ab-angle->ABCF B

Average Error: 31.3 → 21.3

Time: 17.9s

Precision: binary64

Cost: 13568

Math

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

↓

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

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
 (* (+ b a) (* (- b a) (sin (* angle (* PI 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 (b + a) * ((b - a) * sin((angle * (((double) M_PI) * 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 (b + a) * ((b - a) * Math.sin((angle * (Math.PI * 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 (b + a) * ((b - a) * math.sin((angle * (math.pi * 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(Float64(b + a) * Float64(Float64(b - a) * sin(Float64(angle * Float64(pi * 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 = (b + a) * ((b - a) * sin((angle * (pi * 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[(N[(b + a), $MachinePrecision] * N[(N[(b - a), $MachinePrecision] * N[Sin[N[(angle * N[(Pi * 0.011111111111111112), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3

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

2. Simplified 31.3

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

3. Applied egg-rr 36.6

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot b - a \cdot a\right) \cdot \sin \left(2 \cdot \left(\pi \cdot \left(angle \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]

4. Applied egg-rr 21.3

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

5. Taylor expanded in angle around inf 21.3

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

6. Simplified 21.3

   \[\leadsto \left(b + a\right) \cdot \left(\left(b - a\right) \cdot \color{blue}{\sin \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)}\right) \]
   Proof
   (sin.f64 (*.f64 angle (*.f64 (PI.f64) 1/90))): 0 points increase in error, 0 points decrease in error
   (sin.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 angle (PI.f64)) 1/90))): 28 points increase in error, 33 points decrease in error
   (sin.f64 (Rewrite<= *-commutative_binary64 (*.f64 1/90 (*.f64 angle (PI.f64))))): 0 points increase in error, 0 points decrease in error

7. Final simplification 21.3

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

Alternatives

Alternative 1
Error	22.8
Cost	13768

\[\begin{array}{l} t_0 := \left(b + a\right) \cdot \left(a \cdot \left(-\sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\right)\\ \mathbf{if}\;angle \leq -0.36:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 2.25 \cdot 10^{-14}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	22.7
Cost	13704

\[\begin{array}{l} t_0 := \left(b + a\right) \cdot \left(b \cdot \sin \left(0.011111111111111112 \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{if}\;angle \leq -0.032:\\ \;\;\;\;t_0\\ \mathbf{elif}\;angle \leq 22000000000:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	25.0
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 9.2 \cdot 10^{+26}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(\left(\left(b + a\right) \cdot angle\right) \cdot \left(\pi \cdot \left(a - b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + \pi \cdot \left(a \cdot a\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 4
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 1.36 \cdot 10^{+26}:\\ \;\;\;\;\left(b + a\right) \cdot \left(0.011111111111111112 \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + \pi \cdot \left(a \cdot a\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 5
Error	24.9
Cost	7300

\[\begin{array}{l} \mathbf{if}\;angle \leq 1.45 \cdot 10^{+26}:\\ \;\;\;\;\left(b + a\right) \cdot \left(\left(b - a\right) \cdot \left(angle \cdot \left(\pi \cdot 0.011111111111111112\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(angle \cdot \left(\left(1 + \pi \cdot \left(a \cdot a\right)\right) + -1\right)\right)\\ \end{array} \]

Alternative 6
Error	32.5
Cost	7176

\[\begin{array}{l} t_0 := -0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-55}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	32.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -3.6 \cdot 10^{-31}:\\ \;\;\;\;a \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(a \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-53}:\\ \;\;\;\;0.011111111111111112 \cdot \left(angle \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 8
Error	32.5
Cost	7176

\[\begin{array}{l} \mathbf{if}\;a \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;a \cdot \left(\left(angle \cdot -0.011111111111111112\right) \cdot \left(a \cdot \pi\right)\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-52}:\\ \;\;\;\;angle \cdot \left(0.011111111111111112 \cdot \left(\pi \cdot \left(b \cdot b\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.011111111111111112 \cdot \left(a \cdot \left(angle \cdot \left(a \cdot \pi\right)\right)\right)\\ \end{array} \]

Alternative 9
Error	39.9
Cost	6912

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

Reproduce

herbie shell --seed 2022338 
(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)))))