Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.9% → 99.7%

Time: 6.2s

Alternatives: 18

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.9%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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: 83.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;\frac{t\_1}{\mathsf{hypot}\left(t\_1, kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq -0.02:\\ \;\;\;\;\frac{th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)}{\mathsf{hypot}\left(\sin kx, t\_1\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.02)
        (*
         (/
          (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0))))
          (hypot (sin kx) t_1))
         t_1)
        (* (/ (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 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.02) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * pow(th, 2.0)))) / hypot(sin(kx), t_1)) * t_1;
	} 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 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.02) {
		tmp = ((th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0)))) / Math.hypot(Math.sin(kx), t_1)) * t_1;
	} 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 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.02:
		tmp = ((th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0)))) / math.hypot(math.sin(kx), t_1)) * t_1
	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 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.02)
		tmp = Float64(Float64(Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))) / hypot(sin(kx), t_1)) * t_1);
	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 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.02)
		tmp = ((th * (1.0 + (-0.16666666666666666 * (th ^ 2.0)))) / hypot(sin(kx), t_1)) * t_1;
	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, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.02], N[(N[(N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$1), $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))))) < -1

   1. Initial program 93.9%

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

   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 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.8

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

   6. Applied rewrites 57.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.0200000000000000004

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 49.7

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

   6. Applied rewrites 49.7%

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

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

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 3: 83.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))
        (t_3 (/ t_1 t_2)))
   (*
    (copysign 1.0 ky)
    (if (<= t_3 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_3 -0.1)
        (* (/ th t_2) t_1)
        (* (/ (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 = sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)));
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / t_2) * t_1;
	} 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.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)));
	double t_3 = t_1 / t_2;
	double tmp;
	if (t_3 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_3 <= -0.1) {
		tmp = (th / t_2) * t_1;
	} 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.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))
	t_3 = t_1 / t_2
	tmp = 0
	if t_3 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_3 <= -0.1:
		tmp = (th / t_2) * t_1
	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 = sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))
	t_3 = Float64(t_1 / t_2)
	tmp = 0.0
	if (t_3 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_3 <= -0.1)
		tmp = Float64(Float64(th / t_2) * t_1);
	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 = sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)));
	t_3 = t_1 / t_2;
	tmp = 0.0;
	if (t_3 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_3 <= -0.1)
		tmp = (th / t_2) * t_1;
	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[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / t$95$2), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -0.1], N[(N[(th / t$95$2), $MachinePrecision] * t$95$1), $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))))) < -1

   1. Initial program 93.9%

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

   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 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.8

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

   6. Applied rewrites 57.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-sin.f64 47.4

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

   6. Applied rewrites 47.4%

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

   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)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 4: 83.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.1)
        (* (/ 1.0 (/ (hypot (sin kx) t_1) t_1)) 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 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (1.0 / (hypot(sin(kx), t_1) / t_1)) * 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 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (1.0 / (Math.hypot(Math.sin(kx), t_1) / t_1)) * 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 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (1.0 / (math.hypot(math.sin(kx), t_1) / t_1)) * 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 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(1.0 / Float64(hypot(sin(kx), t_1) / t_1)) * 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 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (1.0 / (hypot(sin(kx), t_1) / t_1)) * 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, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(1.0 / N[(N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + t$95$1 ^ 2], $MachinePrecision] / t$95$1), $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 3 regimes

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

   1. Initial program 93.9%

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

   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 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.8

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

   6. Applied rewrites 57.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 93.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in th around 0

