Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 9.4s

Alternatives: 24

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: 94.2% 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(\sin kx, \sin ky\right)}{\sin ky}} \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	return (1.0 * sin(th)) / (hypot(sin(kx), sin(ky)) / sin(ky));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

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

   12. unpow2 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-sin.f64 99.6

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

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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. associate-*l/ N/A

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

   10. lower-/.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lower-hypot.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lift-sin.f64 N/A

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

   16. lift-sin.f64 N/A

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

   17. lift-/.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. associate-*l/ N/A

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

   12. associate-/l* N/A

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

   13. lower-*.f64 N/A

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

   14. lift-sin.f64 N/A

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

   15. lower-/.f64 N/A

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

3. Applied rewrites 99.6%

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

Alternative 4: 74.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.046:\\ \;\;\;\;\frac{\sin ky \cdot th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

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 -1.0)
     (* (/ (sin ky) (fabs (sin ky))) (sin th))
     (if (<= t_1 -0.046)
       (/ (* (sin ky) th) (hypot (sin kx) (sin ky)))
       (/ (* 1.0 (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 <= -1.0) {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	} else if (t_1 <= -0.046) {
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	} else {
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	} else if (t_1 <= -0.046) {
		tmp = (Math.sin(ky) * th) / Math.hypot(Math.sin(kx), Math.sin(ky));
	} else {
		tmp = (1.0 * Math.sin(th)) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -1.0:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	elif t_1 <= -0.046:
		tmp = (math.sin(ky) * th) / math.hypot(math.sin(kx), math.sin(ky))
	else:
		tmp = (1.0 * math.sin(th)) / (math.hypot(ky, math.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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	elseif (t_1 <= -0.046)
		tmp = Float64(Float64(sin(ky) * th) / hypot(sin(kx), sin(ky)));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -1.0)
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	elseif (t_1 <= -0.046)
		tmp = (sin(ky) * th) / hypot(sin(kx), sin(ky));
	else
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.046], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $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))))) < -1

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 44.5

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

   4. Applied rewrites 44.5%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.045999999999999999

   1. Initial program 94.2%

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

   2. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. lower-hypot.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-sin.f64 47.2

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

   4. Applied rewrites 47.2%

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

   if -0.045999999999999999 < (/.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 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

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

Alternative 5: 72.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 2.85e-5)
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
   (/ (* 1.0 (sin th)) (/ (hypot ky (sin kx)) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.85e-5) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.85e-5) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = (1.0 * Math.sin(th)) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 2.85e-5:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = (1.0 * math.sin(th)) / (math.hypot(ky, math.sin(kx)) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 2.85e-5)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(Float64(1.0 * 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 <= 2.85e-5)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 2.85e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $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 < 2.8500000000000002e-5

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.9%

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

   if 2.8500000000000002e-5 < th

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 2.85 \cdot 10^{-5}:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 2.85e-5)
   (* (sin ky) (/ th (hypot (sin kx) (sin ky))))
   (/ (* 1.0 (sin th)) (/ (hypot ky (sin kx)) ky))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.85e-5) {
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	} else {
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 2.85e-5) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(kx), Math.sin(ky)));
	} else {
		tmp = (1.0 * Math.sin(th)) / (Math.hypot(ky, Math.sin(kx)) / ky);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 2.85e-5:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(kx), math.sin(ky)))
	else:
		tmp = (1.0 * math.sin(th)) / (math.hypot(ky, math.sin(kx)) / ky)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 2.85e-5)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))));
	else
		tmp = Float64(Float64(1.0 * 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 <= 2.85e-5)
		tmp = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	else
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 2.85e-5], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $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 < 2.8500000000000002e-5

   1. Initial program 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow2 N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-hypot.f64 50.8

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

   6. Applied rewrites 50.8%

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

   if 2.8500000000000002e-5 < th

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 66.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;ky \leq 25:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= ky 25.0)
     (* (/ t_1 (hypot (sin kx) t_1)) (sin th))
     (* (/ (sin ky) (fabs (sin ky))) (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 <= 25.0) {
		tmp = (t_1 / hypot(sin(kx), t_1)) * sin(th);
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * 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 <= 25.0)
		tmp = Float64(Float64(t_1 / hypot(sin(kx), t_1)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * 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, 25.0], N[(N[(t$95$1 / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ky < 25

