Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 6.4s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

Alternative 2: 82.6% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 0.0065:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\left|th\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|th\right|\right)}^{2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 0.0065)
    (*
     (/ (sin ky) (hypot (sin ky) (sin kx)))
     (* (fabs th) (+ 1.0 (* -0.16666666666666666 (pow (fabs th) 2.0)))))
    (* (/ ky (hypot ky (sin kx))) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 0.0065) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (fabs(th) * (1.0 + (-0.16666666666666666 * pow(fabs(th), 2.0))));
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 0.0065) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * (Math.abs(th) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(th), 2.0))));
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 0.0065:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * (math.fabs(th) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(th), 2.0))))
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 0.0065)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(abs(th) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(th) ^ 2.0)))));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 0.0065)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * (abs(th) * (1.0 + (-0.16666666666666666 * (abs(th) ^ 2.0))));
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 0.0065], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[th], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[th], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 0.0064999999999999997

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.9%

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

   6. Applied rewrites 50.9%

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

   if 0.0064999999999999997 < th

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.4%

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

Alternative 3: 82.6% accurate, 1.1× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 0.0065:\\ \;\;\;\;\frac{\left|th\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|th\right|\right)}^{2}\right)}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 0.0065)
    (*
     (/
      (* (fabs th) (+ 1.0 (* -0.16666666666666666 (pow (fabs th) 2.0))))
      (hypot (sin ky) (sin kx)))
     (sin ky))
    (* (/ ky (hypot ky (sin kx))) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 0.0065) {
		tmp = ((fabs(th) * (1.0 + (-0.16666666666666666 * pow(fabs(th), 2.0)))) / hypot(sin(ky), sin(kx))) * sin(ky);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 0.0065) {
		tmp = ((Math.abs(th) * (1.0 + (-0.16666666666666666 * Math.pow(Math.abs(th), 2.0)))) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.sin(ky);
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 0.0065:
		tmp = ((math.fabs(th) * (1.0 + (-0.16666666666666666 * math.pow(math.fabs(th), 2.0)))) / math.hypot(math.sin(ky), math.sin(kx))) * math.sin(ky)
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 0.0065)
		tmp = Float64(Float64(Float64(abs(th) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(th) ^ 2.0)))) / hypot(sin(ky), sin(kx))) * sin(ky));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 0.0065)
		tmp = ((abs(th) * (1.0 + (-0.16666666666666666 * (abs(th) ^ 2.0)))) / hypot(sin(ky), sin(kx))) * sin(ky);
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 0.0065], N[(N[(N[(N[Abs[th], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[th], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 0.0064999999999999997

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-hypot.f64 N/A

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

      5. pow1/2 N/A

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

      6. pow2 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. pow1/2 N/A

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

      16. lift-hypot.f64 N/A

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

      17. *-commutative N/A

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

   5. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 50.9%

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

   10. Applied rewrites 50.9%

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

   if 0.0064999999999999997 < th

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.4%

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

Alternative 4: 81.7% accurate, 1.3× speedup

Math

\[\mathsf{copysign}\left(1, th\right) \cdot \begin{array}{l} \mathbf{if}\;\left|th\right| \leq 1.15 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left|th\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin \left(\left|th\right|\right)\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (*
  (copysign 1.0 th)
  (if (<= (fabs th) 1.15e-18)
    (* (/ (sin ky) (hypot (sin ky) (sin kx))) (fabs th))
    (* (/ ky (hypot ky (sin kx))) (sin (fabs th))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(th) <= 1.15e-18) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * fabs(th);
	} else {
		tmp = (ky / hypot(ky, sin(kx))) * sin(fabs(th));
	}
	return copysign(1.0, th) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(th) <= 1.15e-18) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * Math.abs(th);
	} else {
		tmp = (ky / Math.hypot(ky, Math.sin(kx))) * Math.sin(Math.abs(th));
	}
	return Math.copySign(1.0, th) * tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(th) <= 1.15e-18:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * math.fabs(th)
	else:
		tmp = (ky / math.hypot(ky, math.sin(kx))) * math.sin(math.fabs(th))
	return math.copysign(1.0, th) * tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(th) <= 1.15e-18)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * abs(th));
	else
		tmp = Float64(Float64(ky / hypot(ky, sin(kx))) * sin(abs(th)));
	end
	return Float64(copysign(1.0, th) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(th) <= 1.15e-18)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * abs(th);
	else
		tmp = (ky / hypot(ky, sin(kx))) * sin(abs(th));
	end
	tmp_2 = (sign(th) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[th]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[th], $MachinePrecision], 1.15e-18], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Abs[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[Abs[th], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if th < 1.15e-18

