Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.0% → 99.7%

Time: 6.9s

Alternatives: 20

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.95:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.88:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin kx) (sin ky))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.95)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 2e-8)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_4 0.88) t_1 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.95) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-8) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.95) {
		tmp = (Math.sin(ky) / Math.sqrt(t_3)) * Math.sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-8) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_2 + t_3))
	tmp = 0
	if t_4 <= -0.95:
		tmp = (math.sin(ky) / math.sqrt(t_3)) * math.sin(th)
	elif t_4 <= -0.2:
		tmp = t_1
	elif t_4 <= 2e-8:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= 0.88:
		tmp = t_1
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.95)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-8)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.88)
		tmp = t_1;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_2 + t_3));
	tmp = 0.0;
	if (t_4 <= -0.95)
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-8)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_4 <= 0.88)
		tmp = t_1;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.95], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.88], t$95$1, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.94999999999999996

   1. Initial program 94.0%

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

   2. Taylor expanded in kx around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sin.f64 40.6

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

   4. Applied rewrites 40.6%

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

   if -0.94999999999999996 < (/.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))))) < -0.20000000000000001 or 2e-8 < (/.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))))) < 0.880000000000000004

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 47.4

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

   8. Applied rewrites 47.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

   if 0.880000000000000004 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 3: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.95:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_3}}\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.88:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin kx) (sin ky))))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.95)
     (/ (* (sin ky) (sin th)) (sqrt t_3))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 2e-8)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_4 0.88) t_1 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.95) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_3);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-8) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.95) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_3);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 2e-8) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_2 + t_3))
	tmp = 0
	if t_4 <= -0.95:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_3)
	elif t_4 <= -0.2:
		tmp = t_1
	elif t_4 <= 2e-8:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= 0.88:
		tmp = t_1
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.95)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_3));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-8)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.88)
		tmp = t_1;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_2 + t_3));
	tmp = 0.0;
	if (t_4 <= -0.95)
		tmp = (sin(ky) * sin(th)) / sqrt(t_3);
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 2e-8)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_4 <= 0.88)
		tmp = t_1;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.95], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.88], t$95$1, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.94999999999999996

   1. Initial program 94.0%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.2

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

   4. Applied rewrites 41.2%

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

   if -0.94999999999999996 < (/.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))))) < -0.20000000000000001 or 2e-8 < (/.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))))) < 0.880000000000000004

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 47.4

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

   8. Applied rewrites 47.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

   if 0.880000000000000004 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 4: 75.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin ky}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_2}}\\ \mathbf{if}\;t\_3 \leq -0.95:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{elif}\;t\_3 \leq 0.88:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (* (sin ky) th) (hypot (sin kx) (sin ky))))
        (t_2 (pow (sin ky) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_2)))))
   (if (<= t_3 -0.95)
     (/ (* (sin ky) (sin th)) (sqrt t_2))
     (if (<= t_3 -0.2)
       t_1
       (if (<= t_3 2e-8)
         (* (/ (sin ky) (fabs (sin kx))) (sin th))
         (if (<= t_3 0.88) t_1 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double t_2 = pow(sin(ky), 2.0);
	double t_3 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.95) {
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = t_1;
	} else if (t_3 <= 2e-8) {
		tmp = (sin(ky) / fabs(sin(kx))) * sin(th);
	} else if (t_3 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_2 = Math.pow(Math.sin(ky), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2));
	double tmp;
	if (t_3 <= -0.95) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt(t_2);
	} else if (t_3 <= -0.2) {
		tmp = t_1;
	} else if (t_3 <= 2e-8) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(kx))) * Math.sin(th);
	} else if (t_3 <= 0.88) {
		tmp = t_1;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	t_2 = math.pow(math.sin(ky), 2.0)
	t_3 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))
	tmp = 0
	if t_3 <= -0.95:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt(t_2)
	elif t_3 <= -0.2:
		tmp = t_1
	elif t_3 <= 2e-8:
		tmp = (math.sin(ky) / math.fabs(math.sin(kx))) * math.sin(th)
	elif t_3 <= 0.88:
		tmp = t_1
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	t_2 = sin(ky) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_2)))
	tmp = 0.0
	if (t_3 <= -0.95)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(t_2));
	elseif (t_3 <= -0.2)
		tmp = t_1;
	elseif (t_3 <= 2e-8)
		tmp = Float64(Float64(sin(ky) / abs(sin(kx))) * sin(th));
	elseif (t_3 <= 0.88)
		tmp = t_1;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	t_2 = sin(ky) ^ 2.0;
	t_3 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_2));
	tmp = 0.0;
	if (t_3 <= -0.95)
		tmp = (sin(ky) * sin(th)) / sqrt(t_2);
	elseif (t_3 <= -0.2)
		tmp = t_1;
	elseif (t_3 <= 2e-8)
		tmp = (sin(ky) / abs(sin(kx))) * sin(th);
	elseif (t_3 <= 0.88)
		tmp = t_1;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.95], N[(N[(N[Sin[ky], $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.2], t$95$1, If[LessEqual[t$95$3, 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.88], t$95$1, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.94999999999999996

