ab-angle->ABCF A

Percentage Accurate: 80.2% → 80.1%

Time: 50.7s

Alternatives: 6

Speedup: N/A×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ {\left(a \cdot \sin t\_0\right)}^{2} + {\left(b \cdot \cos t\_0\right)}^{2} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

   1. clear-num 78.8%

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

   2. un-div-inv 78.8%

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

7. Applied egg-rr 78.8%

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

8. Final simplification 78.8%

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

Alternative 2: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

6. Taylor expanded in angle around 0 78.7%

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

7. Final simplification 78.7%

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

Alternative 3: 80.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 78.8%

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

6. Final simplification 78.8%

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

Alternative 4: 60.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 72.5%

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

   1. *-commutative 72.5%

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

   2. associate-*r* 72.5%

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

   3. metadata-eval 72.5%

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

   4. associate-/r/ 72.5%

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

   5. associate-*l/ 72.5%

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

   6. *-lft-identity 72.5%

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

   7. associate-/l/ 72.5%

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

   8. associate-/r/ 72.5%

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

7. Simplified 72.5%

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

8. Taylor expanded in angle around 0 72.7%

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

   1. unpow2 72.7%

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

   2. *-commutative 72.7%

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

   3. *-commutative 72.7%

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

   4. clear-num 72.7%

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

   5. associate-/r/ 72.7%

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

   6. clear-num 72.7%

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

   7. div-inv 72.7%

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

   8. metadata-eval 72.7%

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

   9. clear-num 72.7%

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

   10. associate-/r/ 72.7%

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

   11. clear-num 72.7%

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

   12. div-inv 72.7%

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

   13. metadata-eval 72.7%

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

10. Applied egg-rr 72.7%

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

    1. add-sqr-sqrt 48.0%

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

    2. sqrt-prod 61.3%

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

    3. rem-sqrt-square 61.3%

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

    4. *-commutative 61.3%

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

    5. associate-*l* 61.3%

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

    6. associate-*l* 61.3%

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

    7. *-commutative 61.3%

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

12. Applied egg-rr 61.3%

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

13. Final simplification 61.3%

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

Alternative 5: 75.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 72.5%

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

   1. *-commutative 72.5%

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

   2. associate-*r* 72.5%

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

   3. metadata-eval 72.5%

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

   4. associate-/r/ 72.5%

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

   5. associate-*l/ 72.5%

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

   6. *-lft-identity 72.5%

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

   7. associate-/l/ 72.5%

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

   8. associate-/r/ 72.5%

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

7. Simplified 72.5%

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

8. Taylor expanded in angle around 0 72.7%

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

   1. unpow2 72.7%

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

   2. *-commutative 72.7%

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

   3. *-commutative 72.7%

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

   4. clear-num 72.7%

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

   5. associate-/r/ 72.7%

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

   6. clear-num 72.7%

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

   7. div-inv 72.7%

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

   8. metadata-eval 72.7%

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

   9. clear-num 72.7%

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

   10. associate-/r/ 72.7%

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

   11. clear-num 72.7%

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

   12. div-inv 72.7%

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

   13. metadata-eval 72.7%

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

10. Applied egg-rr 72.7%

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

11. Taylor expanded in a around 0 72.7%

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

12. Final simplification 72.7%

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

Alternative 6: 75.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := angle \cdot \left(a \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\\ {b}^{2} + t\_0 \cdot t\_0 \end{array} \end{array} \]

FPCore

(FPCore (a b angle)
 :precision binary64
 (let* ((t_0 (* angle (* a (* PI 0.005555555555555556)))))
   (+ (pow b 2.0) (* t_0 t_0))))

C

double code(double a, double b, double angle) {
	double t_0 = angle * (a * (((double) M_PI) * 0.005555555555555556));
	return pow(b, 2.0) + (t_0 * t_0);
}

Java

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

Python

def code(a, b, angle):
	t_0 = angle * (a * (math.pi * 0.005555555555555556))
	return math.pow(b, 2.0) + (t_0 * t_0)

Julia

function code(a, b, angle)
	t_0 = Float64(angle * Float64(a * Float64(pi * 0.005555555555555556)))
	return Float64((b ^ 2.0) + Float64(t_0 * t_0))
end

MATLAB

function tmp = code(a, b, angle)
	t_0 = angle * (a * (pi * 0.005555555555555556));
	tmp = (b ^ 2.0) + (t_0 * t_0);
end

Wolfram

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

Derivation

1. Initial program 78.4%

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

   1. associate-*l/ 78.4%

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

   2. associate-/l* 78.5%

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

   3. cos-neg 78.5%

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

   4. distribute-lft-neg-out 78.5%

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

   5. distribute-frac-neg 78.5%

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

   6. distribute-frac-neg 78.5%

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

   7. distribute-lft-neg-out 78.5%

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

   8. cos-neg 78.5%

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

   9. associate-*l/ 78.4%

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

   10. associate-/l* 78.4%

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

3. Simplified 78.4%

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

5. Taylor expanded in angle around 0 72.5%

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

   1. *-commutative 72.5%

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

   2. associate-*r* 72.5%

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

   3. metadata-eval 72.5%

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

   4. associate-/r/ 72.5%

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

   5. associate-*l/ 72.5%

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

   6. *-lft-identity 72.5%

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

   7. associate-/l/ 72.5%

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

   8. associate-/r/ 72.5%

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

7. Simplified 72.5%

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

8. Taylor expanded in angle around 0 72.7%

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

   1. unpow2 72.7%

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

   2. *-commutative 72.7%

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

   3. *-commutative 72.7%

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

   4. clear-num 72.7%

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

   5. associate-/r/ 72.7%

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

   6. clear-num 72.7%

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

   7. div-inv 72.7%

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

   8. metadata-eval 72.7%

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

   9. clear-num 72.7%

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

   10. associate-/r/ 72.7%

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

   11. clear-num 72.7%

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

   12. div-inv 72.7%

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

   13. metadata-eval 72.7%

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

10. Applied egg-rr 72.7%

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

11. Final simplification 72.7%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))