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 51.1%

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

   if 1.15e-18 < th

   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. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.4%

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

Alternative 5: 80.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 0.102:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left|kx\right|\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin \left(\left|kx\right|\right)\right|}{\sin th \cdot \sin ky}}\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 0.102)
   (* (/ (sin ky) (hypot (sin ky) (fabs kx))) (sin th))
   (/ 1.0 (/ (fabs (sin (fabs kx))) (* (sin th) (sin ky))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 0.102) {
		tmp = (sin(ky) / hypot(sin(ky), fabs(kx))) * sin(th);
	} else {
		tmp = 1.0 / (fabs(sin(fabs(kx))) / (sin(th) * sin(ky)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 0.102) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.abs(kx))) * Math.sin(th);
	} else {
		tmp = 1.0 / (Math.abs(Math.sin(Math.abs(kx))) / (Math.sin(th) * Math.sin(ky)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 0.102:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.fabs(kx))) * math.sin(th)
	else:
		tmp = 1.0 / (math.fabs(math.sin(math.fabs(kx))) / (math.sin(th) * math.sin(ky)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 0.102)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), abs(kx))) * sin(th));
	else
		tmp = Float64(1.0 / Float64(abs(sin(abs(kx))) / Float64(sin(th) * sin(ky))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 0.102)
		tmp = (sin(ky) / hypot(sin(ky), abs(kx))) * sin(th);
	else
		tmp = 1.0 / (abs(sin(abs(kx))) / (sin(th) * sin(ky)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 0.102], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Abs[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.101999999999999993

   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

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

   4. Applied rewrites 52.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. pow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 58.4%

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

   6. Applied rewrites 58.4%

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

   if 0.101999999999999993 < 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

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

      1. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.7%

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

   4. Applied rewrites 41.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 N/A

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

      12. lower-fabs.f64 N/A

          \[\leadsto \frac{1}{\frac{\left|\sin kx\right|}{\mathsf{Rewrite=>}\left(*-commutative, \left(\sin th \cdot \sin ky\right)\right)}} \]

   6. Applied rewrites 42.6%

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

Alternative 6: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.995:\\ \;\;\;\;\frac{\sin th \cdot t\_1}{\mathsf{hypot}\left(kx, t\_1\right)}\\ \mathbf{elif}\;t\_2 \leq 0.62:\\ \;\;\;\;\frac{t\_1}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -0.995)
      (/ (* (sin th) t_1) (hypot kx t_1))
      (if (<= t_2 0.62)
        (* (/ t_1 (fabs (sin kx))) (sin th))
        (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = (sin(th) * t_1) / hypot(kx, t_1);
	} else if (t_2 <= 0.62) {
		tmp = (t_1 / fabs(sin(kx))) * sin(th);
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double tmp;
	if (t_2 <= -0.995) {
		tmp = (Math.sin(th) * t_1) / Math.hypot(kx, t_1);
	} else if (t_2 <= 0.62) {
		tmp = (t_1 / Math.abs(Math.sin(kx))) * Math.sin(th);
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	tmp = 0
	if t_2 <= -0.995:
		tmp = (math.sin(th) * t_1) / math.hypot(kx, t_1)
	elif t_2 <= 0.62:
		tmp = (t_1 / math.fabs(math.sin(kx))) * math.sin(th)
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.995)
		tmp = Float64(Float64(sin(th) * t_1) / hypot(kx, t_1));
	elseif (t_2 <= 0.62)
		tmp = Float64(Float64(t_1 / abs(sin(kx))) * sin(th));
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.995)
		tmp = (sin(th) * t_1) / hypot(kx, t_1);
	elseif (t_2 <= 0.62)
		tmp = (t_1 / abs(sin(kx))) * sin(th);
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$2, -0.995], N[(N[(N[Sin[th], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[kx ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.62], N[(N[(t$95$1 / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 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))))) < -0.994999999999999996

   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

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

   4. Applied rewrites 52.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 50.7%

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-hypot.f64 54.6%

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

   6. Applied rewrites 54.6%

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

   if -0.994999999999999996 < (/.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))))) < 0.619999999999999996

