ab-angle->ABCF C

Average Error: 20.7 → 20.7

Time: 23.4s

Precision: binary64

Cost: 65024

Math

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

↓

\[{\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

FPCore

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

↓

(FPCore (a b angle)
 :precision binary64
 (+
  (pow
   (*
    a
    (expm1
     (log1p (pow (cbrt (cos (* angle (* PI 0.005555555555555556)))) 3.0))))
   2.0)
  (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))

C

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

↓

double code(double a, double b, double angle) {
	return pow((a * expm1(log1p(pow(cbrt(cos((angle * (((double) M_PI) * 0.005555555555555556)))), 3.0)))), 2.0) + pow((b * sin((((double) M_PI) * (angle / 180.0)))), 2.0);
}

Java

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

↓

public static double code(double a, double b, double angle) {
	return Math.pow((a * Math.expm1(Math.log1p(Math.pow(Math.cbrt(Math.cos((angle * (Math.PI * 0.005555555555555556)))), 3.0)))), 2.0) + Math.pow((b * Math.sin((Math.PI * (angle / 180.0)))), 2.0);
}

Julia

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

↓

function code(a, b, angle)
	return Float64((Float64(a * expm1(log1p((cbrt(cos(Float64(angle * Float64(pi * 0.005555555555555556)))) ^ 3.0)))) ^ 2.0) + (Float64(b * sin(Float64(pi * Float64(angle / 180.0)))) ^ 2.0))
end

Wolfram

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

↓

code[a_, b_, angle_] := N[(N[Power[N[(a * N[(Exp[N[Log[1 + N[Power[N[Power[N[Cos[N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * N[Sin[N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.7

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

2. Applied egg-rr 20.7

   \[\leadsto {\left(a \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right)\right)\right)}\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

3. Applied egg-rr 20.7

   \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)}\right)}^{3}}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

4. Taylor expanded in angle around 0 20.7

   \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}}\right)}^{3}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

5. Simplified 20.7

   \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \color{blue}{\left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}}\right)}^{3}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]
   Proof
   (*.f64 angle (*.f64 (PI.f64) 1/180)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/180)): 45 points increase in error, 32 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 1/180 (*.f64 angle (PI.f64)))): 0 points increase in error, 0 points decrease in error

6. Final simplification 20.7

   \[\leadsto {\left(a \cdot \mathsf{expm1}\left(\mathsf{log1p}\left({\left(\sqrt[3]{\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)}\right)}^{3}\right)\right)\right)}^{2} + {\left(b \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right)}^{2} \]

Alternatives

Alternative 1
Error	20.7
Cost	39488

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

Alternative 2
Error	20.7
Cost	39360

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

Alternative 3
Error	20.7
Cost	39360

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

Alternative 4
Error	20.7
Cost	39360

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

Alternative 5
Error	20.7
Cost	39360

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

Alternative 6
Error	20.8
Cost	26368

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

Alternative 7
Error	20.7
Cost	26240

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

Alternative 8
Error	20.7
Cost	26240

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

Alternative 9
Error	22.4
Cost	20424

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.361841837595009 \cdot 10^{+72}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.3240795603042202 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(angle \cdot b, \left(\pi \cdot 0.005555555555555556\right) \cdot \left(b \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right), a \cdot a\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 10
Error	22.4
Cost	14152

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.361841837595009 \cdot 10^{+72}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.3240795603042202 \cdot 10^{+47}:\\ \;\;\;\;a \cdot a + \left(angle \cdot b\right) \cdot \left(\left(\pi \cdot 0.005555555555555556\right) \cdot \left(angle \cdot \left(b \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 11
Error	22.5
Cost	13768

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.361841837595009 \cdot 10^{+72}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.3240795603042202 \cdot 10^{+47}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 12
Error	22.4
Cost	13768

\[\begin{array}{l} \mathbf{if}\;angle \leq -6.361841837595009 \cdot 10^{+72}:\\ \;\;\;\;a \cdot a\\ \mathbf{elif}\;angle \leq 1.3240795603042202 \cdot 10^{+47}:\\ \;\;\;\;a \cdot a + {\left(b \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;a \cdot a\\ \end{array} \]

Alternative 13
Error	32.4
Cost	192

\[a \cdot a \]

Reproduce

herbie shell --seed 2022300 
(FPCore (a b angle)
  :name "ab-angle->ABCF C"
  :precision binary64
  (+ (pow (* a (cos (* PI (/ angle 180.0)))) 2.0) (pow (* b (sin (* PI (/ angle 180.0)))) 2.0)))