ab-angle->ABCF B

Average Error: 30.7 → 21.1

Time: 19.0s

Precision: binary64

Cost: 13824

Math

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

↓

\[2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \]

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) (* (* 0.5 (- b a)) (sin (* PI (* angle 0.011111111111111112)))))))

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) * ((0.5 * (b - a)) * sin((((double) M_PI) * (angle * 0.011111111111111112)))));
}

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) * ((0.5 * (b - a)) * Math.sin((Math.PI * (angle * 0.011111111111111112)))));
}

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) * ((0.5 * (b - a)) * math.sin((math.pi * (angle * 0.011111111111111112)))))

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(2.0 * Float64(Float64(b + a) * Float64(Float64(0.5 * Float64(b - a)) * sin(Float64(pi * Float64(angle * 0.011111111111111112))))))
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) * ((0.5 * (b - a)) * sin((pi * (angle * 0.011111111111111112)))));
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[(2.0 * N[(N[(b + a), $MachinePrecision] * N[(N[(0.5 * N[(b - a), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(Pi * N[(angle * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 30.7

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

2. Simplified 30.7

   \[\leadsto \color{blue}{2 \cdot \left(\left(b \cdot b - a \cdot a\right) \cdot \left(\sin \left(\frac{\pi \cdot angle}{180}\right) \cdot \cos \left(\frac{\pi \cdot angle}{180}\right)\right)\right)} \]
   Proof
   (*.f64 2 (*.f64 (-.f64 (*.f64 b b) (*.f64 a a)) (*.f64 (sin.f64 (/.f64 (*.f64 (PI.f64) angle) 180)) (cos.f64 (/.f64 (*.f64 (PI.f64) angle) 180))))): 0 points increase in error, 0 points decrease in error
   (*.f64 2 (*.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 b 2)) (*.f64 a a)) (*.f64 (sin.f64 (/.f64 (*.f64 (PI.f64) angle) 180)) (cos.f64 (/.f64 (*.f64 (PI.f64) angle) 180))))): 0 points increase in error, 0 points decrease in error
   (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (Rewrite<= unpow2_binary64 (pow.f64 a 2))) (*.f64 (sin.f64 (/.f64 (*.f64 (PI.f64) angle) 180)) (cos.f64 (/.f64 (*.f64 (PI.f64) angle) 180))))): 0 points increase in error, 0 points decrease in error
   (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (*.f64 (sin.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (/.f64 (*.f64 (PI.f64) angle) 180))))): 26 points increase in error, 27 points decrease in error
   (*.f64 2 (*.f64 (-.f64 (pow.f64 b 2) (pow.f64 a 2)) (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (cos.f64 (Rewrite<= associate-*r/_binary64 (*.f64 (PI.f64) (/.f64 angle 180))))))): 13 points increase in error, 17 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (*.f64 (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180)))))): 1 points increase in error, 1 points decrease in error
   (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (*.f64 2 (-.f64 (pow.f64 b 2) (pow.f64 a 2))) (sin.f64 (*.f64 (PI.f64) (/.f64 angle 180)))) (cos.f64 (*.f64 (PI.f64) (/.f64 angle 180))))): 19 points increase in error, 13 points decrease in error

3. Applied egg-rr 26.4

   \[\leadsto 2 \cdot \color{blue}{\mathsf{fma}\left(b + a, \left(b - a\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 0\right) \cdot 0.5\right), \mathsf{fma}\left(a, -a, a \cdot a\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 0\right) \cdot 0.5\right)\right)} \]

4. Taylor expanded in a around 0 21.1

   \[\leadsto 2 \cdot \mathsf{fma}\left(b + a, \left(b - a\right) \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 0\right) \cdot 0.5\right), \color{blue}{0} \cdot \left(\left(\sin \left(\left(\pi \cdot angle\right) \cdot 0.011111111111111112\right) + 0\right) \cdot 0.5\right)\right) \]

5. Applied egg-rr 21.1

   \[\leadsto 2 \cdot \color{blue}{\left(\left(b + a\right) \cdot \left(\sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right) \cdot \left(0.5 \cdot \left(b - a\right)\right)\right)\right)} \]

6. Final simplification 21.1

   \[\leadsto 2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right) \]

Alternatives

Alternative 1
Error	21.4
Cost	14088

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -1.28 \cdot 10^{-6}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin t_0\\ \mathbf{elif}\;angle \leq 0.0033:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot \left(\left(b + a\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 2
Error	21.4
Cost	14088

