Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 5.8s

Alternatives: 22

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: 93.8% 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 92.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_2\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -1.0)
     t_1
     (if (<= t_3 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.05)
         (*
          (/
           (sin ky)
           (sqrt
            (fma
             (fma
              (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
              (* ky ky)
              1.0)
             (* ky ky)
             t_2)))
          (sin th))
         (if (<= t_3 0.9)
           (*
            (* (/ 1.0 (hypot (sin kx) (sin ky))) (sin ky))
            (*
             (fma
              (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
              (* th th)
              1.0)
             th))
           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 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = t_1;
	} else if (t_3 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.05) {
		tmp = (sin(ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_2))) * sin(th);
	} else if (t_3 <= 0.9) {
		tmp = ((1.0 / hypot(sin(kx), sin(ky))) * sin(ky)) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th);
	} 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 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = t_1;
	elseif (t_3 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_2))) * sin(th));
	elseif (t_3 <= 0.9)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), sin(ky))) * sin(ky)) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th));
	else
		tmp = t_1;
	end
	return 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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], t$95$1, If[LessEqual[t$95$3, -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$3, 0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $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))))) < -1 or 0.900000000000000022 < (/.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 83.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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 95.6%

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

   if -1 < (/.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 99.3

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

   4. Applied rewrites 99.3%

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

   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 46.9%

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

   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))))) < 0.050000000000000003

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   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))))) < 0.900000000000000022

   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 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. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. lift-hypot.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-sin.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 64.6

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

   10. Applied rewrites 64.6%

       \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;t\_3 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_2\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.9:\\ \;\;\;\;\left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -1.0)
     t_1
     (if (<= t_3 -0.05)
       (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
       (if (<= t_3 0.02)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (fma
             (fma
              (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
              (* ky ky)
              1.0)
             (* ky ky)
             t_2)))
          (sin th))
         (if (<= t_3 0.9)
           (*
            (* (/ 1.0 (hypot (sin kx) (sin ky))) (sin ky))
            (*
             (fma
              (- (* (* th th) 0.008333333333333333) 0.16666666666666666)
              (* th th)
              1.0)
             th))
           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 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -1.0) {
		tmp = t_1;
	} else if (t_3 <= -0.05) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else if (t_3 <= 0.02) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_2))) * sin(th);
	} else if (t_3 <= 0.9) {
		tmp = ((1.0 / hypot(sin(kx), sin(ky))) * sin(ky)) * (fma((((th * th) * 0.008333333333333333) - 0.16666666666666666), (th * th), 1.0) * th);
	} 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 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = t_1;
	elseif (t_3 <= -0.05)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	elseif (t_3 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_2))) * sin(th));
	elseif (t_3 <= 0.9)
		tmp = Float64(Float64(Float64(1.0 / hypot(sin(kx), sin(ky))) * sin(ky)) * Float64(fma(Float64(Float64(Float64(th * th) * 0.008333333333333333) - 0.16666666666666666), Float64(th * th), 1.0) * th));
	else
		tmp = t_1;
	end
	return 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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1.0], t$95$1, If[LessEqual[t$95$3, -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$3, 0.02], N[(N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.9], N[(N[(N[(1.0 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(th * th), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(th * th), $MachinePrecision] + 1.0), $MachinePrecision] * th), $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))))) < -1 or 0.900000000000000022 < (/.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 83.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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 95.6%

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

   if -1 < (/.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 99.3

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

   4. Applied rewrites 99.3%

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

   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 46.9%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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))))) < 0.900000000000000022

   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 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. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-hypot.f64 N/A

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

      5. sqrt-pow2 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. pow1/2 N/A

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

      12. lower-/.f64 N/A

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

      13. pow2 N/A

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

      14. pow2 N/A

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

      15. lift-hypot.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-sin.f64 99.4