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 55.0

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

   9. Applied rewrites 55.0%

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

   if 25 < ky

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 44.5

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

   4. Applied rewrites 44.5%

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

Alternative 8: 65.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2750:\\ \;\;\;\;\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\left|\sin ky\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2750.0)
   (/ (* 1.0 (sin th)) (/ (hypot ky (sin kx)) ky))
   (* (/ (sin ky) (fabs (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2750.0) {
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	} else {
		tmp = (sin(ky) / fabs(sin(ky))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2750.0) {
		tmp = (1.0 * Math.sin(th)) / (Math.hypot(ky, Math.sin(kx)) / ky);
	} else {
		tmp = (Math.sin(ky) / Math.abs(Math.sin(ky))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2750.0:
		tmp = (1.0 * math.sin(th)) / (math.hypot(ky, math.sin(kx)) / ky)
	else:
		tmp = (math.sin(ky) / math.fabs(math.sin(ky))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2750.0)
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(hypot(ky, sin(kx)) / ky));
	else
		tmp = Float64(Float64(sin(ky) / abs(sin(ky))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2750.0)
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	else
		tmp = (sin(ky) / abs(sin(ky))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2750.0], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2750

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

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

   if 2750 < ky

   1. Initial program 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 44.5

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

   4. Applied rewrites 44.5%

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

Alternative 9: 65.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.02)
   (/ (* (sin th) (sin ky)) (fabs (sin ky)))
   (/ (* 1.0 (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(th) * sin(ky)) / fabs(sin(ky));
	} else {
		tmp = (1.0 * sin(th)) / (hypot(ky, sin(kx)) / ky);
	}
	return tmp;
}

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= -0.02:
		tmp = (math.sin(th) * math.sin(ky)) / math.fabs(math.sin(ky))
	else:
		tmp = (1.0 * math.sin(th)) / (math.hypot(ky, math.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(th) * sin(ky)) / abs(sin(ky)));
	else
		tmp = Float64(Float64(1.0 * sin(th)) / Float64(hypot(ky, sin(kx)) / ky));
	end
	return tmp
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.02], N[(N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / N[Abs[N[Sin[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $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 94.2%

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

   2. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. unpow2 N/A

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

      7. rem-sqrt-square N/A

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

      8. lower-fabs.f64 N/A

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

      9. lift-sin.f64 50.1

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

   4. Applied rewrites 50.1%

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

   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 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 60.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.98)
     (* (/ 1.0 (/ (hypot t_1 kx) t_1)) (sin th))
     (/ (* 1.0 (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) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.98) {
		tmp = (1.0 / (hypot(t_1, kx) / t_1)) * sin(th);
	} else {
		tmp = (1.0 * 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 (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.98)
		tmp = Float64(Float64(1.0 / Float64(hypot(t_1, kx) / t_1)) * sin(th));
	else
		tmp = Float64(Float64(1.0 * 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[(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], -0.98], N[(N[(1.0 / N[(N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 * N[Sin[th], $MachinePrecision]), $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))))) < -0.97999999999999998

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.3

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 36.8%

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

   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)))))

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. associate-*l/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-sin.f64 65.0

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

   11. Applied rewrites 65.0%

       \[\leadsto \color{blue}{\frac{1 \cdot \sin th}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 60.7% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) -0.98)
     (* (/ 1.0 (/ (hypot t_1 kx) t_1)) (sin th))
     (* (/ ky (hypot (sin kx) ky)) (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 ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= -0.98) {
		tmp = (1.0 / (hypot(t_1, kx) / t_1)) * sin(th);
	} else {
		tmp = (ky / hypot(sin(kx), ky)) * 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 (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= -0.98)
		tmp = Float64(Float64(1.0 / Float64(hypot(t_1, kx) / t_1)) * sin(th));
	else
		tmp = Float64(Float64(ky / hypot(sin(kx), ky)) * 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[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], -0.98], N[(N[(1.0 / N[(N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.3

