Average Error: 3.8 → 2.9
Time: 1.1min
Precision: binary64
Cost: 2305
\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + 0.08333333333333333 \cdot \left(ky \cdot \left(kx \cdot kx\right)\right)\right) - 0.16666666666666666 \cdot {ky}^{3}}\\ \end{array}\]
\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th
↓
\begin{array}{l}
\mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\
\;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\

\mathbf{else}:\\
\;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + 0.08333333333333333 \cdot \left(ky \cdot \left(kx \cdot kx\right)\right)\right) - 0.16666666666666666 \cdot {ky}^{3}}\\

\end{array}
(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))
↓
(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1.0)
   (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th))
   (*
    (sin th)
    (/
     (sin ky)
     (-
      (+ ky (* 0.08333333333333333 (* ky (* kx kx))))
      (* 0.16666666666666666 (pow ky 3.0)))))))
double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) * sin(th);
}
↓
double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) <= 1.0) {
		tmp = (sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) * sin(th);
	} else {
		tmp = sin(th) * (sin(ky) / ((ky + (0.08333333333333333 * (ky * (kx * kx)))) - (0.16666666666666666 * pow(ky, 3.0))));
	}
	return tmp;
}

Error

Bits error versus kx

Bits error versus ky

Bits error versus th

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs
Alternative 1
Accuracy3.8
Cost1024
\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]
Alternative 2
Accuracy4.4
Cost1472
\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\left(\sqrt[3]{\sin ky}\right)}^{4} \cdot {\left(\sqrt[3]{\sin ky}\right)}^{2}}} \cdot \sin th\]
Alternative 3
Accuracy33.7
Cost2048
\[\sqrt{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \left(\sqrt{\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th\right)\]

Derivation

  1. Split input into 2 regimes

  2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) < 1

    1. Initial program 2.0

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_782.0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{1 \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \cdot \sin th\]

    4. Applied sqrt-prod_binary64_942.0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th\]

    5. Applied *-un-lft-identity_binary64_782.0

      \[\leadsto \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{1} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]

    6. Applied times-frac_binary64_842.0

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{1}} \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th\]

    7. Applied associate-*l*_binary64_192.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)}\]
    8. Using strategy rm

    9. Applied pow1_binary64_1392.0

      \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\sqrt{\color{blue}{{\left({\sin kx}^{2}\right)}^{1}} + {\sin ky}^{2}}} \cdot \sin th\right)\]

    if 1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2))))

    1. Initial program 63.7

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]
    2. Using strategy rm

    3. Applied *-un-lft-identity_binary64_7863.7

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{1 \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}} \cdot \sin th\]

    4. Applied sqrt-prod_binary64_9463.7

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{1} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}} \cdot \sin th\]

    5. Applied *-un-lft-identity_binary64_7863.7

      \[\leadsto \frac{\color{blue}{1 \cdot \sin ky}}{\sqrt{1} \cdot \sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]

    6. Applied times-frac_binary64_8463.7

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{1}} \cdot \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th\]

    7. Applied associate-*l*_binary64_1963.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\right)}\]

    8. Taylor expanded around 0 32.7

      \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\color{blue}{\left(0.08333333333333333 \cdot \left({kx}^{2} \cdot ky\right) + ky\right) - 0.16666666666666666 \cdot {ky}^{3}}} \cdot \sin th\right)\]

    9. Simplified32.7

      \[\leadsto \frac{1}{\sqrt{1}} \cdot \left(\frac{\sin ky}{\color{blue}{\left(ky + 0.08333333333333333 \cdot \left(ky \cdot \left(kx \cdot kx\right)\right)\right) - 0.16666666666666666 \cdot {ky}^{3}}} \cdot \sin th\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification2.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\left(ky + 0.08333333333333333 \cdot \left(ky \cdot \left(kx \cdot kx\right)\right)\right) - 0.16666666666666666 \cdot {ky}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020322 
(FPCore (kx ky th)
  :name "Toniolo and Linder, Equation (3b), real"
  :precision binary64
  (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))