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 64.6

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

   10. Applied rewrites 64.6%

       \[\leadsto \left(\frac{1}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\left(th \cdot th\right) \cdot 0.008333333333333333 - 0.16666666666666666, th \cdot th, 1\right) \cdot th\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;t\_2 \cdot th\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_3\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.99999998:\\ \;\;\;\;t\_2 \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (hypot (sin ky) (sin kx))))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -1.0)
     t_1
     (if (<= t_4 -0.05)
       (* t_2 th)
       (if (<= t_4 0.02)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (fma
             (fma
              (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
              (* ky ky)
              1.0)
             (* ky ky)
             t_3)))
          (sin th))
         (if (<= t_4 0.99999998)
           (* t_2 (* (fma (* th th) -0.16666666666666666 1.0) th))
           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) / hypot(sin(ky), sin(kx));
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.05) {
		tmp = t_2 * th;
	} else if (t_4 <= 0.02) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_3))) * sin(th);
	} else if (t_4 <= 0.99999998) {
		tmp = t_2 * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} 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) / hypot(sin(ky), sin(kx)))
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.05)
		tmp = Float64(t_2 * th);
	elseif (t_4 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_3))) * sin(th));
	elseif (t_4 <= 0.99999998)
		tmp = Float64(t_2 * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = t_1;
	end
	return 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[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], t$95$1, If[LessEqual[t$95$4, -0.05], N[(t$95$2 * th), $MachinePrecision], If[LessEqual[t$95$4, 0.02], N[(N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.99999998], N[(t$95$2 * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $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))))) < -1 or 0.999999980000000011 < (/.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 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

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

   7. Applied rewrites 99.9%

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

   if -1 < (/.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 99.3

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

   4. Applied rewrites 99.3%

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

   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 46.9%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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))))) < 0.999999980000000011

   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 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 60.0

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

   7. Applied rewrites 60.0%

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

Alternative 5: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.05:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_3\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.9:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -1.0)
     t_1
     (if (<= t_4 -0.05)
       t_2
       (if (<= t_4 0.02)
         (*
          (/
           (*
            (fma
             (-
              (*
               (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
               (* ky ky))
              0.16666666666666666)
             (* ky ky)
             1.0)
            ky)
           (sqrt
            (fma
             (fma
              (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
              (* ky ky)
              1.0)
             (* ky ky)
             t_3)))
          (sin th))
         (if (<= t_4 0.9) t_2 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) / hypot(sin(ky), sin(kx))) * th;
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.05) {
		tmp = t_2;
	} else if (t_4 <= 0.02) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_3))) * sin(th);
	} else if (t_4 <= 0.9) {
		tmp = t_2;
	} 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(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.05)
		tmp = t_2;
	elseif (t_4 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_3))) * sin(th));
	elseif (t_4 <= 0.9)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return 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[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], t$95$1, If[LessEqual[t$95$4, -0.05], t$95$2, If[LessEqual[t$95$4, 0.02], N[(N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.9], t$95$2, 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))))) < -1 or 0.900000000000000022 < (/.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 83.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.9

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

   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 95.6%

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

   if -1 < (/.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 0.0200000000000000004 < (/.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))))) < 0.900000000000000022

   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 99.4

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

   4. Applied rewrites 99.4%

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

   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 54.8%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/
         (*
          (fma
           (-
            (*
             (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
             (* ky ky))
            0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           t_1)))
        (sin th))
       (if (<= t_2 2.0) (sin th) (* (/ (sin ky) (hypot ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $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 + kx ^ 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.5

         \[\leadsto \sin th \]

   5. Applied rewrites 68.5%

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

   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 2.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 99.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 99.6%

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

Alternative 7: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/
         (*
          (fma
           (- (* (* ky ky) 0.008333333333333333) 0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           t_1)))
        (sin th))
       (if (<= t_2 2.0) (sin th) (* (/ (sin ky) (hypot ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = ((fma((((ky * ky) * 0.008333333333333333) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(Float64(ky * ky) * 0.008333333333333333) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.008333333333333333), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $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 + kx ^ 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left(\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}\right) \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot {ky}^{2} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2} \cdot \frac{1}{120} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{1}{120} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{1}{120} - \frac{1}{6}, {ky}^{2}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{1}{120} - \frac{1}{6}, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lift-*.f64 98.6

          \[\leadsto \frac{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.008333333333333333 - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.5

         \[\leadsto \sin th \]

