Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.5% → 99.7%

Time: 7.8s

Alternatives: 21

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Initial Program: 93.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ (sin ky) (hypot (sin ky) (sin kx))) (sin th)))

C

double code(double kx, double ky, double th) {
	return (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	return (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 92.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
5. Add Preprocessing

Alternative 2: 86.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(0.008333333333333333 \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.99)
     (*
      (/
       (sin ky)
       (hypot
        (sin ky)
        (*
         (fma
          (- (* 0.008333333333333333 (* kx kx)) 0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (if (<= t_1 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_1 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_1 0.999)
           (*
            (* (sin ky) th)
            (/ 1.0 (sqrt (- 1.0 (* 0.5 (+ (cos (+ kx kx)) (cos (+ ky ky))))))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(((0.008333333333333333 * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_1 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_1 <= 0.999) {
		tmp = (sin(ky) * th) * (1.0 / sqrt((1.0 - (0.5 * (cos((kx + kx)) + cos((ky + ky)))))));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(Float64(Float64(0.008333333333333333 * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_1 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_1 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) * Float64(1.0 / sqrt(Float64(1.0 - Float64(0.5 * Float64(cos(Float64(kx + kx)) + cos(Float64(ky + ky))))))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 82.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + {kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left({kx}^{2} \cdot \left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) + 1\right) \cdot kx\right)} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}\right) \cdot {kx}^{2} + 1\right) \cdot kx\right)} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot {kx}^{2} - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      8. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot \left(kx \cdot kx\right) - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot \left(kx \cdot kx\right) - \frac{1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      10. pow2 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{1}{120} \cdot \left(kx \cdot kx\right) - \frac{1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

      11. lift-*.f64 93.5

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(0.008333333333333333 \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 93.5%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

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

   if 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. sqr-sin-a N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 86.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.99)
     (*
      (*
       (/
        1.0
        (hypot
         (/ 1.0 (/ (fma 0.16666666666666666 (* kx kx) 1.0) kx))
         (sin ky)))
       (sin ky))
      (sin th))
     (if (<= t_1 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_1 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_1 0.999)
           (*
            (* (sin ky) th)
            (/ 1.0 (sqrt (- 1.0 (* 0.5 (+ (cos (+ kx kx)) (cos (+ ky ky))))))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = ((1.0 / hypot((1.0 / (fma(0.16666666666666666, (kx * kx), 1.0) / kx)), sin(ky))) * sin(ky)) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_1 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_1 <= 0.999) {
		tmp = (sin(ky) * th) * (1.0 / sqrt((1.0 - (0.5 * (cos((kx + kx)) + cos((ky + ky)))))));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(Float64(1.0 / hypot(Float64(1.0 / Float64(fma(0.16666666666666666, Float64(kx * kx), 1.0) / kx)), sin(ky))) * sin(ky)) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_1 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_1 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) * Float64(1.0 / sqrt(Float64(1.0 - Float64(0.5 * Float64(cos(Float64(kx + kx)) + cos(Float64(ky + ky))))))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[(1.0 / N[Sqrt[N[(1.0 / N[(N[(0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] / kx), $MachinePrecision]), $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 82.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around inf

      \[\leadsto \color{blue}{\left(\sin ky \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}\right)} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      9. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      11. lift-sin.f64 99.7

          \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      2. unpow1 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left({\sin kx}^{1}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      3. metadata-eval N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left({\sin kx}^{\left(\mathsf{neg}\left(-1\right)\right)}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      4. pow-neg N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{{\sin kx}^{-1}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      5. inv-pow N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1}{\sin kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1}{\sin kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1}{\sin kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      8. lift-sin.f64 99.7

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1}{\sin kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   7. Applied rewrites 99.7%

      \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1}{\sin kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1 + \frac{1}{6} \cdot {kx}^{2}}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{1 + \frac{1}{6} \cdot {kx}^{2}}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      2. +-commutative N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\frac{1}{6} \cdot {kx}^{2} + 1}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{6}, {kx}^{2}, 1\right)}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      4. pow2 N/A

