ab-angle->ABCF C

Percentage Accurate: 96.7% → 96.7%

Time: 25.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. a = -2.8596904483816153e-171
  2. b = -9.035647969315671e+193
  3. angle = -1.309056001303669e+92

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: 96.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} angle_m = \left|angle\right| \\ \begin{array}{l} t_0 := \pi \cdot \left(angle\_m \cdot 0.005555555555555556\right)\\ t_1 := t\_0 + 1\\ {\left(a \cdot \left(\cos t\_1 \cdot \cos 1 + \sin t\_1 \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(t\_0\right)\right)\right)\right)\right)\right)}^{2} \end{array} \end{array} \]

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* PI (* angle_m 0.005555555555555556))) (t_1 (+ t_0 1.0)))
   (+
    (pow (* a (+ (* (cos t_1) (cos 1.0)) (* (sin t_1) (sin 1.0)))) 2.0)
    (pow (* b (sin (expm1 (expm1 (log1p (log1p t_0)))))) 2.0))))

C

angle_m = fabs(angle);
double code(double a, double b, double angle_m) {
	double t_0 = ((double) M_PI) * (angle_m * 0.005555555555555556);
	double t_1 = t_0 + 1.0;
	return pow((a * ((cos(t_1) * cos(1.0)) + (sin(t_1) * sin(1.0)))), 2.0) + pow((b * sin(expm1(expm1(log1p(log1p(t_0)))))), 2.0);
}

Java

angle_m = Math.abs(angle);
public static double code(double a, double b, double angle_m) {
	double t_0 = Math.PI * (angle_m * 0.005555555555555556);
	double t_1 = t_0 + 1.0;
	return Math.pow((a * ((Math.cos(t_1) * Math.cos(1.0)) + (Math.sin(t_1) * Math.sin(1.0)))), 2.0) + Math.pow((b * Math.sin(Math.expm1(Math.expm1(Math.log1p(Math.log1p(t_0)))))), 2.0);
}

Python

angle_m = math.fabs(angle)
def code(a, b, angle_m):
	t_0 = math.pi * (angle_m * 0.005555555555555556)
	t_1 = t_0 + 1.0
	return math.pow((a * ((math.cos(t_1) * math.cos(1.0)) + (math.sin(t_1) * math.sin(1.0)))), 2.0) + math.pow((b * math.sin(math.expm1(math.expm1(math.log1p(math.log1p(t_0)))))), 2.0)

Julia

angle_m = abs(angle)
function code(a, b, angle_m)
	t_0 = Float64(pi * Float64(angle_m * 0.005555555555555556))
	t_1 = Float64(t_0 + 1.0)
	return Float64((Float64(a * Float64(Float64(cos(t_1) * cos(1.0)) + Float64(sin(t_1) * sin(1.0)))) ^ 2.0) + (Float64(b * sin(expm1(expm1(log1p(log1p(t_0)))))) ^ 2.0))
end

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := Block[{t$95$0 = N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, N[(N[Power[N[(a * N[(N[(N[Cos[t$95$1], $MachinePrecision] * N[Cos[1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$1], $MachinePrecision] * N[Sin[1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Exp[N[(Exp[N[Log[1 + N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 95.9%

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

   1. expm1-log1p-u 90.5%

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

   2. div-inv 90.5%

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

   3. metadata-eval 90.5%

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

4. Applied egg-rr 90.5%

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

   1. expm1-log1p-u 90.6%

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

   2. expm1-undefine 73.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1}\right)\right)\right)}^{2} \]

6. Applied egg-rr 73.0%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1}\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. expm1-define 90.6%

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

8. Simplified 90.6%

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

   1. clear-num 90.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)}^{2} \]

   2. un-div-inv 90.6%

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

10. Applied egg-rr 90.6%

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

    1. div-inv 90.6%

       \[\leadsto {\left(a \cdot \cos \color{blue}{\left(\pi \cdot \frac{1}{\frac{180}{angle}}\right)}\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)}^{2} \]

    2. clear-num 90.6%

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

    3. div-inv 90.6%

       \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\left(angle \cdot \frac{1}{180}\right)}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)}^{2} \]

    4. metadata-eval 90.6%

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

    5. expm1-log1p-u 90.6%

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

    6. expm1-undefine 90.6%

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

    7. cos-diff 90.6%

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

    8. log1p-undefine 90.6%

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

    9. rem-exp-log 90.6%

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

    10. +-commutative 90.6%

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

    11. log1p-undefine 90.6%

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

    12. rem-exp-log 90.6%

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

    13. +-commutative 90.6%

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

12. Applied egg-rr 90.6%

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

13. Final simplification 90.6%

    \[\leadsto {\left(a \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \cos 1 + \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + 1\right) \cdot \sin 1\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow
   (*
    b
    (sin
     (expm1 (expm1 (log1p (log1p (* PI (* angle_m 0.005555555555555556))))))))
   2.0)
  (pow (* a (cos (/ PI (/ 180.0 angle_m)))) 2.0)))

