Toniolo and Linder, Equation (3b), real

Percentage Accurate: 93.8% → 99.7%

Time: 9.2s

Alternatives: 19

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 93.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. 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

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-hypot.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 77.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_2 := ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ t_3 := \frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot th\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 0.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.044444444444444446, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \left(ky \cdot ky\right)}, \sin th\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_2 (* ky (/ (sin th) (hypot ky (sin kx)))))
        (t_3
         (*
          (/
           (sin ky)
           (sqrt
            (+
             (- 0.5 (* (cos (* 2.0 kx)) 0.5))
             (- 0.5 (* 0.5 (cos (* 2.0 ky)))))))
          th)))
   (if (<= t_1 -1.0)
     (* (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* -2.0 ky)))))) (sin th))
     (if (<= t_1 -0.1)
       t_3
       (if (<= t_1 0.5)
         t_2
         (if (<= t_1 0.995)
           t_3
           (if (<= t_1 2.0)
             (fma
              (* -0.5 (* kx kx))
              (/
               (sin th)
               (*
                (fma
                 (fma (* ky ky) 0.044444444444444446 -0.3333333333333333)
                 (* ky ky)
                 1.0)
                (* ky ky)))
              (sin th))
             t_2)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_2 = ky * (sin(th) / hypot(ky, sin(kx)));
	double t_3 = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (0.5 - (0.5 * cos((2.0 * ky))))))) * th;
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((-2.0 * ky)))))) * sin(th);
	} else if (t_1 <= -0.1) {
		tmp = t_3;
	} else if (t_1 <= 0.5) {
		tmp = t_2;
	} else if (t_1 <= 0.995) {
		tmp = t_3;
	} else if (t_1 <= 2.0) {
		tmp = fma((-0.5 * (kx * kx)), (sin(th) / (fma(fma((ky * ky), 0.044444444444444446, -0.3333333333333333), (ky * ky), 1.0) * (ky * ky))), sin(th));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_2 = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))))
	t_3 = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky))))))) * th)
	tmp = 0.0
	if (t_1 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(-2.0 * ky)))))) * sin(th));
	elseif (t_1 <= -0.1)
		tmp = t_3;
	elseif (t_1 <= 0.5)
		tmp = t_2;
	elseif (t_1 <= 0.995)
		tmp = t_3;
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(-0.5 * Float64(kx * kx)), Float64(sin(th) / Float64(fma(fma(Float64(ky * ky), 0.044444444444444446, -0.3333333333333333), Float64(ky * ky), 1.0) * Float64(ky * ky))), sin(th));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -0.1], t$95$3, If[LessEqual[t$95$1, 0.5], t$95$2, If[LessEqual[t$95$1, 0.995], t$95$3, If[LessEqual[t$95$1, 2.0], N[(N[(-0.5 * N[(kx * kx), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(ky * ky), $MachinePrecision] * 0.044444444444444446 + -0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[th], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 85.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 54.4

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

   4. Applied rewrites 54.4%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 53.9%

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

   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 or 0.5 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.994999999999999996

   1. Initial program 99.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 46.6%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 46.6

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

   9. Applied rewrites 46.6%

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

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

   1. Initial program 89.3%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 89.3

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 93.9%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 93.5%

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

   if 0.994999999999999996 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in kx around 0

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in ky around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot \left(kx \cdot kx\right), \frac{\sin th}{{ky}^{2} \cdot \color{blue}{\left(1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)\right)}}, \sin th\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 96.8%

      \[\leadsto \mathsf{fma}\left(-0.5 \cdot \left(kx \cdot kx\right), \frac{\sin th}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, 0.044444444444444446, -0.3333333333333333\right), ky \cdot ky, 1\right) \cdot \color{blue}{\left(ky \cdot ky\right)}}, \sin th\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 3: 51.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.7:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.7)
     (* (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) th)
     (if (<= t_1 0.02) (* (sin ky) (/ (sin th) (sin kx))) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.7) {
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	} else if (t_1 <= 0.02) {
		tmp = sin(ky) * (sin(th) / sin(kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.7d0)) then
        tmp = (sin(ky) / sqrt(((1.0d0 - cos((2.0d0 * ky))) * 0.5d0))) * th
    else if (t_1 <= 0.02d0) then
        tmp = sin(ky) * (sin(th) / sin(kx))
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.7) {
		tmp = (Math.sin(ky) / Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 0.5))) * th;
	} else if (t_1 <= 0.02) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sin(kx));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.7:
		tmp = (math.sin(ky) / math.sqrt(((1.0 - math.cos((2.0 * ky))) * 0.5))) * th
	elif t_1 <= 0.02:
		tmp = math.sin(ky) * (math.sin(th) / math.sin(kx))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.7)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * th);
	elseif (t_1 <= 0.02)
		tmp = Float64(sin(ky) * Float64(sin(th) / sin(kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.7)
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	elseif (t_1 <= 0.02)
		tmp = sin(ky) * (sin(th) / sin(kx));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.7], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.69999999999999996

   1. Initial program 89.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 36.7%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 24.1%

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

   if -0.69999999999999996 < (/.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 99.5%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 48.3%

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

   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 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

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

Alternative 4: 50.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) th)
     (if (<= t_1 0.02) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	} else if (t_1 <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = (sin(ky) / sqrt(((1.0d0 - cos((2.0d0 * ky))) * 0.5d0))) * th
    else if (t_1 <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (Math.sin(ky) / Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 0.5))) * th;
	} else if (t_1 <= 0.02) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = (math.sin(ky) / math.sqrt(((1.0 - math.cos((2.0 * ky))) * 0.5))) * th
	elif t_1 <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * th);
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	elseif (t_1 <= 0.02)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.10000000000000001