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 36.8%

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

   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)))))

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

Alternative 12: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_1 \leq -0.405:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(t\_2, kx\right)}{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\frac{t\_2}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_1 -0.405)
     (* (/ 1.0 (/ (hypot t_2 kx) t_2)) (sin th))
     (if (<= t_1 0.05)
       (* (/ t_2 (fabs (sin kx))) (sin th))
       (* (/ 1.0 (/ (hypot ky kx) ky)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_1 <= -0.405) {
		tmp = (1.0 / (hypot(t_2, kx) / t_2)) * sin(th);
	} else if (t_1 <= 0.05) {
		tmp = (t_2 / fabs(sin(kx))) * sin(th);
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_1 <= -0.405)
		tmp = Float64(Float64(1.0 / Float64(hypot(t_2, kx) / t_2)) * sin(th));
	elseif (t_1 <= 0.05)
		tmp = Float64(Float64(t_2 / abs(sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$1, -0.405], N[(N[(1.0 / N[(N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(N[(t$95$2 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40500000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.3

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 36.8%

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

   if -0.40500000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. rem-sqrt-square N/A

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

      3. lower-fabs.f64 N/A

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

      4. lift-sin.f64 44.3

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

   4. Applied rewrites 44.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 38.7

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

   7. Applied rewrites 38.7%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

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

Alternative 13: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := \mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\\ \mathbf{if}\;t\_1 \leq -0.405:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(t\_2, kx\right)}{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;ky \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* (fma (* ky ky) -0.16666666666666666 1.0) ky)))
   (if (<= t_1 -0.405)
     (* (/ 1.0 (/ (hypot t_2 kx) t_2)) (sin th))
     (if (<= t_1 0.05)
       (* ky (/ (sin th) (fabs (sin kx))))
       (* (/ 1.0 (/ (hypot ky kx) ky)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = fma((ky * ky), -0.16666666666666666, 1.0) * ky;
	double tmp;
	if (t_1 <= -0.405) {
		tmp = (1.0 / (hypot(t_2, kx) / t_2)) * sin(th);
	} else if (t_1 <= 0.05) {
		tmp = ky * (sin(th) / fabs(sin(kx)));
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky)
	tmp = 0.0
	if (t_1 <= -0.405)
		tmp = Float64(Float64(1.0 / Float64(hypot(t_2, kx) / t_2)) * sin(th));
	elseif (t_1 <= 0.05)
		tmp = Float64(ky * Float64(sin(th) / abs(sin(kx))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision]}, If[LessEqual[t$95$1, -0.405], N[(N[(1.0 / N[(N[Sqrt[t$95$2 ^ 2 + kx ^ 2], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.05], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.40500000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 45.3

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

   4. Applied rewrites 45.3%

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

   5. Taylor expanded in ky around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 47.3

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

   7. Applied rewrites 47.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 47.3

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

   9. Applied rewrites 55.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 36.8%

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

   if -0.40500000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.050000000000000003

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-sin.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

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

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

      15. lift-sin.f64 N/A

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

      16. lift-fabs.f64 39.1

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

   6. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

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

Alternative 14: 51.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (* ky (/ (sin th) (fabs (sin kx))))
   (* (/ 1.0 (/ (hypot ky 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)))) <= 0.05) {
		tmp = ky * (sin(th) / fabs(sin(kx)));
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-sin.f64 37.1

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

   4. Applied rewrites 37.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-fabs.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. pow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. pow2 N/A

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

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

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

      15. lift-sin.f64 N/A

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

      16. lift-fabs.f64 39.1

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

   6. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.2% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (* (/ ky (fabs (sin kx))) (sin th))
   (* (/ 1.0 (/ (hypot ky 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)))) <= 0.05) {
		tmp = (ky / fabs(sin(kx))) * sin(th);
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. rem-sqrt-square N/A

