Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.6% → 99.7%

Time: 6.7s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

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: 93.6% accurate, 1.0× speedup

Math

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

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

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

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 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. 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. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

Alternative 2: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \frac{t\_1}{\mathsf{hypot}\left(t\_1, \sin kx\right)} \cdot th\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.37:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (* (/ t_1 (hypot t_1 (sin kx))) th)))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.05)
        t_3
        (if (<= t_2 0.37)
          (* t_1 (/ (sin th) (fabs (sin kx))))
          (if (<= t_2 0.99)
            t_3
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = (t_1 / hypot(t_1, sin(kx))) * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.37) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_2 <= 0.99) {
		tmp = t_3;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = (t_1 / Math.hypot(t_1, Math.sin(kx))) * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.37) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_2 <= 0.99) {
		tmp = t_3;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = (t_1 / math.hypot(t_1, math.sin(kx))) * th
	tmp = 0
	if t_2 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.05:
		tmp = t_3
	elif t_2 <= 0.37:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_2 <= 0.99:
		tmp = t_3
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * th)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.37)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_2 <= 0.99)
		tmp = t_3;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = (t_1 / hypot(t_1, sin(kx))) * th;
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.37)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_2 <= 0.99)
		tmp = t_3;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.37], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99], t$95$3, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -1

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 58.3%

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

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

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.5%

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

   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))))) < 0.37

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.1%

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

   4. Applied rewrites 41.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-fabs.f64 44.2%

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

   6. Applied rewrites 44.2%

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

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

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.7%

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

Alternative 3: 86.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ t_3 := \frac{th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.05:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.37:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_2 \leq 0.99:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))))
        (t_3 (* (/ th (hypot (sin kx) t_1)) t_1)))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.05)
        t_3
        (if (<= t_2 0.37)
          (* t_1 (/ (sin th) (fabs (sin kx))))
          (if (<= t_2 0.99)
            t_3
            (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = (th / hypot(sin(kx), t_1)) * t_1;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.37) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} else if (t_2 <= 0.99) {
		tmp = t_3;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = (th / Math.hypot(Math.sin(kx), t_1)) * t_1;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.05) {
		tmp = t_3;
	} else if (t_2 <= 0.37) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else if (t_2 <= 0.99) {
		tmp = t_3;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = (th / math.hypot(math.sin(kx), t_1)) * t_1
	tmp = 0
	if t_2 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.05:
		tmp = t_3
	elif t_2 <= 0.37:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	elif t_2 <= 0.99:
		tmp = t_3
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	t_3 = Float64(Float64(th / hypot(sin(kx), t_1)) * t_1)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.37)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	elseif (t_2 <= 0.99)
		tmp = t_3;
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = (th / hypot(sin(kx), t_1)) * t_1;
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.05)
		tmp = t_3;
	elseif (t_2 <= 0.37)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	elseif (t_2 <= 0.99)
		tmp = t_3;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.05], t$95$3, If[LessEqual[t$95$2, 0.37], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99], t$95$3, N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < -1

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in kx around 0

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

   6. Applied rewrites 58.3%

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

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

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.4%

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

   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))))) < 0.37

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.1%

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

   4. Applied rewrites 41.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.1%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-fabs.f64 44.2%

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

   6. Applied rewrites 44.2%

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

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

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.7%

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

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 1.02 \cdot 10^{-19}:\\ \;\;\;\;\frac{\left|th\right|}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 1.02e-19)
    (* (/ (fabs th) (hypot (sin kx) (sin ky))) (sin ky))
    (* (/ ky (hypot ky (sin kx))) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 1.02e-19) {
		tmp = (fabs(th) / hypot(sin(kx), sin(ky))) * sin(ky);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 1.02e-19) {
		tmp = (Math.abs(th) / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 1.02e-19:
		tmp = (math.fabs(th) / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 1.02e-19)
		tmp = Float64(Float64(abs(th) / hypot(sin(kx), sin(ky))) * sin(ky));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 1.02e-19)
		tmp = (abs(th) / hypot(sin(kx), sin(ky))) * sin(ky);
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 1.02e-19], N[(N[(N[Abs[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 1.02e-19