   1. Initial program 94.0%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 41.2

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

   4. Applied rewrites 41.2%

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

   if -0.94999999999999996 < (/.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))))) < -0.20000000000000001 or 2e-8 < (/.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))))) < 0.880000000000000004

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 47.4

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

   8. Applied rewrites 47.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.880000000000000004 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 5: 72.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{if}\;t\_1 \leq -0.95:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.88:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))))
   (if (<= t_1 -0.95)
     (* (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* 2.0 ky)))))) (sin th))
     (if (<= t_1 -0.2)
       t_2
       (if (<= t_1 2e-8)
         (* (/ (sin ky) (fabs (sin kx))) (sin th))
         (if (<= t_1 0.88) t_2 (* (/ ky (hypot ky (sin kx))) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	double tmp;
	if (t_1 <= -0.95) {
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((2.0 * ky)))))) * sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = (sin(ky) / fabs(sin(kx))) * sin(th);
	} else if (t_1 <= 0.88) {
		tmp = t_2;
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_2 = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	double tmp;
	if (t_1 <= -0.95) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 * (1.0 - Math.cos((2.0 * ky)))))) * Math.sin(th);
	} else if (t_1 <= -0.2) {
		tmp = t_2;
	} else if (t_1 <= 2e-8) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(kx))) * Math.sin(th);
	} else if (t_1 <= 0.88) {
		tmp = t_2;
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_2 = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	tmp = 0
	if t_1 <= -0.95:
		tmp = (math.sin(ky) / math.sqrt((0.5 * (1.0 - math.cos((2.0 * ky)))))) * math.sin(th)
	elif t_1 <= -0.2:
		tmp = t_2
	elif t_1 <= 2e-8:
		tmp = (math.sin(ky) / math.fabs(math.sin(kx))) * math.sin(th)
	elif t_1 <= 0.88:
		tmp = t_2
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)))
	tmp = 0.0
	if (t_1 <= -0.95)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(2.0 * ky)))))) * sin(th));
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = Float64(Float64(sin(ky) / abs(sin(kx))) * sin(th));
	elseif (t_1 <= 0.88)
		tmp = t_2;
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_2 = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	tmp = 0.0;
	if (t_1 <= -0.95)
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((2.0 * ky)))))) * sin(th);
	elseif (t_1 <= -0.2)
		tmp = t_2;
	elseif (t_1 <= 2e-8)
		tmp = (sin(ky) / abs(sin(kx))) * sin(th);
	elseif (t_1 <= 0.88)
		tmp = t_2;
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.95], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.2], t$95$2, If[LessEqual[t$95$1, 2e-8], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.88], t$95$2, N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.94999999999999996

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 30.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   6. Applied rewrites 30.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

   if -0.94999999999999996 < (/.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))))) < -0.20000000000000001 or 2e-8 < (/.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))))) < 0.880000000000000004

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 47.4

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

   8. Applied rewrites 47.4%

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

   if -0.20000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2e-8

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.880000000000000004 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 6: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in kx around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 30.9

         \[\leadsto \frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th \]