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

      4. lower-fabs.f64 N/A

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

      5. lift-sin.f64 39.1

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

   4. Applied rewrites 39.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.05)
   (/ (* (sin th) ky) (fabs (sin kx)))
   (* (/ 1.0 (/ (hypot ky 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)))) <= 0.05) {
		tmp = (sin(th) * ky) / fabs(sin(kx));
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-sin.f64 37.1

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

   4. Applied rewrites 37.1%

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

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

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 48.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 5.9 \cdot 10^{-6}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 5.9e-6)
   (*
    (/ 1.0 (/ (hypot ky (sin kx)) ky))
    (* (fma (* th th) -0.16666666666666666 1.0) th))
   (* (/ 1.0 (/ (hypot ky kx) ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 5.9e-6) {
		tmp = (1.0 / (hypot(ky, sin(kx)) / ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 5.9e-6)
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, sin(kx)) / ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 5.9e-6], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 5.90000000000000026e-6

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in th around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. unpow2 N/A

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

       7. lower-*.f64 33.9

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

   12. Applied rewrites 33.9%

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

   if 5.90000000000000026e-6 < th

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 1.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 1.1e+14)
   (* (/ 1.0 (/ (hypot ky kx) ky)) (sin th))
   (/ (* th ky) (fabs (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.1e+14) {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	} else {
		tmp = (th * ky) / fabs(sin(kx));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 1.1e+14) {
		tmp = (1.0 / (Math.hypot(ky, kx) / ky)) * Math.sin(th);
	} else {
		tmp = (th * ky) / Math.abs(Math.sin(kx));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 1.1e+14:
		tmp = (1.0 / (math.hypot(ky, kx) / ky)) * math.sin(th)
	else:
		tmp = (th * ky) / math.fabs(math.sin(kx))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 1.1e+14)
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * sin(th));
	else
		tmp = Float64(Float64(th * ky) / abs(sin(kx)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 1.1e+14)
		tmp = (1.0 / (hypot(ky, kx) / ky)) * sin(th);
	else
		tmp = (th * ky) / abs(sin(kx));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 1.1e+14], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(th * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.1e14

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

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

   if 1.1e14 < kx

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. unpow2 N/A

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

      6. rem-sqrt-square N/A

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

      7. lower-fabs.f64 N/A

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

      8. lift-sin.f64 37.1

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

   4. Applied rewrites 37.1%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 18.8%

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

Alternative 19: 36.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{th \cdot ky}{\left|\sin kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 6.4e-7)
   (* (sin th) (/ ky (sqrt (fma ky ky (* kx kx)))))
   (/ (* th ky) (fabs (sin kx)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 6.4e-7) {
		tmp = sin(th) * (ky / sqrt(fma(ky, ky, (kx * kx))));
	} else {
		tmp = (th * ky) / fabs(sin(kx));
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 6.4e-7)
		tmp = Float64(sin(th) * Float64(ky / sqrt(fma(ky, ky, Float64(kx * kx)))));
	else
		tmp = Float64(Float64(th * ky) / abs(sin(kx)));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 6.4e-7], N[(N[Sin[th], $MachinePrecision] * N[(ky / N[Sqrt[N[(ky * ky + N[(kx * kx), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(th * ky), $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 6.4000000000000001e-7

   1. Initial program 94.2%

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

   2. Taylor expanded in ky around 0

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

   4. Applied rewrites 45.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 52.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 52.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 N/A

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

      14. lift-sin.f64 65.0

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

   9. Applied rewrites 65.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 46.5%

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

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lift-sin.f64 46.5

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

       6. lift-/.f64 N/A

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

       7. lift-/.f64 N/A

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

       8. div-flip-rev N/A

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

       9. lower-/.f64 46.5

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

       10. lift-hypot.f64 N/A

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

       11. lower-sqrt.f64 N/A

           \[\leadsto \sin th \cdot \frac{ky}{\color{blue}{\sqrt{ky \cdot ky + kx \cdot kx}}} \]

