Average Error: 3.9 → 3.3
Time: 13.5s
Precision: binary64
\[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq -\infty:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \end{array}\]
\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}}} \cdot \sin th \leq -\infty:\\
\;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\

\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))))
       (sin th))
      (- INFINITY))
   (* (sin th) (/ (sin ky) (sin kx)))
   (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th))))
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))) * sin(th)) <= -((double) INFINITY)) {
		tmp = sin(th) * (sin(ky) / sin(kx));
	} else {
		tmp = (sin(ky) / sqrt(pow(sin(kx), 2.0) + pow(sin(ky), 2.0))) * sin(th);
	}
	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

Derivation

  1. Split input into 2 regimes

  2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) (sin.f64 th)) < -inf.0

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity_binary6464.0

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

    4. Applied associate-*l*_binary6464.0

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

    5. Simplified64.0

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

    6. Taylor expanded around 0 40.8

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

    if -inf.0 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) 2) (pow.f64 (sin.f64 ky) 2)))) (sin.f64 th))

    1. Initial program 2.2

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

    3. Applied *-un-lft-identity_binary642.2

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

  4. Final simplification3.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq -\infty:\\ \;\;\;\;\sin th \cdot \frac{\sin ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th\\ \end{array}\]

Reproduce

herbie shell --seed 2021175 
(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)))