), 2.0)) * (a * a);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (a <= 1.6e-163)
		tmp = Float64(Float64(Float64((Float64(t_2 * a) ^ 2.0) / Float64(b * b)) + (t_1 ^ 2.0)) * Float64(b * b));
	elseif (a <= 1.9e+130)
		tmp = Float64(Float64(Float64((Float64(t_1 * b) ^ 2.0) / Float64(a * a)) + (fma(sin(fma(0.5, pi, Float64(pi / 2.0))), t_1, t_2) ^ 2.0)) * Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a, 1.6e-163], N[(N[(N[(N[Power[N[(t$95$2 * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+130], N[(N[(N[(N[Power[N[(t$95$1 * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.59999999999999994e-163

   1. Initial program 77.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   if 1.59999999999999994e-163 < a < 1.9000000000000001e130

   1. Initial program 67.4%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

   4. Applied rewrites 67.4%

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

   6. Applied rewrites 67.4%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \frac{angle}{180}\right), \cos \left(\frac{\pi}{2}\right), \color{blue}{\frac{\mathsf{fma}\left({\sin \left(\pi \cdot \frac{angle}{180}\right)}^{3}, 0, {\cos \left(\pi \cdot \frac{angle}{180}\right)}^{3} \cdot 1\right) \cdot 2}{\left({\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left({\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} - \left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot 2}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \pi\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)} \]
   10. Step-by-step derivation

       1. lift-cos.f64 N/A

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

       2. lift-PI.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. sin-+PI/2-rev N/A

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

       5. lower-sin.f64 N/A

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

       6. lift-/.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-PI.f64 63.5

          \[\leadsto \left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

   11. Applied rewrites 63.5%

       \[\leadsto \left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right) \]

   if 1.9000000000000001e130 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}}{b \cdot b} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(t\_2 \cdot a\right)}^{2}}{b \cdot b} + {t\_1}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(t\_1 \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \pi\right), t\_1, t\_2\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= a 1.6e-163)
     (* (+ (/ (pow (* t_2 a) 2.0) (* b b)) (pow t_1 2.0)) (* b b))
     (if (<= a 1.9e+130)
       (*
        (+
         (/ (pow (* t_1 b) 2.0) (* a a))
         (pow (fma (cos (* 0.5 PI)) t_1 t_2) 2.0))
        (* a a))
       (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (a <= 1.6e-163) {
		tmp = ((pow((t_2 * a), 2.0) / (b * b)) + pow(t_1, 2.0)) * (b * b);
	} else if (a <= 1.9e+130) {
		tmp = ((pow((t_1 * b), 2.0) / (a * a)) + pow(fma(cos((0.5 * ((double) M_PI))), t_1, t_2), 2.0)) * (a * a);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (a <= 1.6e-163)
		tmp = Float64(Float64(Float64((Float64(t_2 * a) ^ 2.0) / Float64(b * b)) + (t_1 ^ 2.0)) * Float64(b * b));
	elseif (a <= 1.9e+130)
		tmp = Float64(Float64(Float64((Float64(t_1 * b) ^ 2.0) / Float64(a * a)) + (fma(cos(Float64(0.5 * pi)), t_1, t_2) ^ 2.0)) * Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a, 1.6e-163], N[(N[(N[(N[Power[N[(t$95$2 * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+130], N[(N[(N[(N[Power[N[(t$95$1 * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.59999999999999994e-163

   1. Initial program 77.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   if 1.59999999999999994e-163 < a < 1.9000000000000001e130

   1. Initial program 67.4%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

   4. Applied rewrites 67.4%

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

   6. Applied rewrites 67.4%

      \[\leadsto {\left(a \cdot \mathsf{fma}\left(\sin \left(\pi \cdot \frac{angle}{180}\right), \cos \left(\frac{\pi}{2}\right), \color{blue}{\frac{\mathsf{fma}\left({\sin \left(\pi \cdot \frac{angle}{180}\right)}^{3}, 0, {\cos \left(\pi \cdot \frac{angle}{180}\right)}^{3} \cdot 1\right) \cdot 2}{\left({\left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + \left({\left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} - \left(0 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \left(1 \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)\right)\right) \cdot 2}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

   7. Taylor expanded in a around inf

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

   9. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \pi\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)} \]