       12. lower-fma.f64 N/A

           \[\leadsto \sin th \cdot \frac{ky}{\sqrt{\color{blue}{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \]

       13. lower-*.f64 34.4

           \[\leadsto \sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, \color{blue}{kx \cdot kx}\right)}} \]

   14. Applied rewrites 34.4%

       \[\leadsto \color{blue}{\sin th \cdot \frac{ky}{\sqrt{\mathsf{fma}\left(ky, ky, kx \cdot kx\right)}}} \]

   if 6.4000000000000001e-7 < kx

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\color{blue}{\sin kx}}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      6. rem-sqrt-square N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      7. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lift-sin.f64 37.1

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   4. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\left|\sin kx\right|}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 31.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq 10^{-6}:\\ \;\;\;\;th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{ky}{\sqrt{ky \cdot ky}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 1e-6)
     (* th (/ ky (fabs (sin kx))))
     (if (<= t_1 2.0)
       (* (/ ky (sqrt (* ky ky))) (sin th))
       (*
        (/ 1.0 (/ (hypot ky kx) ky))
        (* (fma (* th th) -0.16666666666666666 1.0) th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= 1e-6) {
		tmp = th * (ky / fabs(sin(kx)));
	} else if (t_1 <= 2.0) {
		tmp = (ky / sqrt((ky * ky))) * sin(th);
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= 1e-6)
		tmp = Float64(th * Float64(ky / abs(sin(kx))));
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64(ky / sqrt(Float64(ky * ky))) * sin(th));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-6], N[(th * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(ky / N[Sqrt[N[(ky * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\color{blue}{\sin kx}}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      6. rem-sqrt-square N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      7. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lift-sin.f64 37.1

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   4. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\left|\sin kx\right|}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\color{blue}{\left|\sin kx\right|}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\color{blue}{\sin kx}\right|} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\sin kx\right|} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\sin kx\right|} \]

      5. associate-/l* N/A

         \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

      6. lower-*.f64 N/A

         \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      8. pow2 N/A

         \[\leadsto th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      9. lower-/.f64 N/A

         \[\leadsto th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      10. pow2 N/A

          \[\leadsto th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

      12. lower-fabs.f64 N/A

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lift-sin.f64 20.9

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

   9. Applied rewrites 20.9%

      \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   if 9.99999999999999955e-7 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   3. Step-by-step derivation

   4. Applied rewrites 45.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]

   8. Taylor expanded in kx around 0

      \[\leadsto \frac{ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \frac{ky}{\sqrt{\sin ky \cdot \color{blue}{\sin ky}}} \cdot \sin th \]

      2. sqr-sin-a N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      3. lower--.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot ky\right)}}} \cdot \sin th \]

      4. cos-2 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos ky \cdot \cos ky - \color{blue}{\sin ky \cdot \sin ky}\right)}} \cdot \sin th \]

      5. cos-sum N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(ky + ky\right)}}} \cdot \sin th \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{ky}{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

      8. lower-+.f64 3.3

         \[\leadsto \frac{ky}{\sqrt{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}} \cdot \sin th \]

   10. Applied rewrites 3.3%

       \[\leadsto \frac{ky}{\sqrt{\color{blue}{0.5 - 0.5 \cdot \cos \left(ky + ky\right)}}} \cdot \sin th \]

   11. Taylor expanded in ky around 0

       \[\leadsto \frac{ky}{\sqrt{{ky}^{\color{blue}{2}}}} \cdot \sin th \]
   12. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto \frac{ky}{\sqrt{ky \cdot ky}} \cdot \sin th \]

       2. lower-*.f64 20.1

          \[\leadsto \frac{ky}{\sqrt{ky \cdot ky}} \cdot \sin th \]

   13. Applied rewrites 20.1%

       \[\leadsto \frac{ky}{\sqrt{ky \cdot \color{blue}{ky}}} \cdot \sin th \]

   if 2 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   3. Step-by-step derivation

   4. Applied rewrites 45.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}} \cdot \sin th \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      4. lower-/.f64 52.8