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\mathsf{fma}\left(\frac{1}{6}, kx \cdot kx, 1\right)}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

      5. lift-*.f64 94.8

         \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   10. Applied rewrites 94.8%

       \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\frac{1}{\frac{\mathsf{fma}\left(0.16666666666666666, kx \cdot kx, 1\right)}{kx}}, \sin ky\right)} \cdot \sin ky\right) \cdot \sin th \]

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

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

   if 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. sqr-sin-a N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 86.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.99)
     (*
      (/
       (sin ky)
       (hypot (sin ky) (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
      (sin th))
     (if (<= t_1 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_1 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_1 0.999)
           (*
            (* (sin ky) th)
            (/ 1.0 (sqrt (- 1.0 (* 0.5 (+ (cos (+ kx kx)) (cos (+ ky ky))))))))
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) / hypot(sin(ky), (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	} else if (t_1 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_1 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_1 <= 0.999) {
		tmp = (sin(ky) * th) * (1.0 / sqrt((1.0 - (0.5 * (cos((kx + kx)) + cos((ky + ky)))))));
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.99)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	elseif (t_1 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_1 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_1 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) * Float64(1.0 / sqrt(Float64(1.0 - Float64(0.5 * Float64(cos(Float64(kx + kx)) + cos(Float64(ky + ky))))))));
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 82.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(\frac{-1}{6} \cdot {kx}^{2} + 1\right) \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lift-*.f64 93.2

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 93.2%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

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

   if 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. sqr-sin-a N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}} \]

   if 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 86.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.99)
     t_1
     (if (<= t_2 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_2 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_2 0.999)
           (*
            (* (sin ky) th)
            (/ 1.0 (sqrt (- 1.0 (* 0.5 (+ (cos (+ kx kx)) (cos (+ ky ky))))))))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_2 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_2 <= 0.999) {
		tmp = (sin(ky) * th) * (1.0 / sqrt((1.0 - (0.5 * (cos((kx + kx)) + cos((ky + ky)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else if (t_2 <= 0.4) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.999) {
		tmp = (Math.sin(ky) * th) * (1.0 / Math.sqrt((1.0 - (0.5 * (Math.cos((kx + kx)) + Math.cos((ky + ky)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.99:
		tmp = t_1
	elif t_2 <= -0.05:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	elif t_2 <= 0.4:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.999:
		tmp = (math.sin(ky) * th) * (1.0 / math.sqrt((1.0 - (0.5 * (math.cos((kx + kx)) + math.cos((ky + ky)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_2 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) * Float64(1.0 / sqrt(Float64(1.0 - Float64(0.5 * Float64(cos(Float64(kx + kx)) + cos(Float64(ky + ky))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	elseif (t_2 <= 0.4)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_2 <= 0.999)
		tmp = (sin(ky) * th) * (1.0 / sqrt((1.0 - (0.5 * (cos((kx + kx)) + cos((ky + ky)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99], t$95$1, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 - N[(0.5 * N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999 or 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 96.4%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

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

   if 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. unpow2 N/A

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

      9. sqr-sin-a N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. sqr-sin-a N/A

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

      16. lower--.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lower-cos.f64 N/A

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

      19. lower-*.f64 99.2

          \[\leadsto \frac{\sin ky}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \color{blue}{\left(2 \cdot kx\right)}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot kx\right)\right)}}} \cdot \sin th \]

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\sqrt{1 - 0.5 \cdot \left(\cos \left(kx + kx\right) + \cos \left(ky + ky\right)\right)}}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.99)
     t_1
     (if (<= t_2 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_2 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_2 0.999)
           (* (* (sin ky) th) (/ 1.0 (hypot (sin kx) (sin ky))))
           t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_2 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_2 <= 0.999) {
		tmp = (sin(ky) * th) * (1.0 / hypot(sin(kx), sin(ky)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else if (t_2 <= 0.4) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.999) {
		tmp = (Math.sin(ky) * th) * (1.0 / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_2 <= -0.99:
		tmp = t_1
	elif t_2 <= -0.05:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	elif t_2 <= 0.4:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.999:
		tmp = (math.sin(ky) * th) * (1.0 / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_2 <= 0.999)
		tmp = Float64(Float64(sin(ky) * th) * Float64(1.0 / hypot(sin(kx), sin(ky))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	elseif (t_2 <= 0.4)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_2 <= 0.999)
		tmp = (sin(ky) * th) * (1.0 / hypot(sin(kx), sin(ky)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99], t$95$1, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999 or 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 96.4%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

