ab-angle->ABCF B

Average Error: 31.8 → 26.4

Time: 13.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := angle \cdot \left(\pi \cdot 0.005555555555555556\right)\\ t_1 := 0.005555555555555556 \cdot \left(\pi \cdot angle\right)\\ t_2 := \pi \cdot \frac{angle}{180}\\ t_3 := \left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin t_2\right) \cdot \cos t_2\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\mathsf{fma}\left({\pi}^{2}, -1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), 1\right) \cdot \left(b \cdot \sin t_0\right)\right)\right)\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+301}:\\ \;\;\;\;-2 \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \left(\cos t_1 \cdot \sin t_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\cos t_0 \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]

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
 (let* ((t_0 (* angle (* PI 0.005555555555555556)))
        (t_1 (* 0.005555555555555556 (* PI angle)))
        (t_2 (* PI (/ angle 180.0)))
        (t_3 (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin t_2)) (cos t_2))))
   (if (<= t_3 (- INFINITY))
     (*
      2.0
      (*
       b
       (*
        (fma (pow PI 2.0) (* -1.54320987654321e-5 (* angle angle)) 1.0)
        (* b (sin t_0)))))
     (if (<= t_3 2e+301)
       (* -2.0 (* (- (pow a 2.0) (pow b 2.0)) (* (cos t_1) (sin t_1))))
       (*
        2.0
        (* b (* (cos t_0) (* 0.005555555555555556 (* angle (* b PI))))))))))

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) {
	double t_0 = angle * (((double) M_PI) * 0.005555555555555556);
	double t_1 = 0.005555555555555556 * (((double) M_PI) * angle);
	double t_2 = ((double) M_PI) * (angle / 180.0);
	double t_3 = ((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * sin(t_2)) * cos(t_2);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = 2.0 * (b * (fma(pow(((double) M_PI), 2.0), (-1.54320987654321e-5 * (angle * angle)), 1.0) * (b * sin(t_0))));
	} else if (t_3 <= 2e+301) {
		tmp = -2.0 * ((pow(a, 2.0) - pow(b, 2.0)) * (cos(t_1) * sin(t_1)));
	} else {
		tmp = 2.0 * (b * (cos(t_0) * (0.005555555555555556 * (angle * (b * ((double) M_PI))))));
	}
	return tmp;
}

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)
	t_0 = Float64(angle * Float64(pi * 0.005555555555555556))
	t_1 = Float64(0.005555555555555556 * Float64(pi * angle))
	t_2 = Float64(pi * Float64(angle / 180.0))
	t_3 = Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * sin(t_2)) * cos(t_2))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(2.0 * Float64(b * Float64(fma((pi ^ 2.0), Float64(-1.54320987654321e-5 * Float64(angle * angle)), 1.0) * Float64(b * sin(t_0)))));
	elseif (t_3 <= 2e+301)
		tmp = Float64(-2.0 * Float64(Float64((a ^ 2.0) - (b ^ 2.0)) * Float64(cos(t_1) * sin(t_1))));
	else
		tmp = Float64(2.0 * Float64(b * Float64(cos(t_0) * Float64(0.005555555555555556 * Float64(angle * Float64(b * pi))))));
	end
	return tmp
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_] := Block[{t$95$0 = N[(angle * N[(Pi * 0.005555555555555556), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.005555555555555556 * N[(Pi * angle), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(Pi * N[(angle / 180.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(2.0 * N[(b * N[(N[(N[Power[Pi, 2.0], $MachinePrecision] * N[(-1.54320987654321e-5 * N[(angle * angle), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(b * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2e+301], N[(-2.0 * N[(N[(N[Power[a, 2.0], $MachinePrecision] - N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t$95$1], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(b * N[(N[Cos[t$95$0], $MachinePrecision] * N[(0.005555555555555556 * N[(angle * N[(b * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus a

Bits error versus b

Bits error versus angle

Derivation

1. Split input into 3 regimes

2. if (*.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)))) < -inf.0

   1. Initial program 64.0

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

   2. Simplified 64.0

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

   3. Applied egg-rr 64.0

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(a + b, a - b, \mathsf{fma}\left(-b, b, b \cdot b\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. Taylor expanded in a around 0 63.3

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

   5. Simplified 31.5

      \[\leadsto \color{blue}{2 \cdot \left(b \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in angle around 0 31.6

      \[\leadsto 2 \cdot \left(b \cdot \left(\color{blue}{\left(1 + -1.54320987654321 \cdot 10^{-5} \cdot \left({angle}^{2} \cdot {\pi}^{2}\right)\right)} \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]

   7. Simplified 31.6

      \[\leadsto 2 \cdot \left(b \cdot \left(\color{blue}{\mathsf{fma}\left({\pi}^{2}, -1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), 1\right)} \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right) \]

   if -inf.0 < (*.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)))) < 2.00000000000000011e301

   1. Initial program 25.1

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

   2. Simplified 25.1

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

   3. Taylor expanded in angle around inf 25.1

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

   if 2.00000000000000011e301 < (*.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))))

   1. Initial program 63.8

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

   2. Simplified 63.8

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

   3. Applied egg-rr 63.8

      \[\leadsto \sin \left(\pi \cdot \frac{angle}{180}\right) \cdot \left(\left(-2 \cdot \color{blue}{\mathsf{fma}\left(a + b, a - b, \mathsf{fma}\left(-b, b, b \cdot b\right)\right)}\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right)\right) \]

   4. Taylor expanded in a around 0 63.3

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

   5. Simplified 33.8

      \[\leadsto \color{blue}{2 \cdot \left(b \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)} \]

   6. Taylor expanded in angle around 0 34.0

      \[\leadsto 2 \cdot \left(b \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \color{blue}{\left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)}\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq -\infty:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\mathsf{fma}\left({\pi}^{2}, -1.54320987654321 \cdot 10^{-5} \cdot \left(angle \cdot angle\right), 1\right) \cdot \left(b \cdot \sin \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right)\right)\right)\right)\\ \mathbf{elif}\;\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\pi \cdot \frac{angle}{180}\right)\right) \cdot \cos \left(\pi \cdot \frac{angle}{180}\right) \leq 2 \cdot 10^{+301}:\\ \;\;\;\;-2 \cdot \left(\left({a}^{2} - {b}^{2}\right) \cdot \left(\cos \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right) \cdot \sin \left(0.005555555555555556 \cdot \left(\pi \cdot angle\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(b \cdot \left(\cos \left(angle \cdot \left(\pi \cdot 0.005555555555555556\right)\right) \cdot \left(0.005555555555555556 \cdot \left(angle \cdot \left(b \cdot \pi\right)\right)\right)\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022170 
(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)))))