ab-angle->ABCF B

Average Error: 31.4 → 21.5

Time: 21.0s

Precision: binary64

Cost: 46208

Math

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

↓

\[\frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{a + b}} \cdot \cos \left({\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{180}{angle}}}\right)}^{3}\right) \]

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
 (*
  (/ (* (sin (* PI (/ angle 180.0))) (* -2.0 (- a b))) (/ 1.0 (+ a b)))
  (cos (pow (/ (cbrt PI) (cbrt (/ 180.0 angle))) 3.0))))

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 ((sin((((double) M_PI) * (angle / 180.0))) * (-2.0 * (a - b))) / (1.0 / (a + b))) * cos(pow((cbrt(((double) M_PI)) / cbrt((180.0 / angle))), 3.0));
}

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 ((Math.sin((Math.PI * (angle / 180.0))) * (-2.0 * (a - b))) / (1.0 / (a + b))) * Math.cos(Math.pow((Math.cbrt(Math.PI) / Math.cbrt((180.0 / angle))), 3.0));
}

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(Float64(Float64(sin(Float64(pi * Float64(angle / 180.0))) * Float64(-2.0 * Float64(a - b))) / Float64(1.0 / Float64(a + b))) * cos((Float64(cbrt(pi) / cbrt(Float64(180.0 / angle))) ^ 3.0)))
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[(N[(N[(N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * N[(a - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(a + b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[N[Power[N[(N[Power[Pi, 1/3], $MachinePrecision] / N[Power[N[(180.0 / angle), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.4

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

2. Simplified 31.4

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

   [Start] 31.4	\[ \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.4	\[ \left(\color{blue}{\left(\left({b}^{2} - {a}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub-neg [=>] 31.4	\[ \left(\left(\color{blue}{\left({b}^{2} + \left(-{a}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 31.4	\[ \left(\left(\color{blue}{\left(\left(-{a}^{2}\right) + {b}^{2}\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   neg-sub0 [=>] 31.4	\[ \left(\left(\left(\color{blue}{\left(0 - {a}^{2}\right)} + {b}^{2}\right) \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-+l- [=>] 31.4	\[ \left(\left(\color{blue}{\left(0 - \left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   sub0-neg [=>] 31.4	\[ \left(\left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right)\right)} \cdot 2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-lft-neg-out [=>] 31.4	\[ \left(\color{blue}{\left(-\left({a}^{2} - {b}^{2}\right) \cdot 2\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   distribute-rgt-neg-in [=>] 31.4	\[ \left(\color{blue}{\left(\left({a}^{2} - {b}^{2}\right) \cdot \left(-2\right)\right)} \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.4	\[ \left(\left(\left(\color{blue}{a \cdot a} - {b}^{2}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   unpow2 [=>] 31.4	\[ \left(\left(\left(a \cdot a - \color{blue}{b \cdot b}\right) \cdot \left(-2\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   metadata-eval [=>] 31.4	\[ \left(\left(\left(a \cdot a - b \cdot b\right) \cdot \color{blue}{-2}\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

3. Applied egg-rr 34.9

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

4. Simplified 21.4

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

   [Start] 34.9	\[ \frac{\left(a \cdot a - b \cdot b\right) \cdot \left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right)}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 34.9	\[ \frac{\color{blue}{\left(\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)\right) \cdot \left(a \cdot a - b \cdot b\right)}}{a - b} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-/l* [=>] 31.5	\[ \color{blue}{\frac{\left(a - b\right) \cdot \left(-2 \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}{\frac{a - b}{a \cdot a - b \cdot b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-*r* [=>] 31.5	\[ \frac{\color{blue}{\left(\left(a - b\right) \cdot -2\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.5	\[ \frac{\color{blue}{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(a - b\right) \cdot -2\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.5	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \color{blue}{\left(-2 \cdot \left(a - b\right)\right)}}{\frac{a - b}{a \cdot a - b \cdot b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   difference-of-squares [=>] 31.5	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{a - b}{\color{blue}{\left(a + b\right) \cdot \left(a - b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-commutative [=>] 31.5	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{a - b}{\color{blue}{\left(a - b\right) \cdot \left(a + b\right)}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   associate-/r* [=>] 21.4	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\color{blue}{\frac{\frac{a - b}{a - b}}{a + b}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   *-inverses [=>] 21.4	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{\color{blue}{1}}{a + b}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]
   +-commutative [=>] 21.4	\[ \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{\color{blue}{b + a}}} \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \]

5. Applied egg-rr 21.5

   \[\leadsto \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{b + a}} \cdot \cos \color{blue}{\left({\left(\sqrt[3]{\pi \cdot \frac{angle}{180}}\right)}^{3}\right)} \]

6. Applied egg-rr 21.5

   \[\leadsto \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{b + a}} \cdot \cos \left({\color{blue}{\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{180}{angle}}}\right)}}^{3}\right) \]

7. Final simplification 21.5

   \[\leadsto \frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{a + b}} \cdot \cos \left({\left(\frac{\sqrt[3]{\pi}}{\sqrt[3]{\frac{180}{angle}}}\right)}^{3}\right) \]

Alternatives

Alternative 1
Error	21.5
Cost	39808

\[\begin{array}{l} t_0 := \pi \cdot \frac{angle}{180}\\ \frac{\sin t_0 \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{a + b}} \cdot \cos \left({\left(\sqrt[3]{t_0}\right)}^{3}\right) \end{array} \]

Alternative 2
Error	21.4
Cost	27072

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

Alternative 3
Error	21.4
Cost	26944

\[\frac{\sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(-2 \cdot \left(a - b\right)\right)}{\frac{1}{a + b}} \cdot \cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right) \]

Alternative 4
Error	22.7
Cost	13833

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -20000 \lor \neg \left(\frac{angle}{180} \leq 10^{+20}\right):\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 5
Error	22.0
Cost	13832

\[\begin{array}{l} \mathbf{if}\;\frac{angle}{180} \leq -4 \cdot 10^{-25}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin \left(\frac{\pi \cdot angle}{90}\right)\\ \mathbf{elif}\;\frac{angle}{180} \leq 10^{+20}:\\ \;\;\;\;\left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \mathbf{else}:\\ \;\;\;\;b \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \end{array} \]

Alternative 6
Error	22.4
Cost	13824

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

Alternative 7
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -0.38:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;\left(\left(angle \cdot \left(a + b\right)\right) \cdot \left(b - a\right)\right) \cdot \left(\pi \cdot 0.011111111111111112\right)\\ \end{array} \]

Alternative 8
Error	37.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -7.8 \cdot 10^{-70}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 1.85 \cdot 10^{-41}:\\ \;\;\;\;\left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 9
Error	37.6
Cost	7176

\[\begin{array}{l} \mathbf{if}\;b \leq -1.05 \cdot 10^{-70}:\\ \;\;\;\;angle \cdot \left(b \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot b\right)\right)\right)\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-40}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;angle \cdot \left(\pi \cdot \left(\left(b \cdot b\right) \cdot 0.011111111111111112\right)\right)\\ \end{array} \]

Alternative 10
Error	34.3
Cost	7168

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

Alternative 11
Error	34.4
Cost	7168

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

Alternative 12
Error	43.4
Cost	6912

\[0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Alternative 13
Error	43.4
Cost	6912

\[angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right) \]

Reproduce

herbie shell --seed 2023187 
(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)))))