   5. Applied rewrites 68.5%

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

   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 2.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 99.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 99.6%

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

Alternative 8: 72.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/
         (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           t_1)))
        (sin th))
       (if (<= t_2 2.0) (sin th) (* (/ (sin ky) (hypot ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $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 + kx ^ 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 98.0

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.5

         \[\leadsto \sin th \]

   5. Applied rewrites 68.5%

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

   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 2.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 99.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 99.6%

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

Alternative 9: 72.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/
         ky
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           t_1)))
        (sin th))
       (if (<= t_2 2.0) (sin th) (* (/ (sin ky) (hypot ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = (ky / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(ky / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(ky / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $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 + kx ^ 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

   8. Applied rewrites 97.5%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.5

         \[\leadsto \sin th \]

   5. Applied rewrites 68.5%

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

   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 2.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 99.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 99.6%

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

Alternative 10: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{t\_1}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/ (* (fma (* ky ky) -0.16666666666666666 1.0) ky) (sqrt t_1))
        (sin th))
       (if (<= t_2 2.0) (sin th) (* (/ (sin ky) (hypot ky kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(t_1)) * sin(th);
	} else if (t_2 <= 2.0) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(t_1)) * sin(th));
	elseif (t_2 <= 2.0)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[t$95$1], $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 + kx ^ 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 98.0

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   9. Taylor expanded in ky around 0

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

       1. lift-sin.f64 N/A

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

       2. lift-pow.f64 96.9

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{{\sin kx}^{2}}} \cdot \sin th \]

   11. Applied rewrites 96.9%

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

   if 0.0200000000000000004 < (/.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 99.8%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.5

         \[\leadsto \sin th \]

   5. Applied rewrites 68.5%

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

   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 2.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 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 99.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 99.6%

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

Alternative 11: 70.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, t\_1\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.05)
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th))
     (if (<= t_2 0.02)
       (*
        (/
         (*
          (fma
           (-
            (*
             (fma (* ky ky) -0.0001984126984126984 0.008333333333333333)
             (* ky ky))
            0.16666666666666666)
           (* ky ky)
           1.0)
          ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           t_1)))
        (sin th))
       (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double t_2 = sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.05) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	} else if (t_2 <= 0.02) {
		tmp = ((fma(((fma((ky * ky), -0.0001984126984126984, 0.008333333333333333) * (ky * ky)) - 0.16666666666666666), (ky * ky), 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), t_1))) * sin(th);
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.05)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	elseif (t_2 <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(Float64(fma(Float64(ky * ky), -0.0001984126984126984, 0.008333333333333333) * Float64(ky * ky)) - 0.16666666666666666), Float64(ky * ky), 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), t_1))) * sin(th));
	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[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.02], N[(N[(N[(N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.0001984126984126984 + 0.008333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $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 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 52.5

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

   7. Applied rewrites 52.5%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 98.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 98.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + {ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {ky}^{2}\right) - \frac{1}{6}\right)\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.0001984126984126984, 0.008333333333333333\right) \cdot \left(ky \cdot ky\right) - 0.16666666666666666, ky \cdot ky, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (/.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 85.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

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

   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 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 68.1%

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

Alternative 12: 51.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
     (if (<= t_1 0.002) (* (/ 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((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (t_1 <= 0.002) {
		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((0.5d0 - (0.5d0 * cos((ky + ky)))))) * th
    else if (t_1 <= 0.002d0) 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((0.5 - (0.5 * Math.cos((ky + ky)))))) * th;
	} else if (t_1 <= 0.002) {
		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((0.5 - (0.5 * math.cos((ky + ky)))))) * th
	elif t_1 <= 0.002:
		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(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (t_1 <= 0.002)
		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((0.5 - (0.5 * cos((ky + ky)))))) * th;
	elseif (t_1 <= 0.002)
		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[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.002], 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 90.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.8