   6. Applied rewrites 30.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \cdot \sin th \]

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 7: 64.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (/ (* (sin ky) (sin th)) (sqrt (* 0.5 (- 1.0 (cos (* 2.0 ky))))))
   (* (/ ky (hypot ky (sin kx))) (sin th))))

C

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

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = (Math.sin(ky) * Math.sin(th)) / Math.sqrt((0.5 * (1.0 - Math.cos((2.0 * ky)))));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.02:
		tmp = (math.sin(ky) * math.sin(th)) / math.sqrt((0.5 * (1.0 - math.cos((2.0 * ky)))))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * sin(th)) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(2.0 * ky))))));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.02)
		tmp = (sin(ky) * sin(th)) / sqrt((0.5 * (1.0 - cos((2.0 * ky)))));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.0200000000000000004

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 30.7

         \[\leadsto \frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

   6. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}} \]

   if -0.0200000000000000004 < (sin.f64 ky)

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 8: 61.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.71:\\ \;\;\;\;\frac{\sin ky}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt t_1)) th)
     (if (<= t_2 0.71)
       (* (/ (sin ky) (fabs (sin kx))) (sin th))
       (* (/ ky (hypot ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else if (t_2 <= 0.71) {
		tmp = (sin(ky) / fabs(sin(kx))) * sin(th);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * th;
	} else if (t_2 <= 0.71) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(kx))) * Math.sin(th);
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * th
	elif t_2 <= 0.71:
		tmp = (math.sin(ky) / math.fabs(math.sin(kx))) * math.sin(th)
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	elseif (t_2 <= 0.71)
		tmp = Float64(Float64(sin(ky) / abs(sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (sin(ky) / sqrt(t_1)) * th;
	elseif (t_2 <= 0.71)
		tmp = (sin(ky) / abs(sin(kx))) * sin(th);
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.71], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 14.1%

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

   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 21.7

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

   13. Applied rewrites 21.7%

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

   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))))) < 0.70999999999999996

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.70999999999999996 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 9: 61.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin ky}^{2}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + t\_1}}\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{elif}\;t\_2 \leq 0.71:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin ky) 2.0))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) t_1)))))
   (if (<= t_2 -1.0)
     (* (/ (sin ky) (sqrt t_1)) th)
     (if (<= t_2 0.71)
       (* (sin ky) (/ (sin th) (fabs (sin kx))))
       (* (/ ky (hypot ky (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(ky), 2.0);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (sin(ky) / sqrt(t_1)) * th;
	} else if (t_2 <= 0.71) {
		tmp = sin(ky) * (sin(th) / fabs(sin(kx)));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(ky), 2.0);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_1));
	double tmp;
	if (t_2 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(t_1)) * th;
	} else if (t_2 <= 0.71) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(ky), 2.0)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + t_1))
	tmp = 0
	if t_2 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(t_1)) * th
	elif t_2 <= 0.71:
		tmp = math.sin(ky) * (math.sin(th) / math.fabs(math.sin(kx)))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + t_1)))
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(t_1)) * th);
	elseif (t_2 <= 0.71)
		tmp = Float64(sin(ky) * Float64(sin(th) / abs(sin(kx))));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) ^ 2.0;
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + t_1));
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = (sin(ky) / sqrt(t_1)) * th;
	elseif (t_2 <= 0.71)
		tmp = sin(ky) * (sin(th) / abs(sin(kx)));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 0.71], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -1

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 14.1%

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

   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 21.7

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

   13. Applied rewrites 21.7%

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

   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))))) < 0.70999999999999996

   1. Initial program 94.0%

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

   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 41.7

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

   4. Applied rewrites 41.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 41.7

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

      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 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.70999999999999996 < (/.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 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 10: 60.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 14.1%

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

   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 21.7

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

   13. Applied rewrites 21.7%

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

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

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 11: 57.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq -0.05:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.05], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 16.4

         \[\leadsto \frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

   9. Applied rewrites 16.4%

      \[\leadsto \frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

   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)))))