   if 1.9000000000000001e130 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}}{b \cdot b} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \pi\right), \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right), \cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 57.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(t\_2 \cdot a\right)}^{2}}{b \cdot b} + {t\_1}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(t\_1 \cdot b\right)}^{2}}{a \cdot a} + {t\_2}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556))
        (t_1 (sin t_0))
        (t_2 (cos t_0)))
   (if (<= a 1.6e-163)
     (* (+ (/ (pow (* t_2 a) 2.0) (* b b)) (pow t_1 2.0)) (* b b))
     (if (<= a 1.9e+130)
       (* (+ (/ (pow (* t_1 b) 2.0) (* a a)) (pow t_2 2.0)) (* a a))
       (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double tmp;
	if (a <= 1.6e-163) {
		tmp = ((pow((t_2 * a), 2.0) / (b * b)) + pow(t_1, 2.0)) * (b * b);
	} else if (a <= 1.9e+130) {
		tmp = ((pow((t_1 * b), 2.0) / (a * a)) + pow(t_2, 2.0)) * (a * a);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double tmp;
	if (a <= 1.6e-163) {
		tmp = ((Math.pow((t_2 * a), 2.0) / (b * b)) + Math.pow(t_1, 2.0)) * (b * b);
	} else if (a <= 1.9e+130) {
		tmp = ((Math.pow((t_1 * b), 2.0) / (a * a)) + Math.pow(t_2, 2.0)) * (a * a);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	tmp = 0
	if a <= 1.6e-163:
		tmp = ((math.pow((t_2 * a), 2.0) / (b * b)) + math.pow(t_1, 2.0)) * (b * b)
	elif a <= 1.9e+130:
		tmp = ((math.pow((t_1 * b), 2.0) / (a * a)) + math.pow(t_2, 2.0)) * (a * a)
	else:
		tmp = a * a
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	tmp = 0.0
	if (a <= 1.6e-163)
		tmp = Float64(Float64(Float64((Float64(t_2 * a) ^ 2.0) / Float64(b * b)) + (t_1 ^ 2.0)) * Float64(b * b));
	elseif (a <= 1.9e+130)
		tmp = Float64(Float64(Float64((Float64(t_1 * b) ^ 2.0) / Float64(a * a)) + (t_2 ^ 2.0)) * Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	tmp = 0.0;
	if (a <= 1.6e-163)
		tmp = ((((t_2 * a) ^ 2.0) / (b * b)) + (t_1 ^ 2.0)) * (b * b);
	elseif (a <= 1.9e+130)
		tmp = ((((t_1 * b) ^ 2.0) / (a * a)) + (t_2 ^ 2.0)) * (a * a);
	else
		tmp = a * a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[a, 1.6e-163], N[(N[(N[(N[Power[N[(t$95$2 * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+130], N[(N[(N[(N[Power[N[(t$95$1 * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.59999999999999994e-163

   1. Initial program 77.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   if 1.59999999999999994e-163 < a < 1.9000000000000001e130

   1. Initial program 67.4%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 63.5%

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

   if 1.9000000000000001e130 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}}{b \cdot b} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot b\right)}^{2}}{a \cdot a} + {\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(\cos t\_0 \cdot a\right)}^{2}}{b \cdot b} + {\sin t\_0}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}{a \cdot a} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= a 1.6e-163)
     (* (+ (/ (pow (* (cos t_0) a) 2.0) (* b b)) (pow (sin t_0) 2.0)) (* b b))
     (if (<= a 1.9e+130)
       (*
        (+
         (/
          (pow (* (sin (* (* 0.005555555555555556 angle) PI)) b) 2.0)
          (* a a))
         (pow (sin (fma (* 0.005555555555555556 angle) PI (* 0.5 PI))) 2.0))
        (* a a))
       (* a a)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (a <= 1.6e-163) {
		tmp = ((pow((cos(t_0) * a), 2.0) / (b * b)) + pow(sin(t_0), 2.0)) * (b * b);
	} else if (a <= 1.9e+130) {
		tmp = ((pow((sin(((0.005555555555555556 * angle) * ((double) M_PI))) * b), 2.0) / (a * a)) + pow(sin(fma((0.005555555555555556 * angle), ((double) M_PI), (0.5 * ((double) M_PI)))), 2.0)) * (a * a);
	} else {
		tmp = a * a;
	}
	return tmp;
}

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (a <= 1.6e-163)
		tmp = Float64(Float64(Float64((Float64(cos(t_0) * a) ^ 2.0) / Float64(b * b)) + (sin(t_0) ^ 2.0)) * Float64(b * b));
	elseif (a <= 1.9e+130)
		tmp = Float64(Float64(Float64((Float64(sin(Float64(Float64(0.005555555555555556 * angle) * pi)) * b) ^ 2.0) / Float64(a * a)) + (sin(fma(Float64(0.005555555555555556 * angle), pi, Float64(0.5 * pi))) ^ 2.0)) * Float64(a * a));
	else
		tmp = Float64(a * a);
	end
	return tmp
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[a, 1.6e-163], N[(N[(N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+130], N[(N[(N[(N[Power[N[(N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(0.005555555555555556 * angle), $MachinePrecision] * Pi + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision], N[(a * a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < 1.59999999999999994e-163

