ab-angle->ABCF A

?

Percentage Accurate: 79.8% → 79.8%
Time: 26.4s
Precision: binary64
Cost: 52224

?

\[{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} \]
↓
\[{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{{\left(\sqrt[3]{angle \cdot \pi}\right)}^{3}}{180}\right)\right)}^{2} \]
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (* (/ angle 180.0) PI))) 2.0)
  (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))
↓
(FPCore (a b angle)
 :precision binary64
 (+
  (pow (* a (sin (/ (* angle PI) 180.0))) 2.0)
  (pow (* b (cos (/ (pow (cbrt (* angle PI)) 3.0) 180.0))) 2.0)))
double code(double a, double b, double angle) {
	return pow((a * sin(((angle / 180.0) * ((double) M_PI)))), 2.0) + pow((b * cos(((angle / 180.0) * ((double) M_PI)))), 2.0);
}
↓
double code(double a, double b, double angle) {
	return pow((a * sin(((angle * ((double) M_PI)) / 180.0))), 2.0) + pow((b * cos((pow(cbrt((angle * ((double) M_PI))), 3.0) / 180.0))), 2.0);
}

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle / 180.0) * Math.PI))), 2.0) + Math.pow((b * Math.cos(((angle / 180.0) * Math.PI))), 2.0);
}
↓
public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.sin(((angle * Math.PI) / 180.0))), 2.0) + Math.pow((b * Math.cos((Math.pow(Math.cbrt((angle * Math.PI)), 3.0) / 180.0))), 2.0);
}

function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle / 180.0) * pi))) ^ 2.0) + (Float64(b * cos(Float64(Float64(angle / 180.0) * pi))) ^ 2.0))
end
↓
function code(a, b, angle)
	return Float64((Float64(a * sin(Float64(Float64(angle * pi) / 180.0))) ^ 2.0) + (Float64(b * cos(Float64((cbrt(Float64(angle * pi)) ^ 3.0) / 180.0))) ^ 2.0))
end

code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_, angle_] := N[(N[Power[N[(a * N[Sin[N[(N[(angle * Pi), $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Cos[N[(N[Power[N[Power[N[(angle * Pi), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] / 180.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

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

Local Percentage Accuracy?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Herbie found 11 alternatives:

AlternativeAccuracySpeedup

Accuracy vs Speed

The accuracy (vertical axis) and speed (horizontal axis) of each of Herbie's proposed alternatives. Up and to the right is better. Each dot represents an alternative program; the red square represents the initial program.

Bogosity?

Bogosity

Try it out?

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation?

  1. Initial program 81.5%

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

  2. Simplified81.6%

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

    [Start]81.5

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

    associate-*l/ [=>]81.5

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

    associate-*l/ [=>]81.6

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

  3. Applied egg-rr81.6%

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

    [Start]81.6

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

    add-cube-cbrt [=>]81.6

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

    pow3 [=>]81.6

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

  4. Final simplification81.6%

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

Alternatives

Alternative 1
Accuracy79.9%
Cost39360
\[\begin{array}{l} t_0 := angle \cdot \frac{\pi}{180}\\ {\left(a \cdot \sin t_0\right)}^{2} + {\left(b \cdot \cos t_0\right)}^{2} \end{array} \]
Alternative 2
Accuracy79.8%
Cost39360
\[{\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
Alternative 3
Accuracy79.8%
Cost39360
\[{\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2} \]
Alternative 4
Accuracy79.7%
Cost26240
\[{\left(a \cdot \sin \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2} + {b}^{2} \]
Alternative 5
Accuracy74.4%
Cost19840
\[{b}^{2} + {\left(angle \cdot \left(a \cdot \pi\right)\right)}^{2} \cdot 3.08641975308642 \cdot 10^{-5} \]
Alternative 6
Accuracy74.5%
Cost19840
\[{b}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)}^{2} \]
Alternative 7
Accuracy74.5%
Cost19840
\[{b}^{2} + {\left(a \cdot \left(\left(angle \cdot \pi\right) \cdot 0.005555555555555556\right)\right)}^{2} \]
Alternative 8
Accuracy74.5%
Cost19840
\[{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
Alternative 9
Accuracy74.5%
Cost19840
\[{b}^{2} + {\left(\pi \cdot \left(a \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)}^{2} \]
Alternative 10
Accuracy74.5%
Cost19840
\[{b}^{2} + {\left(\left(a \cdot \pi\right) \cdot \left(angle \cdot 0.005555555555555556\right)\right)}^{2} \]

Reproduce?

herbie shell --seed 2023160 
(FPCore (a b angle)
  :name "ab-angle->ABCF A"
  :precision binary64
  (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)))