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -1.6 \cdot 10^{-7}:\\ \;\;\;\;\left(b \cdot b - a \cdot a\right) \cdot \sin t_0\\ \mathbf{elif}\;angle \leq 0.00035:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right) \cdot \left(\left(b + a\right) \cdot 0.5\right)\right)\right)\\ \end{array} \]

Alternative 3
Error	22.3
Cost	13832

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -0.0036:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \left(a \cdot \left(a \cdot \sin t_0\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0024:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]

Alternative 4
Error	22.3
Cost	13832

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -0.00085:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \left(a \cdot \left(a \cdot \sin t_0\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0013:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left(\left(b \cdot b\right) \cdot \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\right)\right)\\ \end{array} \]

Alternative 5
Error	22.3
Cost	13832

\[\begin{array}{l} t_0 := \sin \left(\pi \cdot \left(angle \cdot 0.011111111111111112\right)\right)\\ \mathbf{if}\;angle \leq -0.0056:\\ \;\;\;\;2 \cdot \left(t_0 \cdot \left(a \cdot \left(a \cdot -0.5\right)\right)\right)\\ \mathbf{elif}\;angle \leq 0.0032:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(0.5 \cdot \left(\left(b \cdot b\right) \cdot t_0\right)\right)\\ \end{array} \]

Alternative 6
Error	21.2
Cost	13832

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ t_1 := \left(b \cdot b - a \cdot a\right) \cdot \sin t_0\\ \mathbf{if}\;angle \leq -8 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;angle \leq 3.9 \cdot 10^{-32}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	22.9
Cost	13704

\[\begin{array}{l} t_0 := \sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\\ \mathbf{if}\;angle \leq -210000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 1.95 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	22.7
Cost	13704

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(\pi \cdot angle\right)\\ \mathbf{if}\;angle \leq -9.8 \cdot 10^{-5}:\\ \;\;\;\;2 \cdot \left(-0.5 \cdot \left(a \cdot \left(a \cdot \sin t_0\right)\right)\right)\\ \mathbf{elif}\;angle \leq 2.8 \cdot 10^{+34}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi \cdot angle\right) \cdot 0.005555555555555556\right) \cdot \left(\left(a \cdot a\right) \cdot -2\right)\\ \end{array} \]

Alternative 9
Error	29.2
Cost	7696

\[\begin{array}{l} t_0 := 0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b - a \cdot a\right)\right)\right)\\ t_1 := \left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(b \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq -7.2 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -9.5 \cdot 10^{-237}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+152}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	29.0
Cost	7696

\[\begin{array}{l} t_0 := b \cdot b - a \cdot a\\ t_1 := \left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(b \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq -2.65 \cdot 10^{+153}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-48}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\pi \cdot \left(angle \cdot t_0\right)\right)\\ \mathbf{elif}\;b \leq -6.6 \cdot 10^{-239}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{elif}\;b \leq 3 \cdot 10^{+153}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	24.9
Cost	7556

\[\begin{array}{l} \mathbf{if}\;angle \leq -1.7 \cdot 10^{+19}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(\left(0.5 \cdot \left(b - a\right)\right) \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \end{array} \]

Alternative 12
Error	25.0
Cost	7428

\[\begin{array}{l} \mathbf{if}\;angle \leq -5 \cdot 10^{+19}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(b + a\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(\pi \cdot \left(b - a\right)\right)\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	25.0
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq -5.2 \cdot 10^{+19}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(b \cdot \left(b \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;0.011111111111111112 \cdot \left(\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(\pi \cdot angle\right)\right)\right)\\ \end{array} \]

Alternative 14
Error	38.5
Cost	7176

\[\begin{array}{l} t_0 := \left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(b \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq -4.2 \cdot 10^{+41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+140}:\\ \;\;\;\;\left(\pi \cdot 0.011111111111111112\right) \cdot \left(angle \cdot \left(b \cdot b\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	32.6
Cost	7176

\[\begin{array}{l} t_0 := \left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(b \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq -1.5 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-72}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(a \cdot a\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	32.0
Cost	7176

\[\begin{array}{l} t_0 := \left(angle \cdot \left(b \cdot \pi\right)\right) \cdot \left(b \cdot 0.011111111111111112\right)\\ \mathbf{if}\;b \leq -2.25 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-67}:\\ \;\;\;\;\left(\pi \cdot \left(a \cdot \left(a \cdot angle\right)\right)\right) \cdot -0.011111111111111112\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 17
Error	43.4
Cost	6912

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

Reproduce

herbie shell --seed 2022330 
(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)))))