ab-angle->ABCF B

Average Error: 30.9 → 30.9

Time: 16.5s

Precision: binary64

Cost: 39488

Math

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

↓

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

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
 (let* ((t_0 (* PI (* 0.005555555555555556 angle))))
   (* 2.0 (* (sin t_0) (* (cos t_0) (- (pow b 2.0) (pow a 2.0)))))))

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) {
	double t_0 = ((double) M_PI) * (0.005555555555555556 * angle);
	return 2.0 * (sin(t_0) * (cos(t_0) * (pow(b, 2.0) - pow(a, 2.0))));
}

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) {
	double t_0 = Math.PI * (0.005555555555555556 * angle);
	return 2.0 * (Math.sin(t_0) * (Math.cos(t_0) * (Math.pow(b, 2.0) - Math.pow(a, 2.0))));
}

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):
	t_0 = math.pi * (0.005555555555555556 * angle)
	return 2.0 * (math.sin(t_0) * (math.cos(t_0) * (math.pow(b, 2.0) - math.pow(a, 2.0))))

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)
	t_0 = Float64(pi * Float64(0.005555555555555556 * angle))
	return Float64(2.0 * Float64(sin(t_0) * Float64(cos(t_0) * Float64((b ^ 2.0) - (a ^ 2.0)))))
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)
	t_0 = pi * (0.005555555555555556 * angle);
	tmp = 2.0 * (sin(t_0) * (cos(t_0) * ((b ^ 2.0) - (a ^ 2.0))));
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_] := Block[{t$95$0 = N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(N[Sin[t$95$0], $MachinePrecision] * N[(N[Cos[t$95$0], $MachinePrecision] * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.9

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

2. Simplified 30.9

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

   [Start] 30.9	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   rational.json-simplify-2 [=>] 30.9	\[ \color{blue}{\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)} \]
   rational.json-simplify-2 [=>] 30.9	\[ \cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 30.9	\[ \cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)}\right) \]
   rational.json-simplify-31 [=>] 30.9	\[ \cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)\right)} \]
   rational.json-simplify-31 [=>] 30.9	\[ \color{blue}{\left({b}^{2} - {a}^{2}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot 2\right)\right)} \]
   rational.json-simplify-2 [=>] 30.9	\[ \left({b}^{2} - {a}^{2}\right) \cdot \left(\cos \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}\right) \]

3. Taylor expanded in angle around inf 31.0

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

4. Taylor expanded in angle around inf 30.9

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

5. Simplified 30.9

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

   [Start] 30.9	\[ 2 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\left({b}^{2} - {a}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \]
   rational.json-simplify-2 [=>] 30.9	\[ 2 \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)}\right) \]
   rational.json-simplify-31 [=>] 30.9	\[ 2 \cdot \color{blue}{\left(\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right)} \]
   rational.json-simplify-2 [=>] 30.9	\[ 2 \cdot \left(\sin \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \]
   rational.json-simplify-31 [=>] 31.0	\[ 2 \cdot \left(\sin \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \left(\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \]
   rational.json-simplify-2 [=>] 31.0	\[ 2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \color{blue}{\left(\pi \cdot angle\right)}\right) \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \]
   rational.json-simplify-31 [=>] 30.9	\[ 2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot \left(\cos \color{blue}{\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right)} \cdot \left({b}^{2} - {a}^{2}\right)\right)\right) \]

6. Final simplification 30.9

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

Alternatives

Alternative 1
Error	31.0
Cost	39488

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

Alternative 2
Error	30.9
Cost	39488

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

Alternative 3
Error	32.1
Cost	33092

\[\begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-2 \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot {b}^{2}\right)\right) \cdot 1\\ \end{array} \]

Alternative 4
Error	32.2
Cost	32964

\[\begin{array}{l} \mathbf{if}\;{b}^{2} - {a}^{2} \leq -2 \cdot 10^{-281}:\\ \;\;\;\;\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right) \cdot \left(-2 \cdot {a}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {b}^{2}\right)\\ \end{array} \]

Alternative 5
Error	31.9
Cost	26368

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

Alternative 6
Error	36.7
Cost	20040

\[\begin{array}{l} t_0 := 2 \cdot \left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot {b}^{2}\right)\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-62}:\\ \;\;\;\;{a}^{2} \cdot \left(angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	37.4
Cost	13512

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{-62}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	37.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -2.5 \cdot 10^{-36}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-62}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot {b}^{2}\right)\right)\\ \end{array} \]

Alternative 9
Error	37.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-37}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 9.3 \cdot 10^{-63}:\\ \;\;\;\;angle \cdot \left(-0.011111111111111112 \cdot \left({a}^{2} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 10
Error	37.5
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{-38}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-62}:\\ \;\;\;\;angle \cdot \left({a}^{2} \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 11
Error	37.5
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -7.6 \cdot 10^{-39}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;\left(angle \cdot \left({a}^{2} \cdot \pi\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 12
Error	37.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -5.5 \cdot 10^{-37}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-62}:\\ \;\;\;\;{a}^{2} \cdot \left(angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left({b}^{2} \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 13
Error	37.4
Cost	13512

\[\begin{array}{l} \mathbf{if}\;b \leq -8 \cdot 10^{-36}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot {b}^{2}\right)\right)\\ \mathbf{elif}\;b \leq 4.1 \cdot 10^{-62}:\\ \;\;\;\;{a}^{2} \cdot \left(angle \cdot \left(\pi \cdot -0.011111111111111112\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 14
Error	43.2
Cost	13248

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

Reproduce

herbie shell --seed 2023053 
(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)))))