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {ky}^{2}}}{ky}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\color{blue}{\sin kx}}^{2} + {ky}^{2}}}{ky}} \cdot \sin th \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2} + {\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2}} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{ky \cdot ky} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      12. pow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + \color{blue}{\sin kx \cdot \sin kx}}}{ky}} \cdot \sin th \]

      13. lower-hypot.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(ky, \sin kx\right)}}{ky}} \cdot \sin th \]

      14. lift-sin.f64 65.0

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{\sin kx}\right)}{ky}} \cdot \sin th \]

   9. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]

   13. Taylor expanded in th around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       7. lower-*.f64 27.9

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   15. Applied rewrites 27.9%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.78:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\left|kx\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.78)
   (*
    (/ 1.0 (/ (hypot ky kx) ky))
    (* (fma (* th th) -0.16666666666666666 1.0) th))
   (/ (* (sin th) ky) (fabs kx))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.78) {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else {
		tmp = (sin(th) * ky) / fabs(kx);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.78)
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	else
		tmp = Float64(Float64(sin(th) * ky) / abs(kx));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.78], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.78000000000000003

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   3. Step-by-step derivation

   4. Applied rewrites 45.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}} \cdot \sin th \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      4. lower-/.f64 52.8

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {ky}^{2}}}{ky}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\color{blue}{\sin kx}}^{2} + {ky}^{2}}}{ky}} \cdot \sin th \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2} + {\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2}} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{ky \cdot ky} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      12. pow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + \color{blue}{\sin kx \cdot \sin kx}}}{ky}} \cdot \sin th \]

      13. lower-hypot.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(ky, \sin kx\right)}}{ky}} \cdot \sin th \]

      14. lift-sin.f64 65.0

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{\sin kx}\right)}{ky}} \cdot \sin th \]

   9. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]

   13. Taylor expanded in th around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       7. lower-*.f64 27.9

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   15. Applied rewrites 27.9%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]

   if 0.78000000000000003 < th

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\color{blue}{\sin kx}}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      6. rem-sqrt-square N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      7. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lift-sin.f64 37.1

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   4. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\left|\sin kx\right|}} \]

   5. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin th \cdot ky}{\left|kx\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 19.7%

      \[\leadsto \frac{\sin th \cdot ky}{\left|kx\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 27.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;th \cdot \frac{ky}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-6)
   (* th (/ ky (fabs (sin kx))))
   (*
    (/ 1.0 (/ (hypot ky kx) ky))
    (* (fma (* th th) -0.16666666666666666 1.0) th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-6) {
		tmp = th * (ky / fabs(sin(kx)));
	} else {
		tmp = (1.0 / (hypot(ky, kx) / ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-6)
		tmp = Float64(th * Float64(ky / abs(sin(kx))));
	else
		tmp = Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * 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], 1e-6], N[(th * N[(ky / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\color{blue}{\sin kx}}^{2}}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      6. rem-sqrt-square N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      7. lower-fabs.f64 N/A

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

      8. lift-sin.f64 37.1

         \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   4. Applied rewrites 37.1%

      \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\left|\sin kx\right|}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   6. Step-by-step derivation

   7. Applied rewrites 18.8%

      \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\color{blue}{\left|\sin kx\right|}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\color{blue}{\sin kx}\right|} \]

      3. lift-fabs.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\sin kx\right|} \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{th \cdot ky}{\left|\sin kx\right|} \]

      5. associate-/l* N/A

         \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

      6. lower-*.f64 N/A

         \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      8. pow2 N/A

         \[\leadsto th \cdot \frac{ky}{\sqrt{{\sin kx}^{2}}} \]

      9. lower-/.f64 N/A

         \[\leadsto th \cdot \frac{ky}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

      10. pow2 N/A

          \[\leadsto th \cdot \frac{ky}{\sqrt{\sin kx \cdot \sin kx}} \]

      11. rem-sqrt-square-rev N/A

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

      12. lower-fabs.f64 N/A

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

      13. lift-sin.f64 20.9

          \[\leadsto th \cdot \frac{ky}{\left|\sin kx\right|} \]