   1. Initial program 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.4%

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

   if 1.02e-19 < th

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.7%

      \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 72.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.01)
      (* (/ t_1 (sqrt (pow t_1 2.0))) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

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

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = (t_1 / Math.sqrt(Math.pow(t_1, 2.0))) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if t_1 <= -0.01:
		tmp = (t_1 / math.sqrt(math.pow(t_1, 2.0))) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(Float64(t_1 / sqrt((t_1 ^ 2.0))) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if (t_1 <= -0.01)
		tmp = (t_1 / sqrt((t_1 ^ 2.0))) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.01], N[(N[(t$95$1 / N[Sqrt[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.01

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 13.5%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 21.9%

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

   13. Applied rewrites 21.9%

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

   if -0.01 < (sin.f64 ky)

   1. Initial program 93.6%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. 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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.7%

      \[\leadsto \frac{ky}{\mathsf{hypot}\left(\color{blue}{ky}, \sin kx\right)} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. 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. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in ky around 0

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

6. Applied rewrites 51.7%

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

7. Taylor expanded in ky around 0

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

9. Applied rewrites 65.7%

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

Alternative 7: 54.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin \left(\left|kx\right|\right)\\ t_3 := \frac{t\_1}{\sqrt{{t\_2}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.37:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|t\_2\right|}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{{\left(\left|kx\right|\right)}^{2} + {\left(\left|ky\right|\right)}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\left|kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (sin (fabs kx)))
        (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 0.37)
      (* (sin th) (/ (fabs ky) (fabs t_2)))
      (if (<= t_3 2.0)
        (*
         (/ (fabs ky) (sqrt (+ (pow (fabs kx) 2.0) (pow (fabs ky) 2.0))))
         (sin th))
        (* (/ (fabs ky) (fabs kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.37) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (fabs(ky) / sqrt((pow(fabs(kx), 2.0) + pow(fabs(ky), 2.0)))) * sin(th);
	} else {
		tmp = (fabs(ky) / fabs(kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.sin(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.37) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (Math.abs(ky) / Math.sqrt((Math.pow(Math.abs(kx), 2.0) + Math.pow(Math.abs(ky), 2.0)))) * Math.sin(th);
	} else {
		tmp = (Math.abs(ky) / Math.abs(kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.37:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = (math.fabs(ky) / math.sqrt((math.pow(math.fabs(kx), 2.0) + math.pow(math.fabs(ky), 2.0)))) * math.sin(th)
	else:
		tmp = (math.fabs(ky) / math.fabs(kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.37)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(abs(ky) / sqrt(Float64((abs(kx) ^ 2.0) + (abs(ky) ^ 2.0)))) * sin(th));
	else
		tmp = Float64(Float64(abs(ky) / abs(kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.37)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = (abs(ky) / sqrt(((abs(kx) ^ 2.0) + (abs(ky) ^ 2.0)))) * sin(th);
	else
		tmp = (abs(ky) / abs(kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 0.37], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Abs[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < 0.37

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.2%

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-fabs.f64 39.1%

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

   6. Applied rewrites 39.1%

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

   if 0.37 < (/.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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 34.4%

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

   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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

