Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 7.9s

Alternatives: 22

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.3% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

3. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lift-sin.f64 N/A

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

5. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lower-hypot.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. *-lft-identity N/A

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

   10. associate-/r/ N/A

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

   11. lower-*.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lift-sin.f64 N/A

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

   14. lower-hypot.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-sin.f64 99.6

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

7. Applied rewrites 99.6%

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

Alternative 2: 99.6% 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 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

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 3: 66.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 5.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 5.5e-12)
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
   (/ (sin th) (/ (hypot ky (sin kx)) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 5.5e-12) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 5.5e-12) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = Math.sin(th) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 5.5e-12:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = math.sin(th) / (math.hypot(ky, math.sin(kx)) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 5.5e-12)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 5.5e-12)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 5.5e-12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.5000000000000004e-12

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.5%

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

   if 5.5000000000000004e-12 < th

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 4: 65.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.08 \cdot 10^{-13}:\\ \;\;\;\;\frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 1.08e-13)
   (* (/ th (hypot (sin kx) (sin ky))) (sin ky))
   (/ (sin th) (/ (hypot ky (sin kx)) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.08e-13) {
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.08e-13) {
		tmp = (th / Math.hypot(Math.sin(kx), Math.sin(ky))) * Math.sin(ky);
	} else {
		tmp = Math.sin(th) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 1.08e-13:
		tmp = (th / math.hypot(math.sin(kx), math.sin(ky))) * math.sin(ky)
	else:
		tmp = math.sin(th) / (math.hypot(ky, math.sin(kx)) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 1.08e-13)
		tmp = Float64(Float64(th / hypot(sin(kx), sin(ky))) * sin(ky));
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 1.08e-13)
		tmp = (th / hypot(sin(kx), sin(ky))) * sin(ky);
	else
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 1.08e-13], N[(N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.0799999999999999e-13

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lift-sin.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-lft-identity N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 50.5%

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

   if 1.0799999999999999e-13 < th

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 5: 65.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 2.1:\\ \;\;\;\;\frac{\sin th}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(ky + ky\right) \cdot 0.5}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 2.1)
     (* (/ (sin th) (hypot (sin kx) t_1)) t_1)
     (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ ky ky)) 0.5)))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (ky <= 2.1) {
		tmp = (sin(th) / hypot(sin(kx), t_1)) * t_1;
	} else {
		tmp = (sin(ky) / sqrt((0.5 - (cos((ky + ky)) * 0.5)))) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (ky <= 2.1)
		tmp = Float64(Float64(sin(th) / hypot(sin(kx), t_1)) * t_1);
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(ky + ky)) * 0.5)))) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[ky, 2.1], N[(N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.10000000000000009

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lift-sin.f64 N/A

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

   5. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. *-lft-identity N/A

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

      10. associate-/r/ N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. lower-hypot.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. lift-sin.f64 N/A

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

      17. lift-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 51.4

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

   10. Applied rewrites 51.4%

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

   11. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 53.2

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

   13. Applied rewrites 53.2%

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

   if 2.10000000000000009 < ky

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in kx around 0

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

      1. pow2 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 31.6

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

   7. Applied rewrites 31.6%

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

Alternative 6: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.98:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\mathsf{fma}\left(kx \cdot kx, -0.16666666666666666, 1\right) \cdot kx, \sin ky\right)}{\sin ky}}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-264}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.98)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/
       (hypot (* (fma (* kx kx) -0.16666666666666666 1.0) kx) (sin ky))
       (sin ky)))
     (if (<= t_1 5e-264)
       (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) (sin th))
       (/ (sin th) (/ (hypot ky (sin kx)) ky))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.98) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (hypot((fma((kx * kx), -0.16666666666666666, 1.0) * kx), sin(ky)) / sin(ky));
	} else if (t_1 <= 5e-264) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * sin(th);
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.98)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(hypot(Float64(fma(Float64(kx * kx), -0.16666666666666666, 1.0) * kx), sin(ky)) / sin(ky)));
	elseif (t_1 <= 5e-264)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * sin(th));
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.98], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[(N[(N[(kx * kx), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-264], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.97999999999999998

