ab-angle->ABCF B

Average Error: 31.2 → 31.3

Time: 25.0s

Precision: binary64

Cost: 65728

Math

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

↓

\[2 \cdot \left(\left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \cos \left(\pi \cdot \left(angle \cdot 0.016666666666666666\right)\right)\right) - \cos \left(angle \cdot \left(\pi \cdot -0.016666666666666666\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (*
  2.0
  (*
   (-
    (+
     (cos (* PI (* angle 0.005555555555555556)))
     (cos (* PI (* angle 0.016666666666666666))))
    (cos (* angle (* PI -0.016666666666666666))))
   (* (- (pow b 2.0) (pow a 2.0)) (sin (* PI (/ angle 180.0)))))))

C

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

↓

double code(double a, double b, double angle) {
	return 2.0 * (((cos((((double) M_PI) * (angle * 0.005555555555555556))) + cos((((double) M_PI) * (angle * 0.016666666666666666)))) - cos((angle * (((double) M_PI) * -0.016666666666666666)))) * ((pow(b, 2.0) - pow(a, 2.0)) * sin((((double) M_PI) * (angle / 180.0)))));
}

Java

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

↓

public static double code(double a, double b, double angle) {
	return 2.0 * (((Math.cos((Math.PI * (angle * 0.005555555555555556))) + Math.cos((Math.PI * (angle * 0.016666666666666666)))) - Math.cos((angle * (Math.PI * -0.016666666666666666)))) * ((Math.pow(b, 2.0) - Math.pow(a, 2.0)) * Math.sin((Math.PI * (angle / 180.0)))));
}

Python

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

↓

def code(a, b, angle):
	return 2.0 * (((math.cos((math.pi * (angle * 0.005555555555555556))) + math.cos((math.pi * (angle * 0.016666666666666666)))) - math.cos((angle * (math.pi * -0.016666666666666666)))) * ((math.pow(b, 2.0) - math.pow(a, 2.0)) * math.sin((math.pi * (angle / 180.0)))))

Julia

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

↓

function code(a, b, angle)
	return Float64(2.0 * Float64(Float64(Float64(cos(Float64(pi * Float64(angle * 0.005555555555555556))) + cos(Float64(pi * Float64(angle * 0.016666666666666666)))) - cos(Float64(angle * Float64(pi * -0.016666666666666666)))) * Float64(Float64((b ^ 2.0) - (a ^ 2.0)) * sin(Float64(pi * Float64(angle / 180.0))))))
end

MATLAB

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

↓

function tmp = code(a, b, angle)
	tmp = 2.0 * (((cos((pi * (angle * 0.005555555555555556))) + cos((pi * (angle * 0.016666666666666666)))) - cos((angle * (pi * -0.016666666666666666)))) * (((b ^ 2.0) - (a ^ 2.0)) * sin((pi * (angle / 180.0)))));
end

Wolfram

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

↓

code[a_, b_, angle_] := N[(2.0 * N[(N[(N[(N[Cos[N[(Pi * N[(angle * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(Pi * N[(angle * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Cos[N[(angle * N[(Pi * -0.016666666666666666), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.2

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

2. Simplified 31.2

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

   [Start] 31.2	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   rational_best-simplify-2 [=>] 31.2	\[ \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   rational_best-simplify-2 [=>] 31.2	\[ \cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   rational_best-simplify-46 [=>] 31.2	\[ \cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(2 \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   rational_best-simplify-46 [=>] 31.2	\[ \color{blue}{2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   rational_best-simplify-2 [=>] 31.2	\[ 2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

3. Applied egg-rr 31.2

   \[\leadsto 2 \cdot \left(\color{blue}{\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right) - \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

4. Simplified 31.3

   \[\leadsto 2 \cdot \left(\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   Proof

   [Start] 31.2	\[ 2 \cdot \left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \cos \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right) - \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   trig-simplify-14 [=>] 31.2	\[ 2 \cdot \left(\color{blue}{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \pi \cdot \left(angle \cdot -0.011111111111111112\right)\right)} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]
   rational_best-simplify-46 [=>] 31.3	\[ 2 \cdot \left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right) + \color{blue}{angle \cdot \left(\pi \cdot -0.011111111111111112\right)}\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

5. Applied egg-rr 31.3

   \[\leadsto 2 \cdot \left(\color{blue}{\left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \cos \left(\pi \cdot \left(angle \cdot 0.016666666666666666\right)\right)\right) - \cos \left(angle \cdot \left(\pi \cdot -0.016666666666666666\right)\right)\right)} \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

6. Final simplification 31.3

   \[\leadsto 2 \cdot \left(\left(\left(\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) + \cos \left(\pi \cdot \left(angle \cdot 0.016666666666666666\right)\right)\right) - \cos \left(angle \cdot \left(\pi \cdot -0.016666666666666666\right)\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \]

Alternatives

Alternative 1
Error	33.2
Cost	39684

\[\begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \left(1 \cdot \left(-1 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle \cdot {b}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	33.2
Cost	39684

\[\begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq 2 \cdot 10^{-285}:\\ \;\;\;\;2 \cdot \left(1 \cdot \left(-1 \cdot \left({a}^{2} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left({b}^{2} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	31.2
Cost	39552

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

Alternative 4
Error	31.3
Cost	39488

\[2 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \cos \left(-0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]

Alternative 5
Error	31.2
Cost	26496

\[2 \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \left(0.5 \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right) \]

Alternative 6
Error	34.1
Cost	19968

\[2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \pi\right)\right)\right) \]

Alternative 7
Error	34.0
Cost	19968

\[2 \cdot \left(angle \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right) \]

Alternative 8
Error	38.0
Cost	13640

\[\begin{array}{l} t_0 := 2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)\right)\\ \mathbf{if}\;b \leq -2.4 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.05 \cdot 10^{+20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	38.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-11}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+20}:\\ \;\;\;\;-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	38.0
Cost	13640

\[\begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-12}:\\ \;\;\;\;2 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left({b}^{2} \cdot \pi\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.05 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left({a}^{2} \cdot \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 11
Error	37.9
Cost	13640

\[\begin{array}{l} \mathbf{if}\;b \leq -2.3 \cdot 10^{-13}:\\ \;\;\;\;2 \cdot \left({b}^{2} \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\\ \mathbf{elif}\;b \leq 5.05 \cdot 10^{+20}:\\ \;\;\;\;2 \cdot \left({a}^{2} \cdot \left(angle \cdot \left(\pi \cdot -0.005555555555555556\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.005555555555555556\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	43.3
Cost	13248

\[-0.011111111111111112 \cdot \left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \]

Alternative 13
Error	43.3
Cost	13248

\[-0.011111111111111112 \cdot \left({a}^{2} \cdot \left(angle \cdot \pi\right)\right) \]

Reproduce

herbie shell --seed 2023097 
(FPCore (a b angle)
  :name "ab-angle->ABCF B"
  :precision binary64
  (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* PI (/ angle 180.0)))) (cos (* PI (/ angle 180.0)))))