   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. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.7%

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

   4. Applied rewrites 41.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 44.8%

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

   6. Applied rewrites 44.8%

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

   if 0.619999999999999996 < (/.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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.4%

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

Alternative 7: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := {t\_1}^{2}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + t\_2}} \leq -0.3:\\ \;\;\;\;\frac{t\_1}{\sqrt{t\_2}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))) (t_2 (pow t_1 2.0)))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) t_2))) -0.3)
      (* (/ t_1 (sqrt t_2)) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = pow(t_1, 2.0);
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + t_2))) <= -0.3) {
		tmp = (t_1 / sqrt(t_2)) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double t_2 = Math.pow(t_1, 2.0);
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + t_2))) <= -0.3) {
		tmp = (t_1 / Math.sqrt(t_2)) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	t_2 = math.pow(t_1, 2.0)
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + t_2))) <= -0.3:
		tmp = (t_1 / math.sqrt(t_2)) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = t_1 ^ 2.0
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + t_2))) <= -0.3)
		tmp = Float64(Float64(t_1 / sqrt(t_2)) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	t_2 = t_1 ^ 2.0;
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + t_2))) <= -0.3)
		tmp = (t_1 / sqrt(t_2)) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.3], N[(N[(t$95$1 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[Abs[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.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))))) < -0.299999999999999989

   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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.1%

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

   7. Applied rewrites 17.1%

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

   8. Taylor expanded in th around 0

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

   10. Applied rewrites 13.6%

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

   11. Taylor expanded in kx around 0

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

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 21.9%

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

   13. Applied rewrites 21.9%

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

   if -0.299999999999999989 < (/.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. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7%

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.5%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.4%

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

Alternative 8: 65.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. +-commutative N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.7%

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in ky around 0

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

6. Applied rewrites 51.5%

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

7. Taylor expanded in ky around 0

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

9. Applied rewrites 65.4%

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

Alternative 9: 56.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \sin \left(\left|kx\right|\right)\\ t_3 := \left|t\_2\right|\\ t_4 := \frac{t\_1}{\sqrt{{t\_2}^{2} + {t\_1}^{2}}}\\ t_5 := {\left(\left|kx\right|\right)}^{2}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -0.1:\\ \;\;\;\;t\_1 \cdot \frac{\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 0.05:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{t\_3}\\ \mathbf{elif}\;t\_4 \leq 2:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{t\_5 + {\left(\left|ky\right|\right)}^{2}}} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right| + 0.16666666666666666 \cdot \left(t\_5 \cdot \left|ky\right|\right)}{\left|kx\right|} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (sin (fabs kx)))
        (t_3 (fabs t_2))
        (t_4 (/ t_1 (sqrt (+ (pow t_2 2.0) (pow t_1 2.0)))))
        (t_5 (pow (fabs kx) 2.0)))
   (*
    (copysign 1.0 ky)
    (if (<= t_4 -0.1)
      (* t_1 (/ (* (fma (* th th) -0.16666666666666666 1.0) th) t_3))
      (if (<= t_4 0.05)
        (* (sin th) (/ (fabs ky) t_3))
        (if (<= t_4 2.0)
          (* (/ (fabs ky) (sqrt (+ t_5 (pow (fabs ky) 2.0)))) (sin th))
          (*
           (/
            (+ (fabs ky) (* 0.16666666666666666 (* t_5 (fabs ky))))
            (fabs kx))
           (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double t_2 = sin(fabs(kx));
	double t_3 = fabs(t_2);
	double t_4 = t_1 / sqrt((pow(t_2, 2.0) + pow(t_1, 2.0)));
	double t_5 = pow(fabs(kx), 2.0);
	double tmp;
	if (t_4 <= -0.1) {
		tmp = t_1 * ((fma((th * th), -0.16666666666666666, 1.0) * th) / t_3);
	} else if (t_4 <= 0.05) {
		tmp = sin(th) * (fabs(ky) / t_3);
	} else if (t_4 <= 2.0) {
		tmp = (fabs(ky) / sqrt((t_5 + pow(fabs(ky), 2.0)))) * sin(th);
	} else {
		tmp = ((fabs(ky) + (0.16666666666666666 * (t_5 * fabs(ky)))) / fabs(kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	t_2 = sin(abs(kx))
	t_3 = abs(t_2)
	t_4 = Float64(t_1 / sqrt(Float64((t_2 ^ 2.0) + (t_1 ^ 2.0))))
	t_5 = abs(kx) ^ 2.0
	tmp = 0.0
	if (t_4 <= -0.1)
		tmp = Float64(t_1 * Float64(Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th) / t_3));
	elseif (t_4 <= 0.05)
		tmp = Float64(sin(th) * Float64(abs(ky) / t_3));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(abs(ky) / sqrt(Float64(t_5 + (abs(ky) ^ 2.0)))) * sin(th));
	else
		tmp = Float64(Float64(Float64(abs(ky) + Float64(0.16666666666666666 * Float64(t_5 * abs(ky)))) / abs(kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Abs[kx], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, -0.1], N[(t$95$1 * N[(N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[(t$95$5 + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[ky], $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$5 * N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   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. lower-sqrt.f64 N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-sin.f64 41.7%