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

   5. Applied rewrites 68.8%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 32.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 22.2

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

   10. Applied rewrites 22.2%

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

   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))))) < 2e-3

   1. Initial program 99.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 67.1

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

   5. Applied rewrites 67.1%

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

   if 2e-3 < (/.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 85.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 45.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (* (/ 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)))) <= 0.002) {
		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)))) <= 0.002d0) 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)))) <= 0.002) {
		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)))) <= 0.002:
		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)))) <= 0.002)
		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)))) <= 0.002)
		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], 0.002], 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))))) < 2e-3

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 40.8

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

   5. Applied rewrites 40.8%

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

   if 2e-3 < (/.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 85.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 44.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (/ (* (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)))) <= 0.002) {
		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)))) <= 0.002d0) 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)))) <= 0.002) {
		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)))) <= 0.002:
		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)))) <= 0.002)
		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)))) <= 0.002)
		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], 0.002], 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))))) < 2e-3

   1. Initial program 95.8%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 39.0

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

   5. Applied rewrites 39.0%

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

   if 2e-3 < (/.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 85.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.002:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, kx \cdot kx\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.002)
   (*
    (/
     (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
     (sqrt
      (fma
       (fma
        (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
        (* ky ky)
        1.0)
       (* ky ky)
       (* kx 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)))) <= 0.002) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), (kx * kx)))) * sin(th);
	} else {
		tmp = 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)))) <= 0.002)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), Float64(kx * kx)))) * sin(th));
	else
		tmp = sin(th);
	end
	return 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], 0.002], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $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))))) < 2e-3

   1. Initial program 95.8%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 65.2

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 65.2%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 64.4

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 64.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   9. Taylor expanded in kx around 0

      \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {kx}^{2}\right)}} \cdot \sin th \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, kx \cdot kx\right)}} \cdot \sin th \]

       2. lower-*.f64 37.3

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, kx \cdot kx\right)}} \cdot \sin th \]

   11. Applied rewrites 37.3%

       \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, kx \cdot kx\right)}} \cdot \sin th \]

   if 2e-3 < (/.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 85.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin th \]

   5. Applied rewrites 63.5%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 15.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-304}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-304)
   (* (* (* th th) -0.16666666666666666) th)
   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)) <= 1e-304) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} 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)) <= 1d-304) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    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)) <= 1e-304) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} 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)) <= 1e-304:
		tmp = ((th * th) * -0.16666666666666666) * th
	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)) <= 1e-304)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	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)) <= 1e-304)
		tmp = ((th * th) * -0.16666666666666666) * th;
	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], 1e-304], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $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)) < 9.99999999999999971e-305

   1. Initial program 93.6%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 21.9

         \[\leadsto \sin th \]

   5. Applied rewrites 21.9%

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

   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 13.7

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 13.7%

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

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       2. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       3. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right) \cdot th \]

       4. lift-*.f64 22.0

          \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   11. Applied rewrites 22.0%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 9.99999999999999971e-305 < (*.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 91.2%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 24.4

         \[\leadsto \sin th \]

   5. Applied rewrites 24.4%

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

   6. Taylor expanded in th around 0

      \[\leadsto th \]
   7. Step-by-step derivation

   8. Applied rewrites 14.3%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-27}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-27)
   (* (* (* th th) -0.16666666666666666) 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)))) <= 5e-27) {
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 5d-27) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * 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)))) <= 5e-27) {
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 5e-27:
		tmp = ((th * th) * -0.16666666666666666) * 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)))) <= 5e-27)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * 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)))) <= 5e-27)
		tmp = ((th * th) * -0.16666666666666666) * 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], 5e-27], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $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))))) < 5.0000000000000002e-27

   1. Initial program 95.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 4.1

         \[\leadsto \sin th \]