   1. Initial program 94.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 51.6%

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

   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 65.1%

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

Alternative 12: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 0.000115:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 0.000115)
   (* (sin th) (/ ky (fabs (sin kx))))
   (/ (* th (sin ky)) (sqrt (* 0.5 (- 1.0 (cos (* 2.0 ky))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.000115) {
		tmp = sin(th) * (ky / fabs(sin(kx)));
	} else {
		tmp = (th * sin(ky)) / sqrt((0.5 * (1.0 - cos((2.0 * ky)))));
	}
	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 (ky <= 0.000115d0) then
        tmp = sin(th) * (ky / abs(sin(kx)))
    else
        tmp = (th * sin(ky)) / sqrt((0.5d0 * (1.0d0 - cos((2.0d0 * ky)))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 0.000115) {
		tmp = Math.sin(th) * (ky / Math.abs(Math.sin(kx)));
	} else {
		tmp = (th * Math.sin(ky)) / Math.sqrt((0.5 * (1.0 - Math.cos((2.0 * ky)))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 0.000115:
		tmp = math.sin(th) * (ky / math.fabs(math.sin(kx)))
	else:
		tmp = (th * math.sin(ky)) / math.sqrt((0.5 * (1.0 - math.cos((2.0 * ky)))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 0.000115)
		tmp = Float64(sin(th) * Float64(ky / abs(sin(kx))));
	else
		tmp = Float64(Float64(th * sin(ky)) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(2.0 * ky))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 0.000115)
		tmp = sin(th) * (ky / abs(sin(kx)));
	else
		tmp = (th * sin(ky)) / sqrt((0.5 * (1.0 - cos((2.0 * ky)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 0.000115], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 1.15e-4

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.7

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

      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 39.8

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

   6. Applied rewrites 39.8%

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

   if 1.15e-4 < ky

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in kx around 0

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

      1. lower--.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 16.4

         \[\leadsto \frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]

   9. Applied rewrites 16.4%

      \[\leadsto \frac{th \cdot \sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot ky\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 41.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.5:\\ \;\;\;\;\sin th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.5)
   (* (sin th) (/ ky (fabs (sin kx))))
   (/ (* ky th) (sqrt (fma 0.5 (- 1.0 (cos (* 2.0 kx))) (pow ky 2.0))))))

C

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

Julia

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.5], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.5

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.7

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

      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 39.8

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

   6. Applied rewrites 39.8%

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

   if 0.5 < (/.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 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 20.3

         \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   9. Applied rewrites 20.3%

      \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 22.9

          \[\leadsto \frac{ky \cdot th}{\sqrt{\mathsf{fma}\left(0.5, \color{blue}{1 - \cos \left(2 \cdot kx\right)}, {ky}^{2}\right)}} \]

   12. Applied rewrites 22.9%

       \[\leadsto \frac{ky \cdot th}{\sqrt{\color{blue}{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 22.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.3 \cdot 10^{-122}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.3e-122)
   (* (/ ky kx) (sin th))
   (/ (* ky th) (sqrt (fma 0.5 (- 1.0 (cos (* 2.0 kx))) (pow ky 2.0))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.3e-122) {
		tmp = (ky / kx) * sin(th);
	} else {
		tmp = (ky * th) / sqrt(fma(0.5, (1.0 - cos((2.0 * kx))), pow(ky, 2.0)));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.3e-122)
		tmp = Float64(Float64(ky / kx) * sin(th));
	else
		tmp = Float64(Float64(ky * th) / sqrt(fma(0.5, Float64(1.0 - cos(Float64(2.0 * kx))), (ky ^ 2.0))));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.3e-122], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Power[ky, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.30000000000000007e-122

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 2.30000000000000007e-122 < ky

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 20.3

         \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   9. Applied rewrites 20.3%

      \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   10. Taylor expanded in ky around 0

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

       1. lower-*.f64 22.9

          \[\leadsto \frac{ky \cdot th}{\sqrt{\mathsf{fma}\left(0.5, \color{blue}{1 - \cos \left(2 \cdot kx\right)}, {ky}^{2}\right)}} \]

