ab-angle->ABCF B

Average Error: 31.6 → 21.6

Time: 19.1s

Precision: binary64

Cost: 39616

Math

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

↓

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

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 (+ b a)) (* (- a b) (sin (* 0.005555555555555556 (* angle PI)))))
  (log (exp (cos (* PI (* 0.005555555555555556 angle)))))))

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 * (b + a)) * ((a - b) * sin((0.005555555555555556 * (angle * ((double) M_PI)))))) * log(exp(cos((((double) M_PI) * (0.005555555555555556 * angle)))));
}

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 * (b + a)) * ((a - b) * Math.sin((0.005555555555555556 * (angle * Math.PI))))) * Math.log(Math.exp(Math.cos((Math.PI * (0.005555555555555556 * angle)))));
}

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 * (b + a)) * ((a - b) * math.sin((0.005555555555555556 * (angle * math.pi))))) * math.log(math.exp(math.cos((math.pi * (0.005555555555555556 * angle)))))

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(Float64(Float64(-2.0 * Float64(b + a)) * Float64(Float64(a - b) * sin(Float64(0.005555555555555556 * Float64(angle * pi))))) * log(exp(cos(Float64(pi * Float64(0.005555555555555556 * angle))))))
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 * (b + a)) * ((a - b) * sin((0.005555555555555556 * (angle * pi))))) * log(exp(cos((pi * (0.005555555555555556 * angle)))));
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[(N[(N[(-2.0 * N[(b + a), $MachinePrecision]), $MachinePrecision] * N[(N[(a - b), $MachinePrecision] * N[Sin[N[(0.005555555555555556 * N[(angle * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.6

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

2. Simplified 31.6

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

   [Start] 31.6	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.6	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub-neg [=>] 31.6	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 31.6	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   neg-sub0 [=>] 31.6	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-+l- [=>] 31.6	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub0-neg [=>] 31.6	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-lft-neg-out [=>] 31.6	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-rgt-neg-in [=>] 31.6	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.6	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.6	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   metadata-eval [=>] 31.6	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Taylor expanded in angle around inf 31.6

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

4. Simplified 21.6

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

   [Start] 31.6	\[ \left(-2 \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.6	\[ \left(-2 \cdot \left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.6	\[ \left(-2 \cdot \left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   difference-of-squares [=>] 31.6	\[ \left(-2 \cdot \left(\color{blue}{\left(\left(a + b\right) \cdot \left(a - b\right)\right)} \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.6	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \color{blue}{\left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.6	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \left(\color{blue}{\left(\pi \cdot angle\right)} \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*r* [<=] 31.6	\[ \left(-2 \cdot \left(\left(\left(a + b\right) \cdot \left(a - b\right)\right) \cdot \sin \color{blue}{\left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*l* [=>] 21.6	\[ \left(-2 \cdot \color{blue}{\left(\left(a + b\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*l* [<=] 21.6	\[ \color{blue}{\left(\left(-2 \cdot \left(a + b\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 21.6	\[ \left(\left(-2 \cdot \color{blue}{\left(b + a\right)}\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*r* [=>] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [<=] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(\color{blue}{\left(angle \cdot \pi\right)} \cdot 0.005555555555555556\right)\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [<=] 21.6	\[ \left(\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Applied egg-rr 21.6

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

6. Final simplification 21.6

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

Alternatives

Alternative 1
Error	21.6
Cost	40329

\[\begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+260} \lor \neg \left(t_0 \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot 2\right) \cdot \left(2 \cdot \left(b \cdot b - a \cdot a\right)\right)}{2}\\ \end{array} \]

Alternative 2
Error	21.7
Cost	40073

\[\begin{array}{l} t_0 := {b}^{2} - {a}^{2}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+219} \lor \neg \left(t_0 \leq 5 \cdot 10^{+304}\right):\\ \;\;\;\;\left(-2 \cdot \left(b + a\right)\right) \cdot \left(\left(a - b\right) \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 3
Error	21.6
Cost	26816

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

Alternative 4
Error	21.6
Cost	26816

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

Alternative 5
Error	21.8
Cost	13833

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.2 \cdot 10^{-7} \lor \neg \left(angle \leq 7 \cdot 10^{-47}\right):\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	22.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.34:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 0.185:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot a\right) \cdot \left(-\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 7
Error	22.7
Cost	13640

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.056:\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{elif}\;angle \leq 0.34:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a \cdot \left(-a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 8
Error	22.8
Cost	13577

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.052 \lor \neg \left(angle \leq 0.0055\right):\\ \;\;\;\;\left(b \cdot b\right) \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	29.9
Cost	7432

\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{+151}:\\ \;\;\;\;\pi \cdot \left(a \cdot \left(a \cdot \left(angle \cdot -0.011111111111111112\right)\right)\right)\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+122}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 10
Error	25.2
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 90:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(\left(b + a\right) \cdot \left(angle \cdot \left(b - a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(a \cdot a\right) \cdot -0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 11
Error	38.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+42}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 8.5 \cdot 10^{-160}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(a \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	38.9
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -2 \cdot 10^{-163}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 10^{+143}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	33.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -4.7 \cdot 10^{-7}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(\pi \cdot \left(a \cdot angle\right)\right)\right)\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-85}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	40.2
Cost	6912

\[-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right) \]

Reproduce

herbie shell --seed 2023045 
(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)))))