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

   6. Applied rewrites 50.0%

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

   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)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 5: 83.1% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky)))
        (t_2 (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0))))))
   (*
    (copysign 1.0 ky)
    (if (<= t_2 -1.0)
      (* (/ t_1 (hypot t_1 kx)) (sin th))
      (if (<= t_2 -0.1)
        (* (/ t_1 (hypot t_1 (sin kx))) 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 <= -1.0) {
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / hypot(t_1, sin(kx))) * 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 <= -1.0) {
		tmp = (t_1 / Math.hypot(t_1, kx)) * Math.sin(th);
	} else if (t_2 <= -0.1) {
		tmp = (t_1 / Math.hypot(t_1, Math.sin(kx))) * 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 <= -1.0:
		tmp = (t_1 / math.hypot(t_1, kx)) * math.sin(th)
	elif t_2 <= -0.1:
		tmp = (t_1 / math.hypot(t_1, math.sin(kx))) * 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 <= -1.0)
		tmp = Float64(Float64(t_1 / hypot(t_1, kx)) * sin(th));
	elseif (t_2 <= -0.1)
		tmp = Float64(Float64(t_1 / hypot(t_1, sin(kx))) * 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 <= -1.0)
		tmp = (t_1 / hypot(t_1, kx)) * sin(th);
	elseif (t_2 <= -0.1)
		tmp = (t_1 / hypot(t_1, sin(kx))) * 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, -1.0], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, -0.1], N[(N[(t$95$1 / N[Sqrt[t$95$1 ^ 2 + N[Sin[kx], $MachinePrecision] ^ 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 3 regimes

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

   1. Initial program 93.9%

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

   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 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.8

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

   6. Applied rewrites 57.8%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.10000000000000001

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 50.0%

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

   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)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 6: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|kx\right| \leq 0.55:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \left|kx\right|\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\left|\sin \left(\left|kx\right|\right)\right|}{\sin ky}} \cdot \sin th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (fabs kx) 0.55)
   (* (/ (sin ky) (hypot (sin ky) (fabs kx))) (sin th))
   (* (/ 1.0 (/ (fabs (sin (fabs kx))) (sin ky))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (fabs(kx) <= 0.55) {
		tmp = (sin(ky) / hypot(sin(ky), fabs(kx))) * sin(th);
	} else {
		tmp = (1.0 / (fabs(sin(fabs(kx))) / sin(ky))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.abs(kx) <= 0.55) {
		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(ky))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.fabs(kx) <= 0.55:
		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(ky))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (abs(kx) <= 0.55)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), abs(kx))) * sin(th));
	else
		tmp = Float64(Float64(1.0 / Float64(abs(sin(abs(kx))) / sin(ky))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (abs(kx) <= 0.55)
		tmp = (sin(ky) / hypot(sin(ky), abs(kx))) * sin(th);
	else
		tmp = (1.0 / (abs(sin(abs(kx))) / sin(ky))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Abs[kx], $MachinePrecision], 0.55], 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[(N[(1.0 / N[(N[Abs[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 0.55000000000000004

   1. Initial program 93.9%

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

   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 52.0%

      \[\leadsto \frac{\sin ky}{\sqrt{{\color{blue}{kx}}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 57.8

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

   6. Applied rewrites 57.8%

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

   if 0.55000000000000004 < kx

   1. Initial program 93.9%

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

   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 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 41.8

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

      5. lift-sqrt.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

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

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

      9. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

Alternative 7: 79.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -0.02:\\ \;\;\;\;\left(\frac{1}{\left|t\_1\right|} \cdot t\_1\right) \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= t_1 -0.02)
      (* (* (/ 1.0 (fabs t_1)) t_1) (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 tmp;
	if (t_1 <= -0.02) {
		tmp = ((1.0 / fabs(t_1)) * t_1) * 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 tmp;
	if (t_1 <= -0.02) {
		tmp = ((1.0 / Math.abs(t_1)) * t_1) * 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))
	tmp = 0
	if t_1 <= -0.02:
		tmp = ((1.0 / math.fabs(t_1)) * t_1) * 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))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = Float64(Float64(Float64(1.0 / abs(t_1)) * t_1) * 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));
	tmp = 0.0;
	if (t_1 <= -0.02)
		tmp = ((1.0 / abs(t_1)) * t_1) * 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]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$1, -0.02], N[(N[(N[(1.0 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $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 2 regimes