Alternative 8: 46.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin \left(\left|kx\right|\right)\\ t_3 := \frac{t\_1}{\sqrt{{t\_2}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.9944213611256512:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|t\_2\right|}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{{\left(\left|kx\right|\right)}^{2} + {\left(\left|ky\right|\right)}^{2}}} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\left|kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (sin (fabs kx)))
        (t_3 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 0.9944213611256512)
      (* (sin th) (/ (fabs ky) (fabs t_2)))
      (if (<= t_3 2.0)
        (*
         (/ (fabs ky) (sqrt (+ (pow (fabs kx) 2.0) (pow (fabs ky) 2.0))))
         (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
        (* (/ (fabs ky) (fabs kx)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.9944213611256512) {
		tmp = sin(th) * (fabs(ky) / fabs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (fabs(ky) / sqrt((pow(fabs(kx), 2.0) + pow(fabs(ky), 2.0)))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else {
		tmp = (fabs(ky) / fabs(kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.sin(Math.abs(kx));
	double t_3 = t_1 / Math.sqrt((Math.pow(t_2, 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 0.9944213611256512) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_2));
	} else if (t_3 <= 2.0) {
		tmp = (Math.abs(ky) / Math.sqrt((Math.pow(Math.abs(kx), 2.0) + Math.pow(Math.abs(ky), 2.0)))) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	} else {
		tmp = (Math.abs(ky) / Math.abs(kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.sin(math.fabs(kx))
	t_3 = t_1 / math.sqrt((math.pow(t_2, 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_3 <= 0.9944213611256512:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_2))
	elif t_3 <= 2.0:
		tmp = (math.fabs(ky) / math.sqrt((math.pow(math.fabs(kx), 2.0) + math.pow(math.fabs(ky), 2.0)))) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	else:
		tmp = (math.fabs(ky) / math.fabs(kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 0.9944213611256512)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_2)));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(abs(ky) / sqrt(Float64((abs(kx) ^ 2.0) + (abs(ky) ^ 2.0)))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	else
		tmp = Float64(Float64(abs(ky) / abs(kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = sin(abs(kx));
	t_3 = t_1 / sqrt(((t_2 ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= 0.9944213611256512)
		tmp = sin(th) * (abs(ky) / abs(t_2));
	elseif (t_3 <= 2.0)
		tmp = (abs(ky) / sqrt(((abs(kx) ^ 2.0) + (abs(ky) ^ 2.0)))) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	else
		tmp = (abs(ky) / abs(kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 0.9944213611256512], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Abs[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < 0.99442136112565116

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.2%

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

      7. rem-sqrt-square-rev N/A

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

      8. lower-fabs.f64 39.1%

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

   6. Applied rewrites 39.1%

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

   if 0.99442136112565116 < (/.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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 18.2%

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

   10. Applied rewrites 18.2%

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

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.2%

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

   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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

Alternative 9: 30.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {\sin \left(\left|kx\right|\right)}^{2}\\ t_3 := \frac{t\_1}{\sqrt{t\_2 + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 1.5 \cdot 10^{-126}:\\ \;\;\;\;\frac{\left|ky\right|}{\frac{\left|kx\right| \cdot \mathsf{fma}\left(\left|kx\right| \cdot \frac{\left|kx\right|}{\sqrt{2}}, -0.3333333333333333, \sqrt{2}\right)}{\sqrt{2}}} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.75:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{t\_2}} \cdot th\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{{\left(\left|kx\right|\right)}^{2} + {\left(\left|ky\right|\right)}^{2}}} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\left|kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (pow (sin (fabs kx)) 2.0))
        (t_3 (/ t_1 (sqrt (+ t_2 (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 1.5e-126)
      (*
       (/
        (fabs ky)
        (/
         (*
          (fabs kx)
          (fma
           (* (fabs kx) (/ (fabs kx) (sqrt 2.0)))
           -0.3333333333333333
           (sqrt 2.0)))
         (sqrt 2.0)))
       (sin th))
      (if (<= t_3 0.75)
        (* (/ (fabs ky) (sqrt t_2)) th)
        (if (<= t_3 2.0)
          (*
           (/ (fabs ky) (sqrt (+ (pow (fabs kx) 2.0) (pow (fabs ky) 2.0))))
           (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
          (* (/ (fabs ky) (fabs kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(sin(fabs(kx)), 2.0);
	double t_3 = t_1 / sqrt((t_2 + pow(t_1, 2.0)));
	double tmp;
	if (t_3 <= 1.5e-126) {
		tmp = (fabs(ky) / ((fabs(kx) * fma((fabs(kx) * (fabs(kx) / sqrt(2.0))), -0.3333333333333333, sqrt(2.0))) / sqrt(2.0))) * sin(th);
	} else if (t_3 <= 0.75) {
		tmp = (fabs(ky) / sqrt(t_2)) * th;
	} else if (t_3 <= 2.0) {
		tmp = (fabs(ky) / sqrt((pow(fabs(kx), 2.0) + pow(fabs(ky), 2.0)))) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else {
		tmp = (fabs(ky) / fabs(kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx)) ^ 2.0
	t_3 = Float64(t_1 / sqrt(Float64(t_2 + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_3 <= 1.5e-126)
		tmp = Float64(Float64(abs(ky) / Float64(Float64(abs(kx) * fma(Float64(abs(kx) * Float64(abs(kx) / sqrt(2.0))), -0.3333333333333333, sqrt(2.0))) / sqrt(2.0))) * sin(th));
	elseif (t_3 <= 0.75)
		tmp = Float64(Float64(abs(ky) / sqrt(t_2)) * th);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(abs(ky) / sqrt(Float64((abs(kx) ^ 2.0) + (abs(ky) ^ 2.0)))) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	else
		tmp = Float64(Float64(abs(ky) / abs(kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Sqrt[N[(t$95$2 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, 1.5e-126], N[(N[(N[Abs[ky], $MachinePrecision] / N[(N[(N[Abs[kx], $MachinePrecision] * N[(N[(N[Abs[kx], $MachinePrecision] * N[(N[Abs[kx], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.75], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Abs[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $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))))) < 1.5000000000000001e-126