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

   if 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\left(th \cdot \sin ky\right) \cdot \sqrt{\frac{1}{{\sin kx}^{2} + {\sin ky}^{2}}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 51.0

          \[\leadsto \left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.999:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot \frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{if}\;t\_2 \leq -0.99:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.999:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)))
   (if (<= t_2 -0.99)
     t_1
     (if (<= t_2 -0.05)
       t_3
       (if (<= t_2 0.4)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_2 0.999) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.4) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_2 <= 0.999) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	double tmp;
	if (t_2 <= -0.99) {
		tmp = t_1;
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.4) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.999) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	tmp = 0
	if t_2 <= -0.99:
		tmp = t_1
	elif t_2 <= -0.05:
		tmp = t_3
	elif t_2 <= 0.4:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.999:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	tmp = 0.0
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_2 <= 0.999)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	tmp = 0.0;
	if (t_2 <= -0.99)
		tmp = t_1;
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.4)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_2 <= 0.999)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, If[LessEqual[t$95$2, -0.99], t$95$1, If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.4], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.999], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999 or 0.998999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 84.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 96.4%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003 or 0.40000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.998999999999999999

   1. Initial program 99.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \color{blue}{th} \]
   6. Step-by-step derivation

   7. Applied rewrites 45.1%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.40000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 93.3%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 70.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 0.068:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 0.068)
   (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
   (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.068) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 0.068) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((kx + kx)) * 0.5)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 0.068:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((kx + kx)) * 0.5)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 0.068)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 0.068)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	else
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 0.068], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.068000000000000005

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx}\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 77.9%

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

   if 0.068000000000000005 < kx

   1. Initial program 99.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 10.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 10.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin kx \cdot \color{blue}{\sin kx}}} \cdot \sin th \]

      2. sqr-sin-a-rev N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      7. count-2-rev N/A

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

      8. lower-+.f64 63.5

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th \]

   8. Applied rewrites 63.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - \cos \left(kx + kx\right) \cdot 0.5}}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 66.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.003:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.003)
   (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
   (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.003) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.003) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.003:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.003)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.003)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	else
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.003], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 0.0030000000000000001

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 63.0%

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

   if 0.0030000000000000001 < ky

   1. Initial program 99.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 64.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 64.4%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-+.f64 64.4

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.38:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.38)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th))
     (if (<= t_1 0.01)
       (* (/ (sin ky) (sin kx)) (sin th))
       (if (<= t_1 2.0)
         (sin th)
         (*
          (/
           (sin ky)
           (hypot ky (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.38) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	} else if (t_1 <= 0.01) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_1 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.38)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_1 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.38], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.38

   1. Initial program 86.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 47.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 47.8%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      3. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      7. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}} \cdot \sin th \]

      8. lower-+.f64 47.8

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th \]

   7. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th} \]

   if -0.38 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 64.2

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 64.2%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.1

         \[\leadsto \sin th \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(\frac{-1}{6} \cdot {kx}^{2} + 1\right) \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lift-*.f64 100.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.38:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.6:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.6)
     (* (/ (sin ky) (sqrt t_1)) th)
     (if (<= t_2 0.01)
       (* (/ (sin ky) (sin kx)) (sin th))
       (if (<= t_2 2.0)
         (sin th)
         (*
          (/
           (sin ky)
           (hypot ky (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.6) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else if (t_2 <= 0.01) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.6)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.6], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.599999999999999978

   1. Initial program 85.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 50.1

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 50.1%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.0%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      5. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 31.2