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

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   if -0.0200000000000000004 < (sin.f64 ky)

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 8: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.96:\\ \;\;\;\;\left(\frac{1}{\left|t\_1\right|} \cdot t\_1\right) \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.28:\\ \;\;\;\;\frac{t\_1}{\left|\sin kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 -0.96)
      (*
       (* (/ 1.0 (fabs t_1)) t_1)
       (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
      (if (<= t_2 0.28)
        (* (/ 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 <= -0.96) {
		tmp = ((1.0 / fabs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else if (t_2 <= 0.28) {
		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 <= -0.96) {
		tmp = ((1.0 / Math.abs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	} else if (t_2 <= 0.28) {
		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 <= -0.96:
		tmp = ((1.0 / math.fabs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	elif t_2 <= 0.28:
		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 <= -0.96)
		tmp = Float64(Float64(Float64(1.0 / abs(t_1)) * t_1) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	elseif (t_2 <= 0.28)
		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 <= -0.96)
		tmp = ((1.0 / abs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	elseif (t_2 <= 0.28)
		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, -0.96], N[(N[(N[(1.0 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.28], 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))))) < -0.95999999999999996

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 22.4

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

   11. Applied rewrites 22.4%

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

   if -0.95999999999999996 < (/.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))))) < 0.28000000000000003

   1. Initial program 93.9%

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

   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 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow2 N/A

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

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

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

      5. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.28000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 9: 73.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.96:\\ \;\;\;\;\left(\frac{1}{\left|t\_1\right|} \cdot t\_1\right) \cdot \left(th \cdot \left(1 + -0.16666666666666666 \cdot {th}^{2}\right)\right)\\ \mathbf{elif}\;t\_2 \leq 0.28:\\ \;\;\;\;t\_1 \cdot \frac{\sin th}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

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 -0.96)
      (*
       (* (/ 1.0 (fabs t_1)) t_1)
       (* th (+ 1.0 (* -0.16666666666666666 (pow th 2.0)))))
      (if (<= t_2 0.28)
        (* t_1 (/ (sin th) (fabs (sin kx))))
        (* (/ (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 <= -0.96) {
		tmp = ((1.0 / fabs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * pow(th, 2.0))));
	} else if (t_2 <= 0.28) {
		tmp = t_1 * (sin(th) / fabs(sin(kx)));
	} 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 <= -0.96) {
		tmp = ((1.0 / Math.abs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * Math.pow(th, 2.0))));
	} else if (t_2 <= 0.28) {
		tmp = t_1 * (Math.sin(th) / Math.abs(Math.sin(kx)));
	} 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 <= -0.96:
		tmp = ((1.0 / math.fabs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * math.pow(th, 2.0))))
	elif t_2 <= 0.28:
		tmp = t_1 * (math.sin(th) / math.fabs(math.sin(kx)))
	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 <= -0.96)
		tmp = Float64(Float64(Float64(1.0 / abs(t_1)) * t_1) * Float64(th * Float64(1.0 + Float64(-0.16666666666666666 * (th ^ 2.0)))));
	elseif (t_2 <= 0.28)
		tmp = Float64(t_1 * Float64(sin(th) / abs(sin(kx))));
	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 <= -0.96)
		tmp = ((1.0 / abs(t_1)) * t_1) * (th * (1.0 + (-0.16666666666666666 * (th ^ 2.0))));
	elseif (t_2 <= 0.28)
		tmp = t_1 * (sin(th) / abs(sin(kx)));
	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, -0.96], N[(N[(N[(1.0 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(th * N[(1.0 + N[(-0.16666666666666666 * N[Power[th, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.28], N[(t$95$1 * N[(N[Sin[th], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $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))))) < -0.95999999999999996

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in th around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-pow.f64 22.4

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

   11. Applied rewrites 22.4%

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

   if -0.95999999999999996 < (/.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))))) < 0.28000000000000003