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. sqrt-div N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\cos 0 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \cdot \sin th \]

      11. cos-0 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-unsound-sqrt.f64 27.1%

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sqrt.f64 16.6%

         \[\leadsto \frac{ky}{\frac{kx \cdot \left(\sqrt{2} + -0.3333333333333333 \cdot \frac{{kx}^{2}}{\sqrt{2}}\right)}{\sqrt{2}}} \cdot \sin th \]

   9. Applied rewrites 16.6%

      \[\leadsto \frac{ky}{\frac{kx \cdot \left(\sqrt{2} + -0.3333333333333333 \cdot \frac{{kx}^{2}}{\sqrt{2}}\right)}{\sqrt{\color{blue}{2}}}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 16.6%

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

       6. lift-/.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 16.6%

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

   11. Applied rewrites 16.6%

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

   if 1.5000000000000001e-126 < (/.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))))) < 0.75

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.3%

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

   if 0.75 < (/.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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 18.2%

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

   10. Applied rewrites 18.2%

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

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 21.2%

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

   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 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

Alternative 10: 27.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 2.65 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left|kx\right|, \left|kx\right|, ky \cdot ky\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\left|kx\right| \leq 6000:\\ \;\;\;\;\frac{ky}{\frac{\left|kx\right| \cdot \mathsf{fma}\left(\left|kx\right| \cdot \frac{\left|kx\right|}{\sqrt{2}}, -0.3333333333333333, \sqrt{2}\right)}{\sqrt{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 2.65e-233)
   (*
    (/ (sin ky) (sqrt (fma (fabs kx) (fabs kx) (* ky ky))))
    (* (fma (* th th) -0.16666666666666666 1.0) th))
   (if (<= (fabs kx) 6000.0)
     (*
      (/
       ky
       (/
        (*
         (fabs kx)
         (fma
          (* (fabs kx) (/ (fabs kx) (sqrt 2.0)))
          -0.3333333333333333
          (sqrt 2.0)))
        (sqrt 2.0)))
      (sin th))
     (* (/ ky (sqrt (pow (sin (fabs kx)) 2.0))) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 2.65e-233) {
		tmp = (sin(ky) / sqrt(fma(fabs(kx), fabs(kx), (ky * ky)))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (fabs(kx) <= 6000.0) {
		tmp = (ky / ((fabs(kx) * fma((fabs(kx) * (fabs(kx) / sqrt(2.0))), -0.3333333333333333, sqrt(2.0))) / sqrt(2.0))) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(fabs(kx)), 2.0))) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 2.65e-233)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(abs(kx), abs(kx), Float64(ky * ky)))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (abs(kx) <= 6000.0)
		tmp = Float64(Float64(ky / Float64(Float64(abs(kx) * fma(Float64(abs(kx) * Float64(abs(kx) / sqrt(2.0))), -0.3333333333333333, sqrt(2.0))) / sqrt(2.0))) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(abs(kx)) ^ 2.0))) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 2.65e-233], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[kx], $MachinePrecision], 6000.0], N[(N[(ky / N[(N[(N[Abs[kx], $MachinePrecision] * N[(N[(N[Abs[kx], $MachinePrecision] * N[(N[Abs[kx], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.6499999999999999e-233