   9. Applied rewrites 20.9%

      \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|\sin kx\right|}} \]

   if 9.99999999999999955e-7 < (/.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 94.2%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   3. Step-by-step derivation

   4. Applied rewrites 45.6%

      \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

   5. Taylor expanded in ky around 0

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   6. Step-by-step derivation

   7. Applied rewrites 52.8%

      \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}} \cdot \sin th \]

      2. div-flip N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      4. lower-/.f64 52.8

         \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

      7. lift-pow.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {ky}^{2}}}{ky}} \cdot \sin th \]

      8. lift-sin.f64 N/A

         \[\leadsto \frac{1}{\frac{\sqrt{{\color{blue}{\sin kx}}^{2} + {ky}^{2}}}{ky}} \cdot \sin th \]

      9. +-commutative N/A

         \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2} + {\sin kx}^{2}}}}{ky}} \cdot \sin th \]

      10. lift-pow.f64 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2}} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      11. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{ky \cdot ky} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

      12. pow2 N/A

          \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + \color{blue}{\sin kx \cdot \sin kx}}}{ky}} \cdot \sin th \]

      13. lower-hypot.f64 N/A

          \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(ky, \sin kx\right)}}{ky}} \cdot \sin th \]

      14. lift-sin.f64 65.0

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{\sin kx}\right)}{ky}} \cdot \sin th \]

   9. Applied rewrites 65.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \cdot \sin th \]

   10. Taylor expanded in kx around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
   11. Step-by-step derivation

   12. Applied rewrites 46.5%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]

   13. Taylor expanded in th around 0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

       3. +-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th\right) \]

       4. *-commutative N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th\right) \]

       6. unpow2 N/A

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

       7. lower-*.f64 27.9

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

   15. Applied rewrites 27.9%

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 26.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (/ 1.0 (/ (hypot ky kx) ky))
  (* (fma (* th th) -0.16666666666666666 1.0) th)))

C

double code(double kx, double ky, double th) {
	return (1.0 / (hypot(ky, kx) / ky)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
}

Julia

function code(kx, ky, th)
	return Float64(Float64(1.0 / Float64(hypot(ky, kx) / ky)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
end

Wolfram

code[kx_, ky_, th_] := N[(N[(1.0 / N[(N[Sqrt[ky ^ 2 + kx ^ 2], $MachinePrecision] / ky), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
3. Step-by-step derivation

4. Applied rewrites 45.6%

   \[\leadsto \frac{\color{blue}{ky}}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

5. Taylor expanded in ky around 0

   \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
6. Step-by-step derivation

7. Applied rewrites 52.8%

   \[\leadsto \frac{ky}{\sqrt{{\sin kx}^{2} + {\color{blue}{ky}}^{2}}} \cdot \sin th \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}} \cdot \sin th \]

   2. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

   4. lower-/.f64 52.8

      \[\leadsto \frac{1}{\color{blue}{\frac{\sqrt{{\sin kx}^{2} + {ky}^{2}}}{ky}}} \cdot \sin th \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\sqrt{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2} + {ky}^{2}}}}{ky}} \cdot \sin th \]

   7. lift-pow.f64 N/A

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{\sin kx}^{2}} + {ky}^{2}}}{ky}} \cdot \sin th \]

   8. lift-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{\sqrt{{\color{blue}{\sin kx}}^{2} + {ky}^{2}}}{ky}} \cdot \sin th \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2} + {\sin kx}^{2}}}}{ky}} \cdot \sin th \]

   10. lift-pow.f64 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{{ky}^{2}} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{\color{blue}{ky \cdot ky} + {\sin kx}^{2}}}{ky}} \cdot \sin th \]

   12. pow2 N/A

       \[\leadsto \frac{1}{\frac{\sqrt{ky \cdot ky + \color{blue}{\sin kx \cdot \sin kx}}}{ky}} \cdot \sin th \]

   13. lower-hypot.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(ky, \sin kx\right)}}{ky}} \cdot \sin th \]