   1. Initial program 93.9%

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

   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 41.9

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

   4. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 41.9

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

      7. lift-sqrt.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

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

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

      11. lower-fabs.f64 44.9

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

   6. Applied rewrites 44.9%

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

   if 0.28000000000000003 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 10: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq -0.1:\\ \;\;\;\;\left(\frac{1}{\left|t\_1\right|} \cdot t\_1\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|ky\right|}{\mathsf{hypot}\left(\left|ky\right|, \sin kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) -0.1)
      (* (* (/ 1.0 (fabs t_1)) t_1) th)
      (* (/ (fabs ky) (hypot (fabs ky) (sin kx))) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= -0.1) {
		tmp = ((1.0 / fabs(t_1)) * t_1) * th;
	} else {
		tmp = (fabs(ky) / hypot(fabs(ky), sin(kx))) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= -0.1) {
		tmp = ((1.0 / Math.abs(t_1)) * t_1) * th;
	} else {
		tmp = (Math.abs(ky) / Math.hypot(Math.abs(ky), Math.sin(kx))) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= -0.1:
		tmp = ((1.0 / math.fabs(t_1)) * t_1) * th
	else:
		tmp = (math.fabs(ky) / math.hypot(math.fabs(ky), math.sin(kx))) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = Float64(Float64(Float64(1.0 / abs(t_1)) * t_1) * th);
	else
		tmp = Float64(Float64(abs(ky) / hypot(abs(ky), sin(kx))) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= -0.1)
		tmp = ((1.0 / abs(t_1)) * t_1) * th;
	else
		tmp = (abs(ky) / hypot(abs(ky), sin(kx))) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], -0.1], N[(N[(N[(1.0 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $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))))) < -0.10000000000000001

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in th around 0

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

   11. Applied rewrites 22.7%

       \[\leadsto \left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \color{blue}{th} \]

   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)))))

   1. Initial program 93.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 51.7%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 65.6%

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

Alternative 11: 68.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ t_2 := \frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}}\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.038:\\ \;\;\;\;\left(\frac{1}{\left|t\_1\right|} \cdot t\_1\right) \cdot th\\ \mathbf{elif}\;t\_2 \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(kx, \left|ky\right|\right)} \cdot \left(\sin th \cdot \left|ky\right|\right)\\ \end{array} \end{array} \]

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 -0.038)
      (* (* (/ 1.0 (fabs t_1)) t_1) th)
      (if (<= t_2 1e-12)
        (* (sin th) (/ (fabs ky) (fabs (sin kx))))
        (* (/ 1.0 (hypot kx (fabs ky))) (* (sin th) (fabs ky))))))))

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 <= -0.038) {
		tmp = ((1.0 / fabs(t_1)) * t_1) * th;
	} else if (t_2 <= 1e-12) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (1.0 / hypot(kx, fabs(ky))) * (sin(th) * fabs(ky));
	}
	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 <= -0.038) {
		tmp = ((1.0 / Math.abs(t_1)) * t_1) * th;
	} else if (t_2 <= 1e-12) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	} else {
		tmp = (1.0 / Math.hypot(kx, Math.abs(ky))) * (Math.sin(th) * Math.abs(ky));
	}
	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 <= -0.038:
		tmp = ((1.0 / math.fabs(t_1)) * t_1) * th
	elif t_2 <= 1e-12:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	else:
		tmp = (1.0 / math.hypot(kx, math.fabs(ky))) * (math.sin(th) * math.fabs(ky))
	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 <= -0.038)
		tmp = Float64(Float64(Float64(1.0 / abs(t_1)) * t_1) * th);
	elseif (t_2 <= 1e-12)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(1.0 / hypot(kx, abs(ky))) * Float64(sin(th) * abs(ky)));
	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 <= -0.038)
		tmp = ((1.0 / abs(t_1)) * t_1) * th;
	elseif (t_2 <= 1e-12)
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	else
		tmp = (1.0 / hypot(kx, abs(ky))) * (sin(th) * abs(ky));
	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, -0.038], N[(N[(N[(1.0 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 1e-12], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[kx ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $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))))) < -0.0379999999999999991