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 18.2%

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

   10. Applied rewrites 18.2%

       \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 18.2%

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 18.2%

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

   12. Applied rewrites 18.2%

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

   if 2.6499999999999999e-233 < kx < 6e3

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. sqrt-div N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\cos 0 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \cdot \sin th \]

      11. cos-0 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-unsound-sqrt.f64 27.1%

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sqrt.f64 16.6%

         \[\leadsto \frac{ky}{\frac{kx \cdot \left(\sqrt{2} + -0.3333333333333333 \cdot \frac{{kx}^{2}}{\sqrt{2}}\right)}{\sqrt{2}}} \cdot \sin th \]

   9. Applied rewrites 16.6%

      \[\leadsto \frac{ky}{\frac{kx \cdot \left(\sqrt{2} + -0.3333333333333333 \cdot \frac{{kx}^{2}}{\sqrt{2}}\right)}{\sqrt{\color{blue}{2}}}} \cdot \sin th \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 16.6%

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

       6. lift-/.f64 N/A

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

       7. lift-pow.f64 N/A

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

       8. unpow2 N/A

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

       9. associate-/l* N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 16.6%

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

   11. Applied rewrites 16.6%

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

   if 6e3 < kx

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.3%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 27.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 2.65 \cdot 10^{-233}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\left|kx\right|, \left|kx\right|, ky \cdot ky\right)}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;\left|kx\right| \leq 760000:\\ \;\;\;\;\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 2.65e-233)
   (*
    (/ (sin ky) (sqrt (fma (fabs kx) (fabs kx) (* ky ky))))
    (* (fma (* th th) -0.16666666666666666 1.0) th))
   (if (<= (fabs kx) 760000.0)
     (* (/ 1.0 (/ (fabs kx) ky)) (sin th))
     (* (/ ky (sqrt (pow (sin (fabs kx)) 2.0))) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 2.65e-233) {
		tmp = (sin(ky) / sqrt(fma(fabs(kx), fabs(kx), (ky * ky)))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (fabs(kx) <= 760000.0) {
		tmp = (1.0 / (fabs(kx) / ky)) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(fabs(kx)), 2.0))) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 2.65e-233)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(abs(kx), abs(kx), Float64(ky * ky)))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (abs(kx) <= 760000.0)
		tmp = Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(abs(kx)) ^ 2.0))) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 2.65e-233], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[kx], $MachinePrecision], 760000.0], N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.6499999999999999e-233

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 18.2%

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

   10. Applied rewrites 18.2%

       \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-+.f64 N/A

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

       2. lift-pow.f64 N/A

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

       3. unpow2 N/A

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

       4. lift-pow.f64 N/A

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

       5. pow2 N/A

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

       6. lift-*.f64 N/A

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

       7. lower-fma.f64 18.2%

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. lower-*.f64 18.2%

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

   12. Applied rewrites 18.2%

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

   if 2.6499999999999999e-233 < kx < 7.6e5

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      4. lower-unsound-/.f64 16.7%

         \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

   9. Applied rewrites 16.7%

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   if 7.6e5 < kx

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.3%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 27.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 2.65 \cdot 10^{-233}:\\ \;\;\;\;\sin ky \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\sqrt{\mathsf{fma}\left(\left|kx\right|, \left|kx\right|, ky \cdot ky\right)}}\\ \mathbf{elif}\;\left|kx\right| \leq 760000:\\ \;\;\;\;\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 2.65e-233)
   (*
    (sin ky)
    (/
     (* (fma (* th th) -0.16666666666666666 1.0) th)
     (sqrt (fma (fabs kx) (fabs kx) (* ky ky)))))
   (if (<= (fabs kx) 760000.0)
     (* (/ 1.0 (/ (fabs kx) ky)) (sin th))
     (* (/ ky (sqrt (pow (sin (fabs kx)) 2.0))) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 2.65e-233) {
		tmp = sin(ky) * ((fma((th * th), -0.16666666666666666, 1.0) * th) / sqrt(fma(fabs(kx), fabs(kx), (ky * ky))));
	} else if (fabs(kx) <= 760000.0) {
		tmp = (1.0 / (fabs(kx) / ky)) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(fabs(kx)), 2.0))) * th;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 2.65e-233)
		tmp = Float64(sin(ky) * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / sqrt(fma(abs(kx), abs(kx), Float64(ky * ky)))));
	elseif (abs(kx) <= 760000.0)
		tmp = Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(abs(kx)) ^ 2.0))) * th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 2.65e-233], N[(N[Sin[ky], $MachinePrecision] * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(N[Abs[kx], $MachinePrecision] * N[Abs[kx], $MachinePrecision] + N[(ky * ky), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[kx], $MachinePrecision], 760000.0], N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if kx < 2.6499999999999999e-233

