ab-angle->ABCF A

Average Error: 20.4 → 20.4

Time: 18.4s

Precision: binary64

Cost: 26240

Math

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

↓

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

FPCore

(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 (* (sin (* PI (* 0.005555555555555556 angle))) a) 2.0) (pow b 2.0)))

C

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((sin((((double) M_PI) * (0.005555555555555556 * angle))) * a), 2.0) + pow(b, 2.0);
}

Java

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((Math.sin((Math.PI * (0.005555555555555556 * angle))) * a), 2.0) + Math.pow(b, 2.0);
}

Python

def code(a, b, 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)

↓

def code(a, b, angle):
	return math.pow((math.sin((math.pi * (0.005555555555555556 * angle))) * a), 2.0) + math.pow(b, 2.0)

Julia

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(sin(Float64(pi * Float64(0.005555555555555556 * angle))) * a) ^ 2.0) + (b ^ 2.0))
end

MATLAB

function tmp = code(a, b, angle)
	tmp = ((a * sin(((angle / 180.0) * pi))) ^ 2.0) + ((b * cos(((angle / 180.0) * pi))) ^ 2.0);
end

↓

function tmp = code(a, b, angle)
	tmp = ((sin((pi * (0.005555555555555556 * angle))) * a) ^ 2.0) + (b ^ 2.0);
end

Wolfram

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[(N[Sin[N[(Pi * N[(0.005555555555555556 * angle), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * a), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.4

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

2. Simplified 20.3

   \[\leadsto \color{blue}{{\left(a \cdot \sin \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(angle \cdot \frac{\pi}{180}\right)\right)}^{2}} \]
   Proof
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 angle (/.f64 (PI.f64) 180)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 angle (/.f64 (PI.f64) 180)))) 2)): 0 points increase in error, 0 points decrease in error
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 angle (PI.f64)) 180)))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 angle (/.f64 (PI.f64) 180)))) 2)): 12 points increase in error, 10 points decrease in error
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 angle 180) (PI.f64))))) 2) (pow.f64 (*.f64 b (cos.f64 (*.f64 angle (/.f64 (PI.f64) 180)))) 2)): 14 points increase in error, 15 points decrease in error
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 angle (PI.f64)) 180)))) 2)): 6 points increase in error, 7 points decrease in error
   (+.f64 (pow.f64 (*.f64 a (sin.f64 (*.f64 (/.f64 angle 180) (PI.f64)))) 2) (pow.f64 (*.f64 b (cos.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 angle 180) (PI.f64))))) 2)): 8 points increase in error, 7 points decrease in error

3. Taylor expanded in angle around 0 20.4

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

4. Applied egg-rr 20.4

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

5. Taylor expanded in a around 0 20.4

   \[\leadsto {\color{blue}{\left(a \cdot \sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]

6. Simplified 20.4

   \[\leadsto {\color{blue}{\left(\sin \left(\pi \cdot \left(0.005555555555555556 \cdot angle\right)\right) \cdot a\right)}}^{2} + {\left(b \cdot 1\right)}^{2} \]
   Proof
   (*.f64 (sin.f64 (*.f64 (PI.f64) (*.f64 1/180 angle))) a): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 1/180 angle) (PI.f64)))) a): 0 points increase in error, 0 points decrease in error
   (*.f64 (sin.f64 (Rewrite<= associate-*r*_binary64 (*.f64 1/180 (*.f64 angle (PI.f64))))) a): 26 points increase in error, 47 points decrease in error
   (Rewrite<= *-commutative_binary64 (*.f64 a (sin.f64 (*.f64 1/180 (*.f64 angle (PI.f64)))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 20.4

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

Alternatives

Alternative 1
Error	26.2
Cost	20360

\[\begin{array}{l} \mathbf{if}\;b \leq 2.065436357235566 \cdot 10^{-278}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}\\ \mathbf{elif}\;b \leq 6.0546078552434845 \cdot 10^{-115}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)}^{2}\\ \end{array} \]

Alternative 2
Error	26.2
Cost	20360

\[\begin{array}{l} t_0 := \pi \cdot \left(angle \cdot a\right)\\ \mathbf{if}\;b \leq 2.065436357235566 \cdot 10^{-278}:\\ \;\;\;\;{b}^{2} + \left(\left(angle \cdot a\right) \cdot \left(\pi \cdot t_0\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{elif}\;b \leq 6.0546078552434845 \cdot 10^{-115}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left({\pi}^{2} \cdot \left(angle \cdot \left(a \cdot \left(angle \cdot a\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + {\left(0.005555555555555556 \cdot t_0\right)}^{2}\\ \end{array} \]

Alternative 3
Error	23.7
Cost	20360

\[\begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{+155}:\\ \;\;\;\;{b}^{2} + {\left(a \cdot \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)}^{2}\\ \mathbf{elif}\;a \leq 1.3178122390825705 \cdot 10^{+141}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot \left(angle \cdot \left(\left(a \cdot a\right) \cdot \left(angle \cdot {\pi}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2}\\ \end{array} \]

Alternative 4
Error	23.1
Cost	20360

\[\begin{array}{l} t_0 := {b}^{2} + {\left(0.005555555555555556 \cdot \left(angle \cdot \left(\left(1 + \pi \cdot a\right) + -1\right)\right)\right)}^{2}\\ \mathbf{if}\;angle \leq -5 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 10^{+20}:\\ \;\;\;\;{b}^{2} + \left(\left(angle \cdot a\right) \cdot \left(\pi \cdot \left(\pi \cdot \left(angle \cdot a\right)\right)\right)\right) \cdot 3.08641975308642 \cdot 10^{-5}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	25.9
Cost	19840

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

Alternative 6
Error	26.0
Cost	19840

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(angle \cdot \left(\pi \cdot a\right)\right)}^{2} \]

Alternative 7
Error	26.0
Cost	19840

\[{b}^{2} + 3.08641975308642 \cdot 10^{-5} \cdot {\left(\pi \cdot \left(angle \cdot a\right)\right)}^{2} \]

Reproduce

herbie shell --seed 2022300 
(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)))