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in th around 0

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

   11. Applied rewrites 22.7%

       \[\leadsto \left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \color{blue}{th} \]

   if -0.0379999999999999991 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999998e-13

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.8

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

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

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

      8. lower-fabs.f64 39.8

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

   6. Applied rewrites 39.8%

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

   if 9.9999999999999998e-13 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 43.0%

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

Alternative 12: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\sin th \cdot \frac{\left|ky\right|}{\left|\sin kx\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(kx, \left|ky\right|\right)} \cdot \left(\sin th \cdot \left|ky\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 1e-12)
      (* (sin th) (/ (fabs ky) (fabs (sin kx))))
      (* (/ 1.0 (hypot kx (fabs ky))) (* (sin th) (fabs ky)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 1e-12) {
		tmp = sin(th) * (fabs(ky) / fabs(sin(kx)));
	} else {
		tmp = (1.0 / hypot(kx, fabs(ky))) * (sin(th) * fabs(ky));
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 1e-12) {
		tmp = Math.sin(th) * (Math.abs(ky) / Math.abs(Math.sin(kx)));
	} else {
		tmp = (1.0 / Math.hypot(kx, Math.abs(ky))) * (Math.sin(th) * Math.abs(ky));
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 1e-12:
		tmp = math.sin(th) * (math.fabs(ky) / math.fabs(math.sin(kx)))
	else:
		tmp = (1.0 / math.hypot(kx, math.fabs(ky))) * (math.sin(th) * math.fabs(ky))
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 1e-12)
		tmp = Float64(sin(th) * Float64(abs(ky) / abs(sin(kx))));
	else
		tmp = Float64(Float64(1.0 / hypot(kx, abs(ky))) * Float64(sin(th) * abs(ky)));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 1e-12)
		tmp = sin(th) * (abs(ky) / abs(sin(kx)));
	else
		tmp = (1.0 / hypot(kx, abs(ky))) * (sin(th) * abs(ky));
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 1e-12], N[(N[Sin[th], $MachinePrecision] * N[(N[Abs[ky], $MachinePrecision] / N[Abs[N[Sin[kx], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sqrt[kx ^ 2 + N[Abs[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Abs[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.9999999999999998e-13

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{ky}{\sqrt{{\sin kx}^{2}}}} \cdot \sin th \]
   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 36.8

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. pow2 N/A

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

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

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

      8. lower-fabs.f64 39.8

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

   6. Applied rewrites 39.8%

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

   if 9.9999999999999998e-13 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 43.0%

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

Alternative 13: 46.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_1 := {\sin kx}^{2}\\ \mathbf{if}\;t\_1 \leq 0.008:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(kx, ky\right)} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{t\_1}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 2.0)))
   (if (<= t_1 0.008)
     (* (/ 1.0 (hypot kx ky)) (* (sin th) ky))
     (* (/ ky (sqrt t_1)) th))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.008) {
		tmp = (1.0 / hypot(kx, ky)) * (sin(th) * ky);
	} else {
		tmp = (ky / sqrt(t_1)) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 2.0);
	double tmp;
	if (t_1 <= 0.008) {
		tmp = (1.0 / Math.hypot(kx, ky)) * (Math.sin(th) * ky);
	} else {
		tmp = (ky / Math.sqrt(t_1)) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 2.0)
	tmp = 0
	if t_1 <= 0.008:
		tmp = (1.0 / math.hypot(kx, ky)) * (math.sin(th) * ky)
	else:
		tmp = (ky / math.sqrt(t_1)) * th
	return tmp

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0
	tmp = 0.0
	if (t_1 <= 0.008)
		tmp = Float64(Float64(1.0 / hypot(kx, ky)) * Float64(sin(th) * ky));
	else
		tmp = Float64(Float64(ky / sqrt(t_1)) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(kx) ^ 2.0;
	tmp = 0.0;
	if (t_1 <= 0.008)
		tmp = (1.0 / hypot(kx, ky)) * (sin(th) * ky);
	else
		tmp = (ky / sqrt(t_1)) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$1, 0.008], N[(N[(1.0 / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0080000000000000002