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

   4. Applied rewrites 41.7%

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

   5. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 21.8%

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

   7. Applied rewrites 21.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 21.8%

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

   9. Applied rewrites 23.4%

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

   if -0.10000000000000001 < (/.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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.7%

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

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

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

      8. lower-fabs.f64 39.7%

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

   6. Applied rewrites 39.7%

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

   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))))) < 2

   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

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 34.6%

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

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

   1. Initial program 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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 11.8%

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

   7. Applied rewrites 11.8%

      \[\leadsto \frac{ky + 0.16666666666666666 \cdot \left({kx}^{2} \cdot ky\right)}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 10: 54.8% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (fabs kx) 2.0))
        (t_2 (sin (fabs ky)))
        (t_3 (sin (fabs kx)))
        (t_4 (/ t_2 (sqrt (+ (pow t_3 2.0) (pow t_2 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_4 0.05)
      (* (sin th) (/ (fabs ky) (fabs t_3)))
      (if (<= t_4 2.0)
        (* (/ (fabs ky) (sqrt (+ t_1 (pow (fabs ky) 2.0)))) (sin th))
        (*
         (/ (+ (fabs ky) (* 0.16666666666666666 (* t_1 (fabs ky)))) (fabs kx))
         (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(fabs(kx), 2.0);
	double t_2 = sin(fabs(ky));
	double t_3 = sin(fabs(kx));
	double t_4 = t_2 / sqrt((pow(t_3, 2.0) + pow(t_2, 2.0)));
	double tmp;
	if (t_4 <= 0.05) {
		tmp = sin(th) * (fabs(ky) / fabs(t_3));
	} else if (t_4 <= 2.0) {
		tmp = (fabs(ky) / sqrt((t_1 + pow(fabs(ky), 2.0)))) * sin(th);
	} else {
		tmp = ((fabs(ky) + (0.16666666666666666 * (t_1 * fabs(ky)))) / fabs(kx)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.abs(kx), 2.0);
	double t_2 = Math.sin(Math.abs(ky));
	double t_3 = Math.sin(Math.abs(kx));
	double t_4 = t_2 / Math.sqrt((Math.pow(t_3, 2.0) + Math.pow(t_2, 2.0)));
	double tmp;
	if (t_4 <= 0.05) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(t_3));
	} else if (t_4 <= 2.0) {
		tmp = (Math.abs(ky) / Math.sqrt((t_1 + Math.pow(Math.abs(ky), 2.0)))) * Math.sin(th);
	} else {
		tmp = ((Math.abs(ky) + (0.16666666666666666 * (t_1 * Math.abs(ky)))) / Math.abs(kx)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.fabs(kx), 2.0)
	t_2 = math.sin(math.fabs(ky))
	t_3 = math.sin(math.fabs(kx))
	t_4 = t_2 / math.sqrt((math.pow(t_3, 2.0) + math.pow(t_2, 2.0)))
	tmp = 0
	if t_4 <= 0.05:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(t_3))
	elif t_4 <= 2.0:
		tmp = (math.fabs(ky) / math.sqrt((t_1 + math.pow(math.fabs(ky), 2.0)))) * math.sin(th)
	else:
		tmp = ((math.fabs(ky) + (0.16666666666666666 * (t_1 * math.fabs(ky)))) / math.fabs(kx)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = abs(kx) ^ 2.0
	t_2 = sin(abs(ky))
	t_3 = sin(abs(kx))
	t_4 = Float64(t_2 / sqrt(Float64((t_3 ^ 2.0) + (t_2 ^ 2.0))))
	tmp = 0.0
	if (t_4 <= 0.05)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(t_3)));
	elseif (t_4 <= 2.0)
		tmp = Float64(Float64(abs(ky) / sqrt(Float64(t_1 + (abs(ky) ^ 2.0)))) * sin(th));
	else
		tmp = Float64(Float64(Float64(abs(ky) + Float64(0.16666666666666666 * Float64(t_1 * abs(ky)))) / abs(kx)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = abs(kx) ^ 2.0;
	t_2 = sin(abs(ky));
	t_3 = sin(abs(kx));
	t_4 = t_2 / sqrt(((t_3 ^ 2.0) + (t_2 ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= 0.05)
		tmp = sin(th) * (abs(ky) / abs(t_3));
	elseif (t_4 <= 2.0)
		tmp = (abs(ky) / sqrt((t_1 + (abs(ky) ^ 2.0)))) * sin(th);
	else
		tmp = ((abs(ky) + (0.16666666666666666 * (t_1 * abs(ky)))) / abs(kx)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Abs[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 / N[Sqrt[N[(N[Power[t$95$3, 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, 0.05], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 2.0], N[(N[(N[Abs[ky], $MachinePrecision] / N[Sqrt[N[(t$95$1 + N[Power[N[Abs[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Abs[ky], $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$1 * N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 36.7%