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lift-sin.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 32.7

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

   11. Applied rewrites 32.7%

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

   if -0.97999999999999998 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000001e-264

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 31.3

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

   7. Applied rewrites 31.3%

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

   if 5.0000000000000001e-264 < (/.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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 7: 64.7% accurate, 0.7× speedup

Math

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

FPCore

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

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)))) <= -0.02) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (hypot(kx, sin(ky)) / sin(ky));
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	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)))) <= -0.02)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(hypot(kx, sin(ky)) / sin(ky)));
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	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], -0.02], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $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))))) < -0.0200000000000000004

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lift-sin.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in kx around 0

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

   11. Applied rewrites 32.9%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 8: 59.7% accurate, 0.8× speedup

Math

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

FPCore

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

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)))) <= -0.02) {
		tmp = (sin(ky) / sqrt(fma(kx, kx, (0.5 - (0.5 * cos((2.0 * ky))))))) * th;
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	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)))) <= -0.02)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(kx, kx, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky))))))) * th);
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	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], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(kx * kx + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $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))))) < -0.0200000000000000004

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 26.2%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 9: 59.5% accurate, 0.8× speedup

Math

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

FPCore

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

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)))) <= -0.02) {
		tmp = (sin(ky) * th) / sqrt((fma(kx, kx, 0.5) - (cos((ky + ky)) * 0.5)));
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	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)))) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) / sqrt(Float64(fma(kx, kx, 0.5) - Float64(cos(Float64(ky + ky)) * 0.5))));
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	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], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[(N[(kx * kx + 0.5), $MachinePrecision] - N[(N[Cos[N[(ky + ky), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $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))))) < -0.0200000000000000004

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 42.5

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

      11. lift-fma.f64 N/A

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

   6. Applied rewrites 34.8%

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

   7. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 21.6

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

   9. Applied rewrites 21.6%

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

   if -0.0200000000000000004 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 10: 57.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;\sin ky \leq -0.3:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, t\_1\right)}{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= (sin ky) -0.3)
     (/
      (* (fma (* th th) -0.16666666666666666 1.0) th)
      (/ (hypot (sin kx) t_1) t_1))
     (/ (sin th) (/ (hypot ky (sin kx)) ky)))))

C

double code(double kx, double ky, double th) {
	double t_1 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (sin(ky) <= -0.3) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (hypot(sin(kx), t_1) / t_1);
	} else {
		tmp = sin(th) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (sin(ky) <= -0.3)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(hypot(sin(kx), t_1) / t_1));
	else
		tmp = Float64(sin(th) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[N[Sin[ky], $MachinePrecision], -0.3], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sin[th], $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 ky) < -0.299999999999999989

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lift-sin.f64 N/A

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in th around 0

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

      1. *-lft-identity N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 50.2

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

   8. Applied rewrites 50.2%

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

   9. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 26.5

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

   11. Applied rewrites 26.5%

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

   12. Taylor expanded in ky around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 27.4

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

   14. Applied rewrites 27.4%

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

   if -0.299999999999999989 < (sin.f64 ky)

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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-sin.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. division-flip N/A

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

      10. lower-/.f64 N/A

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

      11. lower-special-/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.8%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 64.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. *-lft-identity N/A

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

       6. lower-/.f64 N/A

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

       7. lift-sin.f64 64.7

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

   11. Applied rewrites 64.7%

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

Alternative 11: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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-sin.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. division-flip N/A

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

   10. lower-/.f64 N/A

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

   11. lower-special-/ N/A

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

   12. lower-/.f64 N/A

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

3. Applied rewrites 99.6%

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

4. Taylor expanded in ky around 0

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

6. Applied rewrites 51.8%

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

7. Taylor expanded in ky around 0

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

9. Applied rewrites 64.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-sin.f64 N/A

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

    4. associate-*l/ N/A

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

    5. *-lft-identity N/A

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

    6. lower-/.f64 N/A

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

    7. lift-sin.f64 64.7

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

11. Applied rewrites 64.7%

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

Alternative 12: 56.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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. lift-pow.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