   5. Applied rewrites 4.1%

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

   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 3.8

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   8. Applied rewrites 3.8%

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

   9. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       2. lower-*.f64 N/A

          \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

       3. pow2 N/A

          \[\leadsto \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right) \cdot th \]

       4. lift-*.f64 18.8

          \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   11. Applied rewrites 18.8%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 5.0000000000000002e-27 < (/.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 85.9%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 61.6

         \[\leadsto \sin th \]

   5. Applied rewrites 61.6%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\sqrt{{\sin ky}^{2}}} \cdot th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (* (/ (sin ky) (sqrt (pow (sin ky) 2.0))) th)
   (if (<= (sin ky) 4e-170)
     (* (/ (sin ky) (hypot ky kx)) (sin th))
     (if (<= (sin ky) 0.02)
       (*
        (/
         (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           (- 0.5 (* 0.5 (cos (+ kx kx)))))))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = (sin(ky) / sqrt(pow(sin(ky), 2.0))) * th;
	} else if (sin(ky) <= 4e-170) {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	} else if (sin(ky) <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), (0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(Float64(sin(ky) / sqrt((sin(ky) ^ 2.0))) * th);
	elseif (sin(ky) <= 4e-170)
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-170], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -2e-3

   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

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 28.6%

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

   if -2e-3 < (sin.f64 ky) < 3.99999999999999993e-170

   1. Initial program 81.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 69.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 3.99999999999999993e-170 < (sin.f64 ky) < 0.0200000000000000004

   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 ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 99.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      10. lower-+.f64 82.7

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   10. Applied rewrites 82.7%

       \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (sin.f64 ky)

   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 64.3

         \[\leadsto \sin th \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 58.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;\sin ky \leq 4 \cdot 10^{-170}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
   (if (<= (sin ky) 4e-170)
     (* (/ (sin ky) (hypot ky kx)) (sin th))
     (if (<= (sin ky) 0.02)
       (*
        (/
         (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           (- 0.5 (* 0.5 (cos (+ kx kx)))))))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (sin(ky) <= 4e-170) {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	} else if (sin(ky) <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), (0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (sin(ky) <= 4e-170)
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 4e-170], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -2e-3

   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

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 28.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 28.2

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

   10. Applied rewrites 28.2%

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

   if -2e-3 < (sin.f64 ky) < 3.99999999999999993e-170

   1. Initial program 81.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. 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 69.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 69.6%

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

   if 3.99999999999999993e-170 < (sin.f64 ky) < 0.0200000000000000004

   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 ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 99.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      10. lower-+.f64 82.7

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   10. Applied rewrites 82.7%

       \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (sin.f64 ky)

   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 64.3

         \[\leadsto \sin th \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 20: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.002:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-134}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 0.02:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.002)
   (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) th)
   (if (<= (sin ky) 1e-134)
     (* (/ ky (sin kx)) (sin th))
     (if (<= (sin ky) 0.02)
       (*
        (/
         (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
         (sqrt
          (fma
           (fma
            (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
            (* ky ky)
            1.0)
           (* ky ky)
           (- 0.5 (* 0.5 (cos (+ kx kx)))))))
        (sin th))
       (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.002) {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * th;
	} else if (sin(ky) <= 1e-134) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (sin(ky) <= 0.02) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), (0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.002)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * th);
	elseif (sin(ky) <= 1e-134)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (sin(ky) <= 0.02)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.002], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-134], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 0.02], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (sin.f64 ky) < -2e-3

   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

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.5

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

   5. Applied rewrites 68.5%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 28.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 28.2

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

   10. Applied rewrites 28.2%

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

   if -2e-3 < (sin.f64 ky) < 1.00000000000000004e-134

   1. Initial program 83.2%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 56.2

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

   5. Applied rewrites 56.2%

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

   if 1.00000000000000004e-134 < (sin.f64 ky) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 99.6

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 99.6%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 98.6

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 98.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      10. lower-+.f64 91.7

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   10. Applied rewrites 91.7%

       \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   if 0.0200000000000000004 < (sin.f64 ky)

   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 64.3

         \[\leadsto \sin th \]