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 43.0%

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

   if 0.0080000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

   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 19.2%

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

Alternative 14: 46.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;{\sin kx}^{2} \leq 0.008:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(kx, ky\right)} \cdot \left(\sin th \cdot ky\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{ky}{\sqrt{1 - \cos \left(kx + kx\right)} \cdot \sqrt{0.5}} \cdot th\\ \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (pow (sin kx) 2.0) 0.008)
   (* (/ 1.0 (hypot kx ky)) (* (sin th) ky))
   (* (/ ky (* (sqrt (- 1.0 (cos (+ kx kx)))) (sqrt 0.5))) th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (pow(sin(kx), 2.0) <= 0.008) {
		tmp = (1.0 / hypot(kx, ky)) * (sin(th) * ky);
	} else {
		tmp = (ky / (sqrt((1.0 - cos((kx + kx)))) * sqrt(0.5))) * th;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.pow(Math.sin(kx), 2.0) <= 0.008) {
		tmp = (1.0 / Math.hypot(kx, ky)) * (Math.sin(th) * ky);
	} else {
		tmp = (ky / (Math.sqrt((1.0 - Math.cos((kx + kx)))) * Math.sqrt(0.5))) * th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.pow(math.sin(kx), 2.0) <= 0.008:
		tmp = (1.0 / math.hypot(kx, ky)) * (math.sin(th) * ky)
	else:
		tmp = (ky / (math.sqrt((1.0 - math.cos((kx + kx)))) * math.sqrt(0.5))) * th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if ((sin(kx) ^ 2.0) <= 0.008)
		tmp = Float64(Float64(1.0 / hypot(kx, ky)) * Float64(sin(th) * ky));
	else
		tmp = Float64(Float64(ky / Float64(sqrt(Float64(1.0 - cos(Float64(kx + kx)))) * sqrt(0.5))) * th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(kx) ^ 2.0) <= 0.008)
		tmp = (1.0 / hypot(kx, ky)) * (sin(th) * ky);
	else
		tmp = (ky / (sqrt((1.0 - cos((kx + kx)))) * sqrt(0.5))) * th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision], 0.008], N[(N[(1.0 / N[Sqrt[kx ^ 2 + ky ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision]), $MachinePrecision], N[(N[(ky / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) < 0.0080000000000000002

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in ky around 0

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

   6. Applied rewrites 49.4%

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

   7. Taylor expanded in ky around 0

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

   9. Applied rewrites 62.0%

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

   10. Taylor expanded in kx around 0

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

   12. Applied rewrites 43.0%

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

   if 0.0080000000000000002 < (pow.f64 (sin.f64 kx) #s(literal 2 binary64))

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. sin-mult N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. sqrt-prod N/A