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 50.5%

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

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 64.7%

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

Alternative 13: 56.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(kx + kx\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.17:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.1)
   (* (/ (sin ky) (sqrt (- 0.5 (* (cos (+ kx kx)) 0.5)))) th)
   (if (<= (sin kx) 0.17)
     (*
      (/
       ky
       (hypot
        ky
        (*
         (fma
          (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (/ (* (sin th) ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.1) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((kx + kx)) * 0.5)))) * th;
	} else if (sin(kx) <= 0.17) {
		tmp = (ky / hypot(ky, (fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(th) * ky) / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(kx + kx)) * 0.5)))) * th);
	elseif (sin(kx) <= 0.17)
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.17], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.10000000000000001

   1. Initial program 94.3%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. sqr-sin-a N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 43.6

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in ky around 0

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

      1. pow2 N/A

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

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

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f64 31.3

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

   7. Applied rewrites 31.3%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 16.6%

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

   if -0.10000000000000001 < (sin.f64 kx) < 0.170000000000000012

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-flip N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 45.9

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

   12. Applied rewrites 45.9%

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

   if 0.170000000000000012 < (sin.f64 kx)

   1. Initial program 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 23.8

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

   4. Applied rewrites 23.8%

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

Alternative 14: 48.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-306}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.17:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(\mathsf{fma}\left(0.008333333333333333, kx \cdot kx, -0.16666666666666666\right), kx \cdot kx, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 1e-306)
   (* (/ ky (hypot ky (sin kx))) th)
   (if (<= (sin kx) 0.17)
     (*
      (/
       ky
       (hypot
        ky
        (*
         (fma
          (fma 0.008333333333333333 (* kx kx) -0.16666666666666666)
          (* kx kx)
          1.0)
         kx)))
      (sin th))
     (/ (* (sin th) ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 1e-306) {
		tmp = (ky / hypot(ky, sin(kx))) * th;
	} else if (sin(kx) <= 0.17) {
		tmp = (ky / hypot(ky, (fma(fma(0.008333333333333333, (kx * kx), -0.16666666666666666), (kx * kx), 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(th) * ky) / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 1e-306)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * th);
	elseif (sin(kx) <= 0.17)
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(fma(0.008333333333333333, Float64(kx * kx), -0.16666666666666666), Float64(kx * kx), 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-306], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.17], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(0.008333333333333333 * N[(kx * kx), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < 1.00000000000000003e-306

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 33.4%

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

   if 1.00000000000000003e-306 < (sin.f64 kx) < 0.170000000000000012

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. sub-flip N/A

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

       7. metadata-eval N/A

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

       8. lower-fma.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 45.9

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

   12. Applied rewrites 45.9%

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

   if 0.170000000000000012 < (sin.f64 kx)

   1. Initial program 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 23.8

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

   4. Applied rewrites 23.8%

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

Alternative 15: 47.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 10^{-306}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot th\\ \mathbf{elif}\;\sin kx \leq 0.01:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \mathsf{fma}\left(kx \cdot kx, -0.16666666666666666, 1\right) \cdot kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 1e-306)
   (* (/ ky (hypot ky (sin kx))) th)
   (if (<= (sin kx) 0.01)
     (*
      (/ ky (hypot ky (* (fma (* kx kx) -0.16666666666666666 1.0) kx)))
      (sin th))
     (/ (* (sin th) ky) (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 1e-306) {
		tmp = (ky / hypot(ky, sin(kx))) * th;
	} else if (sin(kx) <= 0.01) {
		tmp = (ky / hypot(ky, (fma((kx * kx), -0.16666666666666666, 1.0) * kx))) * sin(th);
	} else {
		tmp = (sin(th) * ky) / sin(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 1e-306)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * th);
	elseif (sin(kx) <= 0.01)
		tmp = Float64(Float64(ky / hypot(ky, Float64(fma(Float64(kx * kx), -0.16666666666666666, 1.0) * kx))) * sin(th));
	else
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 1e-306], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 0.01], N[(N[(ky / N[Sqrt[ky ^ 2 + N[(N[(N[(kx * kx), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * kx), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < 1.00000000000000003e-306