   5. Applied rewrites 64.3%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 51.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 4 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;ky \leq 0.026:\\ \;\;\;\;\frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 4e-169)
   (* (/ (sin ky) (hypot ky kx)) (sin th))
   (if (<= ky 0.026)
     (*
      (/
       (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
       (sqrt
        (fma
         (fma
          (- (* (* ky ky) 0.044444444444444446) 0.3333333333333333)
          (* ky ky)
          1.0)
         (* ky ky)
         (- 0.5 (* 0.5 (cos (+ kx kx)))))))
      (sin th))
     (* (/ (sin ky) (sqrt (- 0.5 (* 0.5 (cos (+ ky ky)))))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 4e-169) {
		tmp = (sin(ky) / hypot(ky, kx)) * sin(th);
	} else if (ky <= 0.026) {
		tmp = ((fma((ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma((((ky * ky) * 0.044444444444444446) - 0.3333333333333333), (ky * ky), 1.0), (ky * ky), (0.5 - (0.5 * cos((kx + kx))))))) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (0.5 * cos((ky + ky)))))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 4e-169)
		tmp = Float64(Float64(sin(ky) / hypot(ky, kx)) * sin(th));
	elseif (ky <= 0.026)
		tmp = Float64(Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) / sqrt(fma(fma(Float64(Float64(Float64(ky * ky) * 0.044444444444444446) - 0.3333333333333333), Float64(ky * ky), 1.0), Float64(ky * ky), Float64(0.5 - Float64(0.5 * cos(Float64(kx + kx))))))) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(0.5 * cos(Float64(ky + ky)))))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 4e-169], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[ky, 0.026], N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] / N[Sqrt[N[(N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(0.5 * N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ky < 4.00000000000000008e-169

   1. Initial program 88.4%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.7

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

   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 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 65.6%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 44.5%

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

   if 4.00000000000000008e-169 < ky < 0.0259999999999999988

   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 ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{ky}^{2} \cdot \left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) + {\sin kx}^{2}}}} \cdot \sin th \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right), \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right) \cdot {ky}^{2} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2} \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, {ky}^{2}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      13. unpow2 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      14. lower-*.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      15. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      16. lift-pow.f64 99.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   5. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

   6. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky \cdot \left(1 + \frac{-1}{6} \cdot {ky}^{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {ky}^{2}\right) \cdot \color{blue}{ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {ky}^{2} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\left({ky}^{2} \cdot \frac{-1}{6} + 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({ky}^{2}, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lift-*.f64 99.0

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   8. Applied rewrites 99.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]
   9. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. lift-sin.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. pow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. sqr-sin-a N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. cos-2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos kx \cdot \cos kx - \sin kx \cdot \sin kx\right)\right)}} \cdot \sin th \]

      8. cos-sum N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{6}, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot \frac{2}{45} - \frac{1}{3}, ky \cdot ky, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

      10. lower-+.f64 82.7

          \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   10. Applied rewrites 82.7%

       \[\leadsto \frac{\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\left(ky \cdot ky\right) \cdot 0.044444444444444446 - 0.3333333333333333, ky \cdot ky, 1\right), ky \cdot ky, 0.5 - 0.5 \cdot \cos \left(kx + kx\right)\right)}} \cdot \sin th \]

   if 0.0259999999999999988 < 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. lift-sin.f64 N/A

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

      2. lift-pow.f64 68.7

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

   5. Applied rewrites 68.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. pow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. cos-2 N/A

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

      8. cos-sum N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-+.f64 67.8

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

   7. Applied rewrites 67.8%

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

Alternative 22: 13.3% accurate, 632.0× 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 92.5%

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

3. Taylor expanded in kx around 0

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

   1. lift-sin.f64 23.0

      \[\leadsto \sin th \]

5. Applied rewrites 23.0%

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

6. Taylor expanded in th around 0

   \[\leadsto th \]
7. Step-by-step derivation

8. Applied rewrites 14.1%

   \[\leadsto th \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))