C

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

Java

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

Python

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

Julia

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(Exp[N[(Exp[N[Log[1 + N[Log[1 + N[(Pi * N[(angle$95$m * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(a * N[Cos[N[(Pi / N[(180.0 / angle$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

   1. expm1-log1p-u 90.5%

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

   2. div-inv 90.5%

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

   3. metadata-eval 90.5%

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

4. Applied egg-rr 90.5%

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

   1. expm1-log1p-u 90.6%

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

   2. expm1-undefine 73.0%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1}\right)\right)\right)}^{2} \]

6. Applied egg-rr 73.0%

   \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\color{blue}{e^{\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)} - 1}\right)\right)\right)}^{2} \]
7. Step-by-step derivation

   1. expm1-define 90.6%

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

8. Simplified 90.6%

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

   1. clear-num 90.6%

      \[\leadsto {\left(a \cdot \cos \left(\pi \cdot \color{blue}{\frac{1}{\frac{180}{angle}}}\right)\right)}^{2} + {\left(b \cdot \sin \left(\mathsf{expm1}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)\right)\right)}^{2} \]

   2. un-div-inv 90.6%

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

10. Applied egg-rr 90.6%

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

11. Final simplification 90.6%

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

Alternative 3: 96.8% accurate, 1.0× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* angle_m (/ PI -180.0))))
   (+ (pow (* a (cos t_0)) 2.0) (pow (* b (sin t_0)) 2.0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Final simplification 96.0%

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

Alternative 4: 96.2% accurate, 1.3× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+ (pow (* b (sin (* angle_m (/ PI -180.0)))) 2.0) (pow a 2.0)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[N[(b * N[Sin[N[(angle$95$m * N[(Pi / -180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in angle around 0 94.8%

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

6. Final simplification 94.8%

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

Alternative 5: 88.9% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* angle_m -0.005555555555555556)
   (* (* PI b) (* -0.005555555555555556 (* angle_m (* PI b)))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(angle$95$m * -0.005555555555555556), $MachinePrecision] * N[(N[(Pi * b), $MachinePrecision] * N[(-0.005555555555555556 * N[(angle$95$m * N[(Pi * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in angle around 0 94.8%

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

6. Taylor expanded in angle around 0 91.0%

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

   1. unpow2 91.0%

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

   2. associate-*r* 91.0%

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

   3. associate-*l* 88.1%

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

   4. associate-*r* 88.1%

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

   5. *-commutative 88.1%

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

8. Applied egg-rr 88.1%

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

9. Taylor expanded in b around 0 88.1%

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

    1. *-commutative 88.1%

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

11. Simplified 88.1%

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

12. Final simplification 88.1%

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

Alternative 6: 91.8% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (let* ((t_0 (* (* PI b) (* angle_m -0.005555555555555556))))
   (+ (pow a 2.0) (* t_0 t_0))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in angle around 0 94.8%

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

6. Taylor expanded in angle around 0 91.0%

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

   1. unpow2 91.0%

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

   2. associate-*r* 91.0%

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

   3. *-commutative 91.0%

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

   4. associate-*r* 91.0%

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

   5. *-commutative 91.0%

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

8. Applied egg-rr 91.0%

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

9. Final simplification 91.0%

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

Alternative 7: 91.9% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   (* b (* PI angle_m))
   (* -0.005555555555555556 (* (* PI b) (* angle_m -0.005555555555555556))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision] * N[(-0.005555555555555556 * N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in angle around 0 94.8%

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

6. Taylor expanded in angle around 0 91.0%

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

   1. unpow2 91.0%

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

   2. associate-*r* 91.0%

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

   3. associate-*r* 91.1%

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

   4. *-commutative 91.1%

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

   5. *-commutative 91.1%

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

   6. associate-*l* 91.1%

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

8. Applied egg-rr 91.1%

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

9. Final simplification 91.1%

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

Alternative 8: 91.8% accurate, 3.5× speedup

Math

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

FPCore

angle_m = (fabs.f64 angle)
(FPCore (a b angle_m)
 :precision binary64
 (+
  (pow a 2.0)
  (*
   -0.005555555555555556
   (* (* (* PI b) (* angle_m -0.005555555555555556)) (* b (* PI angle_m))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

angle_m = N[Abs[angle], $MachinePrecision]
code[a_, b_, angle$95$m_] := N[(N[Power[a, 2.0], $MachinePrecision] + N[(-0.005555555555555556 * N[(N[(N[(Pi * b), $MachinePrecision] * N[(angle$95$m * -0.005555555555555556), $MachinePrecision]), $MachinePrecision] * N[(b * N[(Pi * angle$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

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

3. Simplified 96.0%

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

5. Taylor expanded in angle around 0 94.8%

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

6. Taylor expanded in angle around 0 91.0%

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

   1. unpow2 91.0%

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

   2. *-commutative 91.0%

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

   3. associate-*r* 91.0%

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

   4. associate-*r* 91.0%

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

   5. *-commutative 91.0%

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

   6. *-commutative 91.0%

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

   7. associate-*l* 91.1%

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

8. Applied egg-rr 91.1%

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

9. Final simplification 91.1%

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

Reproduce

herbie shell --seed 2024052 
(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)))