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

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

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

      8. lower-fabs.f64 39.7%

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

   6. Applied rewrites 39.7%

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

   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))))) < 2

   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

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

   4. Applied rewrites 52.9%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 34.6%

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

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

   1. Initial program 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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 11.8%

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

   7. Applied rewrites 11.8%

      \[\leadsto \frac{ky + 0.16666666666666666 \cdot \left({kx}^{2} \cdot ky\right)}{\color{blue}{kx}} \cdot \sin th \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 39.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.7%

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

4. Applied rewrites 36.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 36.7%

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

   4. lift-sqrt.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. pow2 N/A

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

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

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

   8. lower-fabs.f64 39.7%

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

6. Applied rewrites 39.7%

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

Alternative 12: 25.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 1.4 \cdot 10^{+78}:\\ \;\;\;\;\frac{ky}{\left|kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2}}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 1.4e+78)
   (* (/ ky (fabs kx)) (sin th))
   (* (/ ky (sqrt (pow (sin (fabs kx)) 2.0))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 1.4e+78) {
		tmp = (ky / fabs(kx)) * sin(th);
	} else {
		tmp = (ky / sqrt(pow(sin(fabs(kx)), 2.0))) * th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (abs(kx) <= 1.4d+78) then
        tmp = (ky / abs(kx)) * sin(th)
    else
        tmp = (ky / sqrt((sin(abs(kx)) ** 2.0d0))) * th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 1.4e+78) {
		tmp = (ky / Math.abs(kx)) * Math.sin(th);
	} else {
		tmp = (ky / Math.sqrt(Math.pow(Math.sin(Math.abs(kx)), 2.0))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 1.4e+78:
		tmp = (ky / math.fabs(kx)) * math.sin(th)
	else:
		tmp = (ky / math.sqrt(math.pow(math.sin(math.fabs(kx)), 2.0))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 1.4e+78)
		tmp = Float64(Float64(ky / abs(kx)) * sin(th));
	else
		tmp = Float64(Float64(ky / sqrt((sin(abs(kx)) ^ 2.0))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 1.4e+78)
		tmp = (ky / abs(kx)) * sin(th);
	else
		tmp = (ky / sqrt((sin(abs(kx)) ^ 2.0))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 1.4e+78], N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 1.4000000000000001e78

   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. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 17.1%

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

   7. Applied rewrites 17.1%

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

   if 1.4000000000000001e78 < 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

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 36.7%

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

   4. Applied rewrites 36.7%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 19.5%

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

Alternative 13: 21.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.7%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.1%

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

7. Applied rewrites 17.1%

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

Alternative 14: 15.7% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    code = (ky / abs(kx)) * th
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sin.f64 36.7%

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

4. Applied rewrites 36.7%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 17.1%

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

7. Applied rewrites 17.1%

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

8. Taylor expanded in th around 0

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

10. Applied rewrites 13.6%

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

Reproduce

herbie shell --seed 2025185 
(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)))