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 33.4%

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

   if 1.00000000000000003e-306 < (sin.f64 kx) < 0.0100000000000000002

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in kx around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 46.0

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

   12. Applied rewrites 46.0%

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

   if 0.0100000000000000002 < (sin.f64 kx)

   1. Initial program 94.3%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 23.8

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

   4. Applied rewrites 23.8%

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

Alternative 16: 46.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 1.7e-9)
   (* (/ ky (hypot ky (sin kx))) th)
   (* (/ ky (hypot ky kx)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.7e-9) {
		tmp = (ky / hypot(ky, sin(kx))) * th;
	} else {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 1.7e-9) {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * th;
	} else {
		tmp = (ky / Math.hypot(ky, kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 1.7e-9:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * th
	else:
		tmp = (ky / math.hypot(ky, kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 1.7e-9)
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * th);
	else
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 1.7e-9)
		tmp = (ky / hypot(ky, sin(kx))) * th;
	else
		tmp = (ky / hypot(ky, kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 1.7e-9], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 1.6999999999999999e-9

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in th around 0

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

   12. Applied rewrites 33.4%

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

   if 1.6999999999999999e-9 < th

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.7%

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

Alternative 17: 42.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq 0.197:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sin kx}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) 0.197)
   (* (/ ky (hypot ky kx)) (sin th))
   (/ (* (fma (* th th) -0.16666666666666666 1.0) th) (/ (sin kx) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= 0.197) {
		tmp = (ky / hypot(ky, kx)) * sin(th);
	} else {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sin(kx) / ky);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= 0.197)
		tmp = Float64(Float64(ky / hypot(ky, kx)) * sin(th));
	else
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sin(kx) / ky));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], 0.197], N[(N[(ky / N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 kx) < 0.19700000000000001

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 50.5%

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

   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 64.7%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.7%

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

   if 0.19700000000000001 < (sin.f64 kx)

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

      6. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
   7. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{th} \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      8. lower-*.f64 50.2

         \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   8. Applied rewrites 50.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sin kx}{\color{blue}{ky}}} \]

       2. lift-sin.f64 15.3

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sin kx}{ky}} \]

   11. Applied rewrites 15.3%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\color{blue}{\frac{\sin kx}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 35.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sin kx}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 2e-10)
   (/ (* (fma (* th th) -0.16666666666666666 1.0) th) (/ (sin kx) ky))
   (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)))) <= 2e-10) {
		tmp = (fma((th * th), -0.16666666666666666, 1.0) * th) / (sin(kx) / ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 2e-10)
		tmp = Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / Float64(sin(kx) / ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2e-10], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / N[(N[Sin[kx], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2.00000000000000007e-10

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}} + {\sin ky}^{2}}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{\sin kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

      5. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + \color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]

      6. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{\sin ky}}^{2}}} \cdot \sin th \]

      7. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2} + {\sin kx}^{2}}}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\sin ky \cdot \sin ky} + {\sin kx}^{2}}} \cdot \sin th \]

      9. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \color{blue}{\sin kx \cdot \sin kx}}} \cdot \sin th \]

      10. lower-hypot.f64 N/A

          \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      11. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      12. lift-sin.f64 99.7

          \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

   3. Applied rewrites 99.7%

      \[\leadsto \frac{\sin ky}{\color{blue}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\color{blue}{\sin ky}}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\color{blue}{\sin ky}, \sin kx\right)} \cdot \sin th \]

      5. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \color{blue}{\sin kx}\right)} \cdot \sin th \]

      6. lower-hypot.f64 N/A

         \[\leadsto \frac{\sin ky}{\color{blue}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}}} \cdot \sin th \]

      7. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\sin ky \cdot \sin ky + \sin kx \cdot \sin kx}} \cdot \color{blue}{\sin th} \]

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}}} \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]
   7. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \frac{\color{blue}{th} \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      4. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

      8. lower-*.f64 50.2

         \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   8. Applied rewrites 50.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\color{blue}{\frac{\sin kx}{ky}}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th}{\frac{\sin kx}{\color{blue}{ky}}} \]