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

   10. Applied rewrites 31.2%

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

   if -0.599999999999999978 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-sin.f64 60.8

         \[\leadsto \frac{\sin ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 60.8%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.1

         \[\leadsto \sin th \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(\frac{-1}{6} \cdot {kx}^{2} + 1\right) \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lift-*.f64 100.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.6:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt t_1)) th)
     (if (<= t_2 0.01)
       (* (/ ky (sin kx)) (sin th))
       (if (<= t_2 2.0)
         (sin th)
         (*
          (/
           (sin ky)
           (hypot ky (* (fma -0.16666666666666666 (* kx kx) 1.0) kx)))
          (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else if (t_2 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, (fma(-0.16666666666666666, (kx * kx), 1.0) * kx))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, Float64(fma(-0.16666666666666666, Float64(kx * kx), 1.0) * kx))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[(N[(-0.16666666666666666 * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 46.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 46.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.3%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      5. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 28.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

   10. Applied rewrites 28.9%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 66.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.1

         \[\leadsto \sin th \]

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\sin th} \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 2.7%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{kx \cdot \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right)}\right)} \cdot \sin th \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(1 + \frac{-1}{6} \cdot {kx}^{2}\right) \cdot \color{blue}{kx}\right)} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left(\frac{-1}{6} \cdot {kx}^{2} + 1\right) \cdot kx\right)} \cdot \sin th \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, {kx}^{2}, 1\right) \cdot kx\right)} \cdot \sin th \]

      5. pow2 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

      6. lift-*.f64 100.0

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\right)} \cdot \sin th \]

   8. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(\frac{-1}{6}, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
   9. Step-by-step derivation

   10. Applied rewrites 100.0%

       \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{ky}, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(-0.16666666666666666, kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt t_1)) th)
     (if (<= t_2 0.01) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else if (t_2 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sin(ky) ** 2.0d0
    t_2 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + t_1))
    if (t_2 <= (-0.05d0)) then
        tmp = (sin(ky) / sqrt(t_1)) * th
    else if (t_2 <= 0.01d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * th;
	} else if (t_2 <= 0.01) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -0.05:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * th
	elif t_2 <= 0.01:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	elseif (t_2 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -0.05)
		tmp = (sin(ky) / sqrt(t_1)) * th;
	elseif (t_2 <= 0.01)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 46.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 46.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.3%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      5. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      6. pow2 N/A

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

      7. lower-pow.f64 N/A

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

      8. lift-sin.f64 28.9

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th \]

   10. Applied rewrites 28.9%

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

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 66.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (* (/ (sin ky) (sqrt (/ 1.0 (/ 1.0 (- 0.5 (* (cos (+ ky ky)) 0.5)))))) th)
     (if (<= t_1 0.01) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (sin(ky) / sqrt((1.0 / (1.0 / (0.5 - (cos((ky + ky)) * 0.5)))))) * th;
	} else if (t_1 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.05d0)) then
        tmp = (sin(ky) / sqrt((1.0d0 / (1.0d0 / (0.5d0 - (cos((ky + ky)) * 0.5d0)))))) * th
    else if (t_1 <= 0.01d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (Math.sin(ky) / Math.sqrt((1.0 / (1.0 / (0.5 - (Math.cos((ky + ky)) * 0.5)))))) * th;
	} else if (t_1 <= 0.01) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.05:
		tmp = (math.sin(ky) / math.sqrt((1.0 / (1.0 / (0.5 - (math.cos((ky + ky)) * 0.5)))))) * th
	elif t_1 <= 0.01:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(1.0 / Float64(1.0 / Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))))) * th);
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (sin(ky) / sqrt((1.0 / (1.0 / (0.5 - (cos((ky + ky)) * 0.5)))))) * th;
	elseif (t_1 <= 0.01)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(1.0 / N[(1.0 / N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 46.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 46.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.3%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      4. lift-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot th \]

      5. sqr-sin-a-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot th \]

      6. pow2 N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin ky}^{\left(\mathsf{neg}\left(-2\right)\right)}}} \cdot th \]

      8. pow-neg N/A

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

      9. metadata-eval N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{{\sin ky}^{\left(\mathsf{neg}\left(2\right)\right)}}}} \cdot th \]