   12. Applied rewrites 22.9%

       \[\leadsto \frac{ky \cdot th}{\sqrt{\color{blue}{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 21.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;\frac{\sin ky}{\sqrt{t\_1 + {\sin ky}^{2}}} \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\frac{ky}{\sqrt{t\_1}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot \sin ky}{\sqrt{{ky}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= (/ (sin ky) (sqrt (+ t_1 (pow (sin ky) 2.0)))) 4e-5)
     (* (/ ky (sqrt t_1)) th)
     (/ (* th (sin ky)) (sqrt (pow ky 2.0))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if ((sin(ky) / sqrt((t_1 + pow(sin(ky), 2.0)))) <= 4e-5) {
		tmp = (ky / sqrt(t_1)) * th;
	} else {
		tmp = (th * sin(ky)) / sqrt(pow(ky, 2.0));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(kx) ** 2.0d0
    if ((sin(ky) / sqrt((t_1 + (sin(ky) ** 2.0d0)))) <= 4d-5) then
        tmp = (ky / sqrt(t_1)) * th
    else
        tmp = (th * sin(ky)) / sqrt((ky ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((t_1 + Math.pow(Math.sin(ky), 2.0)))) <= 4e-5) {
		tmp = (ky / Math.sqrt(t_1)) * th;
	} else {
		tmp = (th * Math.sin(ky)) / Math.sqrt(Math.pow(ky, 2.0));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	tmp = 0
	if (math.sin(ky) / math.sqrt((t_1 + math.pow(math.sin(ky), 2.0)))) <= 4e-5:
		tmp = (ky / math.sqrt(t_1)) * th
	else:
		tmp = (th * math.sin(ky)) / math.sqrt(math.pow(ky, 2.0))
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64(t_1 + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = Float64(Float64(ky / sqrt(t_1)) * th);
	else
		tmp = Float64(Float64(th * sin(ky)) / sqrt((ky ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	tmp = 0.0;
	if ((sin(ky) / sqrt((t_1 + (sin(ky) ^ 2.0)))) <= 4e-5)
		tmp = (ky / sqrt(t_1)) * th;
	else
		tmp = (th * sin(ky)) / sqrt((ky ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 4e-5], N[(N[(ky / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(th * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Power[ky, 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 4.00000000000000033e-5

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   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 19.9%

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

   if 4.00000000000000033e-5 < (/.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 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 20.3

         \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   9. Applied rewrites 20.3%

      \[\leadsto \frac{th \cdot \sin ky}{\sqrt{\mathsf{fma}\left(0.5, 1 - \cos \left(2 \cdot kx\right), {ky}^{2}\right)}} \]

   10. Taylor expanded in kx around 0

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

       1. lower-pow.f64 9.1

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

   12. Applied rewrites 9.1%

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

Alternative 16: 19.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin kx}^{2}}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.8e+68)
   (* (/ ky kx) (sin th))
   (* (/ ky (sqrt (pow (sin kx) 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(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 (kx <= 4.8d+68) then
        tmp = (ky / kx) * sin(th)
    else
        tmp = (ky / sqrt((sin(kx) ** 2.0d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(kx), 2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.8e+68:
		tmp = (ky / kx) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(math.pow(math.sin(kx), 2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.8e+68)
		tmp = Float64(Float64(ky / kx) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(kx) ^ 2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.8e+68)
		tmp = (ky / kx) * sin(th);
	else
		tmp = (ky / sqrt((sin(kx) ^ 2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.8e+68], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.80000000000000016e68

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 4.80000000000000016e68 < kx

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   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 19.9%