       2. lift-sin.f64 15.3

          \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\frac{\sin kx}{ky}} \]

   11. Applied rewrites 15.3%

       \[\leadsto \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{\color{blue}{\frac{\sin kx}{ky}}} \]

   if 2.00000000000000007e-10 < (/.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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 31.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 1.55 \cdot 10^{-23}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      1.55e-23)
   (* (* (* th th) th) -0.16666666666666666)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1.55e-23) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1.55d-23) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1.55e-23) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1.55e-23:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.55e-23)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1.55e-23)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.55e-23], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 1.5499999999999999e-23

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 13.5

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 13.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      4. pow2 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      6. pow2 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      7. lift-*.f64 10.9

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 10.9%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 1.5499999999999999e-23 < (/.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 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 15.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-299}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-299)
   (* (* (* th th) -0.16666666666666666) th)
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-299) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 1d-299) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 1e-299) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 1e-299:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-299)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-299)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-299], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.99999999999999992e-300

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 13.5

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 13.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6}\right) \cdot th \]

      3. pow2 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot \frac{-1}{6}\right) \cdot th \]

      4. lift-*.f64 10.9

         \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   10. Applied rewrites 10.9%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 9.99999999999999992e-300 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \]
   6. Step-by-step derivation

   7. Applied rewrites 13.8%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 15.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-299}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-299)
   (* (* (* th th) th) -0.16666666666666666)
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-299) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 1d-299) then
        tmp = ((th * th) * th) * (-0.16666666666666666d0)
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 1e-299) {
		tmp = ((th * th) * th) * -0.16666666666666666;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 1e-299:
		tmp = ((th * th) * th) * -0.16666666666666666
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-299)
		tmp = Float64(Float64(Float64(th * th) * th) * -0.16666666666666666);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-299)
		tmp = ((th * th) * th) * -0.16666666666666666;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-299], N[(N[(N[(th * th), $MachinePrecision] * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.99999999999999992e-300

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {th}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot th \]

      3. +-commutative N/A

         \[\leadsto \left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th \]

      4. *-commutative N/A

         \[\leadsto \left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th \]

      6. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th \]

      7. lower-*.f64 13.5

         \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th \]

   7. Applied rewrites 13.5%

      \[\leadsto \mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot \color{blue}{th} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{6} \cdot {th}^{\color{blue}{3}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      2. lower-*.f64 N/A

         \[\leadsto {th}^{3} \cdot \frac{-1}{6} \]

      3. unpow3 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      4. pow2 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      5. lower-*.f64 N/A

         \[\leadsto \left({th}^{2} \cdot th\right) \cdot \frac{-1}{6} \]

      6. pow2 N/A

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot \frac{-1}{6} \]

      7. lift-*.f64 10.9

         \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   10. Applied rewrites 10.9%

       \[\leadsto \left(\left(th \cdot th\right) \cdot th\right) \cdot -0.16666666666666666 \]

   if 9.99999999999999992e-300 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 94.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   3. Step-by-step derivation

      1. lift-sin.f64 24.2

         \[\leadsto \sin th \]

   4. Applied rewrites 24.2%

      \[\leadsto \color{blue}{\sin th} \]

   5. Taylor expanded in th around 0

      \[\leadsto th \]
   6. Step-by-step derivation

   7. Applied rewrites 13.8%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 13.8% accurate, 170.4× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 th)

C

double code(double kx, double ky, double th) {
	return th;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = th
end function

Java

public static double code(double kx, double ky, double th) {
	return th;
}

Python

def code(kx, ky, th):
	return th

Julia

function code(kx, ky, th)
	return th
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th;
end

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 94.3%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
3. Step-by-step derivation

   1. lift-sin.f64 24.2

      \[\leadsto \sin th \]

4. Applied rewrites 24.2%

   \[\leadsto \color{blue}{\sin th} \]

5. Taylor expanded in th around 0

   \[\leadsto th \]
6. Step-by-step derivation

7. Applied rewrites 13.8%

   \[\leadsto th \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025132 
(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)))