   14. lift-sin.f64 65.0

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{\sin kx}\right)}{ky}} \cdot \sin th \]

9. Applied rewrites 65.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(ky, \sin kx\right)}{ky}}} \cdot \sin th \]

10. Taylor expanded in kx around 0

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]
11. Step-by-step derivation

12. Applied rewrites 46.5%

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, \color{blue}{kx}\right)}{ky}} \cdot \sin th \]

13. Taylor expanded in th around 0

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(th \cdot \left(1 + \frac{-1}{6} \cdot {th}^{2}\right)\right)} \]
14. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(1 + \frac{-1}{6} \cdot {th}^{2}\right) \cdot \color{blue}{th}\right) \]

    3. +-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left(\frac{-1}{6} \cdot {th}^{2} + 1\right) \cdot th\right) \]

    4. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\left({th}^{2} \cdot \frac{-1}{6} + 1\right) \cdot th\right) \]

    5. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left({th}^{2}, \frac{-1}{6}, 1\right) \cdot th\right) \]

    6. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, \frac{-1}{6}, 1\right) \cdot th\right) \]

    7. lower-*.f64 27.9

       \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right) \]

15. Applied rewrites 27.9%

    \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(ky, kx\right)}{ky}} \cdot \color{blue}{\left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)} \]
16. Add Preprocessing

Alternative 24: 15.8% accurate, 20.0× speedup

Math

\[\begin{array}{l} \\ th \cdot \frac{ky}{\left|kx\right|} \end{array} \]

FPCore

(FPCore (kx ky th) :precision binary64 (* th (/ ky (fabs kx))))

C

double code(double kx, double ky, double th) {
	return th * (ky / fabs(kx));
}

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 * (ky / abs(kx))
end function

Java

public static double code(double kx, double ky, double th) {
	return th * (ky / Math.abs(kx));
}

Python

def code(kx, ky, th):
	return th * (ky / math.fabs(kx))

Julia

function code(kx, ky, th)
	return Float64(th * Float64(ky / abs(kx)))
end

MATLAB

function tmp = code(kx, ky, th)
	tmp = th * (ky / abs(kx));
end

Wolfram

code[kx_, ky_, th_] := N[(th * N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]

2. Taylor expanded in ky around 0

   \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sqrt{{\sin kx}^{2}}}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{ky \cdot \sin th}{\color{blue}{\sqrt{{\sin kx}^{2}}}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \]

   4. lift-sin.f64 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{{\color{blue}{\sin kx}}^{2}}} \]

   5. unpow2 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\sqrt{\sin kx \cdot \sin kx}} \]

   6. rem-sqrt-square N/A

      \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   7. lower-fabs.f64 N/A

      \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

   8. lift-sin.f64 37.1

      \[\leadsto \frac{\sin th \cdot ky}{\left|\sin kx\right|} \]

4. Applied rewrites 37.1%

   \[\leadsto \color{blue}{\frac{\sin th \cdot ky}{\left|\sin kx\right|}} \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]
6. Step-by-step derivation

7. Applied rewrites 18.8%

   \[\leadsto \frac{th \cdot ky}{\left|\sin \color{blue}{kx}\right|} \]

8. Taylor expanded in kx around 0

   \[\leadsto \frac{th \cdot ky}{\left|kx\right|} \]
9. Step-by-step derivation

10. Applied rewrites 13.8%

    \[\leadsto \frac{th \cdot ky}{\left|kx\right|} \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

       \[\leadsto \frac{th \cdot ky}{\color{blue}{\left|kx\right|}} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{th \cdot ky}{\left|\color{blue}{kx}\right|} \]

    3. associate-/l* N/A

       \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|kx\right|}} \]

    4. lower-*.f64 N/A

       \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|kx\right|}} \]

    5. lower-/.f64 15.8

       \[\leadsto th \cdot \frac{ky}{\color{blue}{\left|kx\right|}} \]

12. Applied rewrites 15.8%

    \[\leadsto th \cdot \color{blue}{\frac{ky}{\left|kx\right|}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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)))