      10. pow-flip N/A

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

      11. lower-/.f64 N/A

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

      12. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{\sin ky \cdot \color{blue}{\sin ky}}}}} \cdot th \]

      13. sqr-sin-a-rev N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{\frac{1}{2} - \cos \left(2 \cdot ky\right) \cdot \frac{1}{2}}}}} \cdot th \]

      18. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}}}} \cdot th \]

      19. lift-+.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{\frac{1}{2} - \cos \left(ky + ky\right) \cdot \frac{1}{2}}}}} \cdot th \]

      20. lift-*.f64 23.2

          \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{0.5 - \cos \left(ky + ky\right) \cdot \color{blue}{0.5}}}}} \cdot th \]

   10. Applied rewrites 23.2%

       \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{\color{blue}{\frac{1}{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}}}} \cdot th \]

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 66.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\frac{1}{\frac{1}{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}}} \cdot th\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.05)
     (/ (* (sin ky) th) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5))))
     (if (<= t_1 0.01) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (sin(ky) * th) / sqrt((0.5 - (cos((ky + ky)) * 0.5)));
	} else if (t_1 <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.05d0)) then
        tmp = (sin(ky) * th) / sqrt((0.5d0 - (cos((ky + ky)) * 0.5d0)))
    else if (t_1 <= 0.01d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.05) {
		tmp = (Math.sin(ky) * th) / Math.sqrt((0.5 - (Math.cos((ky + ky)) * 0.5)));
	} else if (t_1 <= 0.01) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.05:
		tmp = (math.sin(ky) * th) / math.sqrt((0.5 - (math.cos((ky + ky)) * 0.5)))
	elif t_1 <= 0.01:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.05)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5))));
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.05)
		tmp = (sin(ky) * th) / sqrt((0.5 - (cos((ky + ky)) * 0.5)));
	elseif (t_1 <= 0.01)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.050000000000000003

   1. Initial program 87.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

      6. lower-*.f64 46.7

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \sin th \]

   5. Applied rewrites 46.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   7. Step-by-step derivation

   8. Applied rewrites 23.3%

      \[\leadsto \frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \cdot \color{blue}{th} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-sin.f64 23.2

         \[\leadsto \frac{\color{blue}{\sin ky} \cdot th}{\sqrt{0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)}} \]

      8. lift--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \]

      11. lift-cos.f64 N/A

          \[\leadsto \frac{\sin ky \cdot th}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}} \]

   10. Applied rewrites 23.2%

       \[\leadsto \color{blue}{\frac{\sin ky \cdot th}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}} \]

   if -0.050000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 98.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 66.1

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 66.1%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky \cdot th}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}}\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky}{\color{blue}{\sin kx}} \cdot \sin th \]

      2. lift-sin.f64 36.2

         \[\leadsto \frac{ky}{\sin kx} \cdot \sin th \]

   5. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{ky}{\sin kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.01)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.01) {
		tmp = (sin(th) * ky) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.01d0) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.01) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.01:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.01)
		tmp = (sin(th) * ky) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.01], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 92.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sin kx}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin \color{blue}{kx}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

      5. lift-sin.f64 33.8

         \[\leadsto \frac{\sin th \cdot ky}{\sin kx} \]

   5. Applied rewrites 33.8%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\sin kx}} \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 33.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-170}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;t\_1 \leq 0.01:\\ \;\;\;\;\frac{ky}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 1e-170)
     (* (* (* th th) th) -0.16666666666666666)
     (if (<= t_1 0.01)
       (*
        (/ ky (sqrt (* (* (fma (* kx kx) -0.3333333333333333 1.0) kx) kx)))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-170) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else if (t_1 <= 0.01) {
		tmp = (ky / sqrt(((fma((kx * kx), -0.3333333333333333, 1.0) * kx) * kx))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 1e-170)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	elseif (t_1 <= 0.01)
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(fma(Float64(kx * kx), -0.3333333333333333, 1.0) * kx) * kx))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-170], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], If[LessEqual[t$95$1, 0.01], N[(N[(ky / N[Sqrt[N[(N[(N[(N[(kx * kx), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * kx), $MachinePrecision] * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999983e-171