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

      16. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.2%

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

Alternative 15: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin kx}^{2} + {t\_1}^{2}}} \leq 4 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left|ky\right|}{\sqrt{1 - \cos \left(kx + kx\right)} \cdot \sqrt{0.5}} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\left|\left|ky\right|\right|} \cdot \left|ky\right|\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin kx) 2.0) (pow t_1 2.0)))) 4e-16)
      (* (/ (fabs ky) (* (sqrt (- 1.0 (cos (+ kx kx)))) (sqrt 0.5))) th)
      (* (* (/ 1.0 (fabs (fabs ky))) (fabs ky)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(kx), 2.0) + pow(t_1, 2.0)))) <= 4e-16) {
		tmp = (fabs(ky) / (sqrt((1.0 - cos((kx + kx)))) * sqrt(0.5))) * th;
	} else {
		tmp = ((1.0 / fabs(fabs(ky))) * fabs(ky)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(t_1, 2.0)))) <= 4e-16) {
		tmp = (Math.abs(ky) / (Math.sqrt((1.0 - Math.cos((kx + kx)))) * Math.sqrt(0.5))) * th;
	} else {
		tmp = ((1.0 / Math.abs(Math.abs(ky))) * Math.abs(ky)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(t_1, 2.0)))) <= 4e-16:
		tmp = (math.fabs(ky) / (math.sqrt((1.0 - math.cos((kx + kx)))) * math.sqrt(0.5))) * th
	else:
		tmp = ((1.0 / math.fabs(math.fabs(ky))) * math.fabs(ky)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4e-16)
		tmp = Float64(Float64(abs(ky) / Float64(sqrt(Float64(1.0 - cos(Float64(kx + kx)))) * sqrt(0.5))) * th);
	else
		tmp = Float64(Float64(Float64(1.0 / abs(abs(ky))) * abs(ky)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(kx) ^ 2.0) + (t_1 ^ 2.0)))) <= 4e-16)
		tmp = (abs(ky) / (sqrt((1.0 - cos((kx + kx)))) * sqrt(0.5))) * th;
	else
		tmp = ((1.0 / abs(abs(ky))) * abs(ky)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[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], 4e-16], N[(N[(N[Abs[ky], $MachinePrecision] / N[(N[Sqrt[N[(1.0 - N[Cos[N[(kx + kx), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(N[(N[(1.0 / N[Abs[N[Abs[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $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))))) < 3.9999999999999999e-16

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

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

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. rem-square-sqrt N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. sin-mult N/A

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

      13. mult-flip N/A

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

      14. metadata-eval N/A

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

      15. sqrt-prod N/A

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

      16. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 27.1%

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

   7. Taylor expanded in th around 0

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

   9. Applied rewrites 14.2%

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

   if 3.9999999999999999e-16 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\frac{1}{\left|ky\right|} \cdot \sin ky\right) \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 17.3%

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

   12. Taylor expanded in ky around 0

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

   14. Applied rewrites 29.9%

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

Alternative 16: 39.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|ky\right|\right)\\ \mathsf{copysign}\left(1, ky\right) \cdot \begin{array}{l} \mathbf{if}\;\frac{t\_1}{\sqrt{{\sin \left(\left|kx\right|\right)}^{2} + {t\_1}^{2}}} \leq 6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left|ky\right|}{\left|kx\right|} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\left|\left|ky\right|\right|} \cdot \left|ky\right|\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (sin (fabs ky))))
   (*
    (copysign 1.0 ky)
    (if (<= (/ t_1 (sqrt (+ (pow (sin (fabs kx)) 2.0) (pow t_1 2.0)))) 6e-18)
      (* (/ (fabs ky) (fabs kx)) (sin th))
      (* (* (/ 1.0 (fabs (fabs ky))) (fabs ky)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(fabs(ky));
	double tmp;
	if ((t_1 / sqrt((pow(sin(fabs(kx)), 2.0) + pow(t_1, 2.0)))) <= 6e-18) {
		tmp = (fabs(ky) / fabs(kx)) * sin(th);
	} else {
		tmp = ((1.0 / fabs(fabs(ky))) * fabs(ky)) * sin(th);
	}
	return copysign(1.0, ky) * tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(Math.abs(ky));
	double tmp;
	if ((t_1 / Math.sqrt((Math.pow(Math.sin(Math.abs(kx)), 2.0) + Math.pow(t_1, 2.0)))) <= 6e-18) {
		tmp = (Math.abs(ky) / Math.abs(kx)) * Math.sin(th);
	} else {
		tmp = ((1.0 / Math.abs(Math.abs(ky))) * Math.abs(ky)) * Math.sin(th);
	}
	return Math.copySign(1.0, ky) * tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(math.fabs(ky))
	tmp = 0
	if (t_1 / math.sqrt((math.pow(math.sin(math.fabs(kx)), 2.0) + math.pow(t_1, 2.0)))) <= 6e-18:
		tmp = (math.fabs(ky) / math.fabs(kx)) * math.sin(th)
	else:
		tmp = ((1.0 / math.fabs(math.fabs(ky))) * math.fabs(ky)) * math.sin(th)
	return math.copysign(1.0, ky) * tmp