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-pow.f64 46.8%

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

   4. Applied rewrites 46.8%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 28.5%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 18.2%

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

   10. Applied rewrites 18.2%

       \[\leadsto \frac{\sin ky}{\sqrt{{kx}^{2} + {ky}^{2}}} \cdot \color{blue}{\left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 18.2%

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

   12. Applied rewrites 18.2%

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

   if 2.6499999999999999e-233 < kx < 7.6e5

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      4. lower-unsound-/.f64 16.7%

         \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

   9. Applied rewrites 16.7%

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   if 7.6e5 < kx

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.3%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 26.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 760000:\\ \;\;\;\;\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 760000.0)
   (* (/ 1.0 (/ (fabs kx) ky)) (sin th))
   (* (/ ky (sqrt (pow (sin (fabs kx)) 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 760000.0) {
		tmp = (1.0 / (fabs(kx) / ky)) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(fabs(kx)), 2.0))) * 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 (abs(kx) <= 760000.0d0) then
        tmp = (1.0d0 / (abs(kx) / ky)) * sin(th)
    else
        tmp = (ky / sqrt((sin(abs(kx)) ** 2.0d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 760000.0) {
		tmp = (1.0 / (Math.abs(kx) / ky)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(Math.abs(kx)), 2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 760000.0:
		tmp = (1.0 / (math.fabs(kx) / ky)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(math.pow(math.sin(math.fabs(kx)), 2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 760000.0)
		tmp = Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(abs(kx)) ^ 2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 760000.0)
		tmp = (1.0 / (abs(kx) / ky)) * sin(th);
	else
		tmp = (ky / sqrt((sin(abs(kx)) ^ 2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 760000.0], N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.6e5

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      4. lower-unsound-/.f64 16.7%

         \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

   9. Applied rewrites 16.7%

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   if 7.6e5 < kx

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.3%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot \color{blue}{th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 26.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 760000:\\ \;\;\;\;\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\frac{\sqrt{1 - \cos \left(\left|kx\right| + \left|kx\right|\right)}}{\sqrt{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 760000.0)
   (* (/ 1.0 (/ (fabs kx) ky)) (sin th))
   (* (/ ky (/ (sqrt (- 1.0 (cos (+ (fabs kx) (fabs kx))))) (sqrt 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 760000.0) {
		tmp = (1.0 / (fabs(kx) / ky)) * sin(th);
	} else {
		tmp = (ky / (sqrt((1.0 - cos((fabs(kx) + fabs(kx))))) / sqrt(2.0))) * 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 (abs(kx) <= 760000.0d0) then
        tmp = (1.0d0 / (abs(kx) / ky)) * sin(th)
    else
        tmp = (ky / (sqrt((1.0d0 - cos((abs(kx) + abs(kx))))) / sqrt(2.0d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 760000.0) {
		tmp = (1.0 / (Math.abs(kx) / ky)) * Math.sin(th);
	} else {
		tmp = (ky / (Math.sqrt((1.0 - Math.cos((Math.abs(kx) + Math.abs(kx))))) / Math.sqrt(2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 760000.0:
		tmp = (1.0 / (math.fabs(kx) / ky)) * math.sin(th)
	else:
		tmp = (ky / (math.sqrt((1.0 - math.cos((math.fabs(kx) + math.fabs(kx))))) / math.sqrt(2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 760000.0)
		tmp = Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(th));
	else
		tmp = Float64(Float64(ky / Float64(sqrt(Float64(1.0 - cos(Float64(abs(kx) + abs(kx))))) / sqrt(2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 760000.0)
		tmp = (1.0 / (abs(kx) / ky)) * sin(th);
	else
		tmp = (ky / (sqrt((1.0 - cos((abs(kx) + abs(kx))))) / sqrt(2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 760000.0], N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[(N[Sqrt[N[(1.0 - N[Cos[N[(N[Abs[kx], $MachinePrecision] + N[Abs[kx], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 7.6e5