   1. Initial program 92.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 3.1

         \[\leadsto \sin th \]

   5. Applied rewrites 3.1%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 3.1

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 3.1%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       2. lower-*.f64 N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       4. pow2 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       5. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       6. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       7. lift-*.f64 12.3

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   11. Applied rewrites 12.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 9.99999999999999983e-171 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0100000000000000002

   1. Initial program 94.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{kx}^{2} \cdot \left(1 + \frac{-1}{3} \cdot {kx}^{2}\right) + {\sin ky}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), {kx}^{2}, {\sin ky}^{2}\right)}} \cdot \sin th \]

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \sin ky \cdot \sin ky\right)}} \cdot \sin th \]

      10. sqr-sin-a N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      11. lower--.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      13. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{3}, kx \cdot kx, 1\right), kx \cdot kx, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

      14. lower-*.f64 44.5

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   5. Applied rewrites 44.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, kx \cdot kx, 1\right), kx \cdot kx, 0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{3} \cdot {kx}^{2}\right)}}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\frac{-1}{3} \cdot \left(kx \cdot kx\right) + 1\right) \cdot \left(kx \cdot kx\right)}} \cdot \sin th \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\left(\frac{-1}{3} \cdot \left(kx \cdot kx\right) + 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\left(\frac{-1}{3} \cdot \left(kx \cdot kx\right) + 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(\left(\frac{-1}{3} \cdot \left(kx \cdot kx\right) + 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

      8. pow2 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left({kx}^{2}, \frac{-1}{3}, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, \frac{-1}{3}, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

      12. lift-*.f64 44.5

          \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

   8. Applied rewrites 44.5%

      \[\leadsto \frac{\sin ky}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right) \cdot kx\right) \cdot \color{blue}{kx}}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, \frac{-1}{3}, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 44.5%

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th \]

   if 0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 70.0

         \[\leadsto \sin th \]

   5. Applied rewrites 70.0%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-170}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{elif}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.01:\\ \;\;\;\;\frac{ky}{\sqrt{\left(\mathsf{fma}\left(kx \cdot kx, -0.3333333333333333, 1\right) \cdot kx\right) \cdot kx}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 31.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-24}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-24)
   (* (* (* th th) th) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-24) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-24) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1e-24) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1e-24:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-24)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-24)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-24], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999924e-25

   1. Initial program 92.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 3.5

         \[\leadsto \sin th \]

   5. Applied rewrites 3.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 3.3

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 3.3%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       2. lower-*.f64 N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       4. pow2 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       5. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       6. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       7. lift-*.f64 13.6

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   11. Applied rewrites 13.6%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 9.99999999999999924e-25 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 91.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 68.1

         \[\leadsto \sin th \]

   5. Applied rewrites 68.1%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-24}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 15.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 4 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      4e-300)
   (* (* (* th th) th) -0.16666666666666666)
   th))

C

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)) <= 4e-300) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 4d-300) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 4e-300) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 4e-300:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 4e-300)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 4e-300)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 4e-300], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 4.0000000000000001e-300

   1. Initial program 91.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 27.8

         \[\leadsto \sin th \]

   5. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 15.9

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 15.9%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   9. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       2. lower-*.f64 N/A

          \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

       3. unpow3 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       4. pow2 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       5. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

       6. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

       7. lift-*.f64 16.4

          \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   11. Applied rewrites 16.4%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 4.0000000000000001e-300 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 93.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

      1. lift-sin.f64 25.6

         \[\leadsto \sin th \]

   5. Applied rewrites 25.6%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \]
   7. Step-by-step derivation

   8. Applied rewrites 12.8%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 4 \cdot 10^{-300}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 14.1% accurate, 170.4× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

   1. lift-sin.f64 26.7

      \[\leadsto \sin th \]

5. Applied rewrites 26.7%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \]
7. Step-by-step derivation

8. Applied rewrites 14.5%

   \[\leadsto th \]

9. Final simplification 14.5%

   \[\leadsto th \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(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)))