Julia

function code(kx, ky, th)
	t_1 = sin(abs(ky))
	tmp = 0.0
	if (Float64(t_1 / sqrt(Float64((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 6e-18)
		tmp = Float64(Float64(abs(ky) / abs(kx)) * sin(th));
	else
		tmp = Float64(Float64(Float64(1.0 / abs(abs(ky))) * abs(ky)) * sin(th));
	end
	return Float64(copysign(1.0, ky) * tmp)
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(abs(ky));
	tmp = 0.0;
	if ((t_1 / sqrt(((sin(abs(kx)) ^ 2.0) + (t_1 ^ 2.0)))) <= 6e-18)
		tmp = (abs(ky) / abs(kx)) * sin(th);
	else
		tmp = ((1.0 / abs(abs(ky))) * abs(ky)) * sin(th);
	end
	tmp_2 = (sign(ky) * abs(1.0)) * tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sin[N[Abs[ky], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[ky]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$1 / N[Sqrt[N[(N[Power[N[Sin[N[Abs[kx], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 6e-18], N[(N[(N[Abs[ky], $MachinePrecision] / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[Abs[N[Abs[ky], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[ky], $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))))) < 5.99999999999999966e-18

   1. Initial program 93.9%

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

   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 36.8

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

   4. Applied rewrites 36.8%

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

   5. Taylor expanded in kx around 0

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

      1. lower-/.f64 16.9

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

   7. Applied rewrites 16.9%

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

   if 5.99999999999999966e-18 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))

   1. Initial program 93.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 N/A

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

      15. *-commutative N/A

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

   3. Applied rewrites 95.9%

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

   4. Taylor expanded in kx around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 40.8

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

   6. Applied rewrites 40.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\left(\frac{1}{\left|\sin ky\right|} \cdot \sin ky\right) \cdot \sin th} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \left(\frac{1}{\left|ky\right|} \cdot \sin ky\right) \cdot \sin th \]
   10. Step-by-step derivation

   11. Applied rewrites 17.3%

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

   12. Taylor expanded in ky around 0

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

   14. Applied rewrites 29.9%

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

Alternative 17: 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 93.9%

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

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 36.8

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

4. Applied rewrites 36.8%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.9

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

7. Applied rewrites 16.9%

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

Alternative 18: 14.8% accurate, 8.8× speedup

Math

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

FPCore

(FPCore (kx ky th)
 :precision binary64
 (* (/ ky (fabs kx)) (* th (fma (* -0.16666666666666666 th) th 1.0))))

C

double code(double kx, double ky, double th) {
	return (ky / fabs(kx)) * (th * fma((-0.16666666666666666 * th), th, 1.0));
}

Julia

function code(kx, ky, th)
	return Float64(Float64(ky / abs(kx)) * Float64(th * fma(Float64(-0.16666666666666666 * th), th, 1.0)))
end

Wolfram

code[kx_, ky_, th_] := N[(N[(ky / N[Abs[kx], $MachinePrecision]), $MachinePrecision] * N[(th * N[(N[(-0.16666666666666666 * th), $MachinePrecision] * th + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.9%

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

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 36.8

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

4. Applied rewrites 36.8%

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

5. Taylor expanded in kx around 0

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

   1. lower-/.f64 16.9

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

7. Applied rewrites 16.9%

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

8. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 12.8

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

10. Applied rewrites 12.8%

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

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

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

    3. lift-*.f64 N/A

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

    4. lift-pow.f64 N/A

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

    5. unpow2 N/A

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

    6. associate-*r* N/A

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

    7. lower-fma.f64 N/A

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

    8. lower-*.f64 12.8

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

12. Applied rewrites 12.8%

    \[\leadsto \frac{ky}{kx} \cdot \left(th \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot th, \color{blue}{th}, 1\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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)))