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.7%

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

   7. Applied rewrites 16.7%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      3. lower-unsound-/.f64 N/A

         \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

      4. lower-unsound-/.f64 16.7%

         \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

   9. Applied rewrites 16.7%

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   if 7.6e5 < kx

   1. Initial program 93.6%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.2%

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

   4. Applied rewrites 36.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. sin-mult N/A

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

      7. sqrt-div N/A

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

      8. lower-unsound-/.f64 N/A

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

      9. lower-unsound-sqrt.f64 N/A

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

      10. +-inverses N/A

          \[\leadsto \frac{ky}{\frac{\sqrt{\cos 0 - \cos \left(kx + kx\right)}}{\sqrt{2}}} \cdot \sin th \]

      11. cos-0 N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-+.f64 N/A

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

      15. lower-unsound-sqrt.f64 27.1%

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.7%

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

Alternative 15: 21.5% accurate, 3.9× speedup

Math

\[\frac{1}{\frac{\left|kx\right|}{ky}} \cdot \sin th \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ 1.0 (/ (fabs kx) ky)) (sin th)))

C

double code(double kx, double ky, double th) {
	return (1.0 / (fabs(kx) / ky)) * 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 = (1.0d0 / (abs(kx) / ky)) * sin(th)
end function

Java

public static double code(double kx, double ky, double th) {
	return (1.0 / (Math.abs(kx) / ky)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	return (1.0 / (math.fabs(kx) / ky)) * math.sin(th)

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(abs(kx) / ky)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (1.0 / (abs(kx) / ky)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

   1. lift-/.f64 N/A

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

   2. div-flip N/A

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   3. lower-unsound-/.f64 N/A

      \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]

   4. lower-unsound-/.f64 16.7%

      \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot \sin th \]

9. Applied rewrites 16.7%

   \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot \sin th \]
10. Add Preprocessing

Alternative 16: 21.5% accurate, 4.2× speedup

Math

\[\frac{ky}{\left|kx\right|} \cdot \sin th \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * 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 = (ky / abs(kx)) * sin(th)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

   \[\leadsto \frac{ky}{\color{blue}{kx}} \cdot \sin th \]
8. Add Preprocessing

Alternative 17: 15.3% accurate, 14.4× speedup

Math

\[\frac{1}{\frac{\left|kx\right|}{ky}} \cdot th \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ 1.0 (/ (fabs kx) ky)) th))

C

double code(double kx, double ky, double th) {
	return (1.0 / (fabs(kx) / ky)) * 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 = (1.0d0 / (abs(kx) / ky)) * th
end function

Java

public static double code(double kx, double ky, double th) {
	return (1.0 / (Math.abs(kx) / ky)) * th;
}

Python

def code(kx, ky, th):
	return (1.0 / (math.fabs(kx) / ky)) * th

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(abs(kx) / ky)) * th)
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = (1.0 / (abs(kx) / ky)) * th;
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(N[Abs[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.5%

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

    1. lift-/.f64 N/A

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

    2. div-flip N/A

       \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]

    3. lower-unsound-/.f64 N/A

       \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]

    4. lower-unsound-/.f64 13.5%

       \[\leadsto \frac{1}{\frac{kx}{ky}} \cdot th \]

12. Applied rewrites 13.5%

    \[\leadsto \frac{1}{\frac{kx}{\color{blue}{ky}}} \cdot th \]
13. Add Preprocessing

Alternative 18: 15.3% accurate, 20.0× speedup

Math

\[\frac{ky}{\left|kx\right|} \cdot th \]

FPCore

(FPCore (kx ky th) :precision binary64 (* (/ ky (fabs kx)) th))

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * 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 = (ky / abs(kx)) * th
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]

Derivation

1. Initial program 93.6%

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

2. Taylor expanded in ky around 0

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.2%

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

4. Applied rewrites 36.2%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.7%

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

7. Applied rewrites 16.7%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.5%

    \[\leadsto \frac{ky}{kx} \cdot \color{blue}{th} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))