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

Alternative 17: 19.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{\frac{\cos \left(kx + kx\right) - 1}{-2}}} \cdot th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.8e+68)
   (* (/ ky kx) (sin th))
   (* (/ ky (sqrt (/ (- (cos (+ kx kx)) 1.0) -2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * sin(th);
	} else {
		tmp = (ky / sqrt(((cos((kx + kx)) - 1.0) / -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 (kx <= 4.8d+68) then
        tmp = (ky / kx) * sin(th)
    else
        tmp = (ky / sqrt(((cos((kx + kx)) - 1.0d0) / (-2.0d0)))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(((Math.cos((kx + kx)) - 1.0) / -2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.8e+68:
		tmp = (ky / kx) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(((math.cos((kx + kx)) - 1.0) / -2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.8e+68)
		tmp = Float64(Float64(ky / kx) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt(Float64(Float64(cos(Float64(kx + kx)) - 1.0) / -2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.8e+68)
		tmp = (ky / kx) * sin(th);
	else
		tmp = (ky / sqrt(((cos((kx + kx)) - 1.0) / -2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.8e+68], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[(N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] - 1.0), $MachinePrecision] / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.80000000000000016e68

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 4.80000000000000016e68 < kx

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 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. sin-mult N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. sub-flip N/A

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

      11. cos-diff N/A

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

      12. cos-sin-sum N/A

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

      13. sub-flip N/A

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

      14. sub-negate-rev N/A

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

      15. lower--.f64 N/A

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

      16. metadata-eval 26.9

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

   6. Applied rewrites 26.9%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.8%

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

Alternative 18: 19.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 4.8 \cdot 10^{+68}:\\ \;\;\;\;\frac{ky}{kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky \cdot th}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 4.8e+68)
   (* (/ ky kx) (sin th))
   (/ (* ky th) (sqrt (* 0.5 (- 1.0 (cos (* 2.0 kx))))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * sin(th);
	} else {
		tmp = (ky * th) / sqrt((0.5 * (1.0 - cos((2.0 * kx)))));
	}
	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 (kx <= 4.8d+68) then
        tmp = (ky / kx) * sin(th)
    else
        tmp = (ky * th) / sqrt((0.5d0 * (1.0d0 - cos((2.0d0 * kx)))))
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 4.8e+68) {
		tmp = (ky / kx) * Math.sin(th);
	} else {
		tmp = (ky * th) / Math.sqrt((0.5 * (1.0 - Math.cos((2.0 * kx)))));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 4.8e+68:
		tmp = (ky / kx) * math.sin(th)
	else:
		tmp = (ky * th) / math.sqrt((0.5 * (1.0 - math.cos((2.0 * kx)))))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 4.8e+68)
		tmp = Float64(Float64(ky / kx) * sin(th));
	else
		tmp = Float64(Float64(ky * th) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(2.0 * kx))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 4.8e+68)
		tmp = (ky / kx) * sin(th);
	else
		tmp = (ky * th) / sqrt((0.5 * (1.0 - cos((2.0 * kx)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 4.8e+68], N[(N[(ky / kx), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky * th), $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 4.80000000000000016e68

   1. Initial program 94.0%

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

   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 36.7

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.4

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

   7. Applied rewrites 17.4%

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

   if 4.80000000000000016e68 < kx

   1. Initial program 94.0%

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

      1. lift-+.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. add-flip N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sin-mult N/A

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

      10. remove-double-neg N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. sin-mult N/A

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

      16. distribute-neg-frac2 N/A

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

      17. sub-negate-rev N/A

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

      18. frac-2neg-rev N/A

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

      19. sub-div N/A

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

   3. Applied rewrites 75.5%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 38.8

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

   6. Applied rewrites 38.8%

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

   7. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 14.8

         \[\leadsto \frac{ky \cdot th}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}} \]

   9. Applied rewrites 14.8%

      \[\leadsto \frac{ky \cdot th}{\color{blue}{\sqrt{0.5 \cdot \left(1 - \cos \left(2 \cdot kx\right)\right)}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 17.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{kx} \cdot \sin th \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.4

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

7. Applied rewrites 17.4%

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

Alternative 20: 14.1% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \frac{ky}{kx} \cdot th \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.0%

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

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 36.7

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.4

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

7. Applied rewrites 17.4%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 14.1%

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

Reproduce

herbie shell --seed 2025156 
(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)))