   1. Initial program 77.8%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.5%

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

   if 1.59999999999999994e-163 < a < 1.9000000000000001e130

   1. Initial program 67.4%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-PI.f64 N/A

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

   4. Applied rewrites 67.4%

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

   5. Taylor expanded in a around inf

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

   7. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\left(\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}{a \cdot a} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)} \]

   if 1.9000000000000001e130 < a

   1. Initial program 87.1%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{a \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 1.6 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}}{b \cdot b} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(b \cdot b\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+130}:\\ \;\;\;\;\left(\frac{{\left(\sin \left(\left(0.005555555555555556 \cdot angle\right) \cdot \pi\right) \cdot b\right)}^{2}}{a \cdot a} + {\sin \left(\mathsf{fma}\left(0.005555555555555556 \cdot angle, \pi, 0.5 \cdot \pi\right)\right)}^{2}\right) \cdot \left(a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\pi \cdot angle\right) \cdot 0.005555555555555556\\ \mathbf{if}\;b \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(\cos t\_0 \cdot a\right)}^{2}}{b \cdot b} + {\sin t\_0}^{2}\right) \cdot \left(b \cdot b\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* (* PI angle) 0.005555555555555556)))
   (if (<= b 1.1e-53)
     (* a a)
     (*
      (+ (/ (pow (* (cos t_0) a) 2.0) (* b b)) (pow (sin t_0) 2.0))
      (* b b)))))

C

double code(double a, double b, double angle) {
	double t_0 = (((double) M_PI) * angle) * 0.005555555555555556;
	double tmp;
	if (b <= 1.1e-53) {
		tmp = a * a;
	} else {
		tmp = ((pow((cos(t_0) * a), 2.0) / (b * b)) + pow(sin(t_0), 2.0)) * (b * b);
	}
	return tmp;
}

Java

public static double code(double a, double b, double angle) {
	double t_0 = (Math.PI * angle) * 0.005555555555555556;
	double tmp;
	if (b <= 1.1e-53) {
		tmp = a * a;
	} else {
		tmp = ((Math.pow((Math.cos(t_0) * a), 2.0) / (b * b)) + Math.pow(Math.sin(t_0), 2.0)) * (b * b);
	}
	return tmp;
}

Python

def code(a, b, angle):
	t_0 = (math.pi * angle) * 0.005555555555555556
	tmp = 0
	if b <= 1.1e-53:
		tmp = a * a
	else:
		tmp = ((math.pow((math.cos(t_0) * a), 2.0) / (b * b)) + math.pow(math.sin(t_0), 2.0)) * (b * b)
	return tmp

Julia

function code(a, b, angle)
	t_0 = Float64(Float64(pi * angle) * 0.005555555555555556)
	tmp = 0.0
	if (b <= 1.1e-53)
		tmp = Float64(a * a);
	else
		tmp = Float64(Float64(Float64((Float64(cos(t_0) * a) ^ 2.0) / Float64(b * b)) + (sin(t_0) ^ 2.0)) * Float64(b * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, angle)
	t_0 = (pi * angle) * 0.005555555555555556;
	tmp = 0.0;
	if (b <= 1.1e-53)
		tmp = a * a;
	else
		tmp = ((((cos(t_0) * a) ^ 2.0) / (b * b)) + (sin(t_0) ^ 2.0)) * (b * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, angle_] := Block[{t$95$0 = N[(N[(Pi * angle), $MachinePrecision] * 0.005555555555555556), $MachinePrecision]}, If[LessEqual[b, 1.1e-53], N[(a * a), $MachinePrecision], N[(N[(N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(b * b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000009e-53

   1. Initial program 74.9%

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

   3. Taylor expanded in angle around 0

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

      1. unpow2 N/A

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

      2. lower-*.f64 58.3

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

   5. Applied rewrites 58.3%

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

   if 1.10000000000000009e-53 < b

   1. Initial program 81.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 57.8%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{-53}:\\ \;\;\;\;a \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{{\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot a\right)}^{2}}{b \cdot b} + {\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}^{2}\right) \cdot \left(b \cdot b\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.1% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = a * a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

6. Final simplification 55.6%

   \[\leadsto a \cdot a \]
7. Add Preprocessing

Alternative 11: 28.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ e^{\log a \cdot 2} \end{array} \]

FPCore

(FPCore (a b angle) :precision binary64 (exp (* (log a) 2.0)))

C

double code(double a, double b, double angle) {
	return exp((log(a) * 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, angle)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    code = exp((log(a) * 2.0d0))
end function

Java

public static double code(double a, double b, double angle) {
	return Math.exp((Math.log(a) * 2.0));
}

Python

def code(a, b, angle):
	return math.exp((math.log(a) * 2.0))

Julia

function code(a, b, angle)
	return exp(Float64(log(a) * 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = exp((log(a) * 2.0));
end

Wolfram

code[a_, b_, angle_] := N[Exp[N[(N[Log[a], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 76.7%

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

3. Taylor expanded in angle around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

   \[\leadsto \color{blue}{a \cdot a} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. pow2 N/A

      \[\leadsto {a}^{\color{blue}{2}} \]

   3. pow-to-exp N/A

      \[\leadsto e^{\log a \cdot 2} \]

   4. lower-exp.f64 N/A

      \[\leadsto e^{\log a \cdot 2} \]

   5. lower-*.f64 N/A

      \[\leadsto e^{\log a \cdot 2} \]

   6. lower-log.f64 28.1

      \[\leadsto e^{\log a \cdot 2} \]

7. Applied rewrites 28.1%

   \[\leadsto e^{\log a \cdot 2} \]

8. Final simplification 28.1%

   \[\leadsto e^{\log a \cdot 2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))