   1. Initial program 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 20.9%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

   5. Applied rewrites 57.2%

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

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

   1. Initial program 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

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

Alternative 5: 50.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.1:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(2 \cdot ky\right)}} \cdot \sin ky\right) \cdot th\\ \mathbf{elif}\;t\_1 \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_1 -0.1)
     (* (* (sqrt (/ 2.0 (- 1.0 (cos (* 2.0 ky))))) (sin ky)) th)
     (if (<= t_1 0.02) (* (/ ky (sin kx)) (sin th)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * sin(ky)) * th;
	} else if (t_1 <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))
    if (t_1 <= (-0.1d0)) then
        tmp = (sqrt((2.0d0 / (1.0d0 - cos((2.0d0 * ky))))) * sin(ky)) * th
    else if (t_1 <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.1) {
		tmp = (Math.sqrt((2.0 / (1.0 - Math.cos((2.0 * ky))))) * Math.sin(ky)) * th;
	} else if (t_1 <= 0.02) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_1 <= -0.1:
		tmp = (math.sqrt((2.0 / (1.0 - math.cos((2.0 * ky))))) * math.sin(ky)) * th
	elif t_1 <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_1 <= -0.1)
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky))))) * sin(ky)) * th);
	elseif (t_1 <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.1)
		tmp = (sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * sin(ky)) * th;
	elseif (t_1 <= 0.02)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.1], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$1, 0.02], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.10000000000000001

   1. Initial program 91.4%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 73.8

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 38.3%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 20.8%

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

   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))))) < 0.0200000000000000004

   1. Initial program 99.6%

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

   3. Taylor expanded in ky around 0

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

   5. Applied rewrites 57.2%

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

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

   1. Initial program 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

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

Alternative 6: 44.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.3%

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

   3. Taylor expanded in ky around 0

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

   5. Applied rewrites 30.3%

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

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

   1. Initial program 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.02:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

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

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.02) {
		tmp = (sin(th) * ky) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.02d0) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

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

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.02:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.02)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 30.4%

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

   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 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 15.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 2 \cdot 10^{-293}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      2e-293)
   (* (* (* th th) -0.16666666666666666) th)
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 2e-293) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 2d-293) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 2e-293) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 2e-293:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-293)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 2e-293)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 2e-293], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 2.0000000000000001e-293

   1. Initial program 93.7%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 22.5%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 14.3%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 2.0000000000000001e-293 < (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th))

   1. Initial program 90.7%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 23.8%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

      \[\leadsto th \]
   7. Step-by-step derivation

   8. Applied rewrites 12.1%

      \[\leadsto th \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 35.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-12}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-12)
   (* (/ ky (sin kx)) th)
   (sin th)))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-12) then
        tmp = (ky / sin(kx)) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-12], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 95.3%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 86.2

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

   4. Applied rewrites 86.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 43.0%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 18.3%

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

   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 85.5%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 30.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 8.8 \cdot 10^{-104}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
      8.8e-104)
   (* (* (* th th) -0.16666666666666666) th)
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 8.8e-104) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 8.8d-104) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 8.8e-104) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 8.8e-104:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 8.8e-104)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 8.8e-104)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 8.8e-104], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 8.80000000000000047e-104

   1. Initial program 94.9%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 3.2%

      \[\leadsto \color{blue}{\sin th} \]

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 3.2%

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

   9. Taylor expanded in th around inf

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

   11. Applied rewrites 13.2%

       \[\leadsto \left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th \]

   if 8.80000000000000047e-104 < (/.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 88.2%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 57.8%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 72.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.06:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-5}:\\ \;\;\;\;\left(\mathsf{fma}\left(ky \cdot ky, -0.16666666666666666, 1\right) \cdot ky\right) \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.06)
   (* (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) th)
   (if (<= (sin ky) 1e-5)
     (*
      (* (fma (* ky ky) -0.16666666666666666 1.0) ky)
      (/ (sin th) (hypot ky (sin kx))))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.06) {
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	} else if (sin(ky) <= 1e-5) {
		tmp = (fma((ky * ky), -0.16666666666666666, 1.0) * ky) * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.06)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * th);
	elseif (sin(ky) <= 1e-5)
		tmp = Float64(Float64(fma(Float64(ky * ky), -0.16666666666666666, 1.0) * ky) * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.06], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-5], N[(N[(N[(N[(ky * ky), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * ky), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.059999999999999998

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 52.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 27.7%

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

   if -0.059999999999999998 < (sin.f64 ky) < 1.00000000000000008e-5

   1. Initial program 86.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 97.4%

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

   if 1.00000000000000008e-5 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.8%

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

Alternative 12: 72.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.06:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(1 - \cos \left(2 \cdot ky\right)\right) \cdot 0.5}} \cdot th\\ \mathbf{elif}\;\sin ky \leq 10^{-5}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.06)
   (* (/ (sin ky) (sqrt (* (- 1.0 (cos (* 2.0 ky))) 0.5))) th)
   (if (<= (sin ky) 1e-5) (* ky (/ (sin th) (hypot ky (sin kx)))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.06) {
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	} else if (sin(ky) <= 1e-5) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.06) {
		tmp = (Math.sin(ky) / Math.sqrt(((1.0 - Math.cos((2.0 * ky))) * 0.5))) * th;
	} else if (Math.sin(ky) <= 1e-5) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(ky) <= -0.06:
		tmp = (math.sin(ky) / math.sqrt(((1.0 - math.cos((2.0 * ky))) * 0.5))) * th
	elif math.sin(ky) <= 1e-5:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.06)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(1.0 - cos(Float64(2.0 * ky))) * 0.5))) * th);
	elseif (sin(ky) <= 1e-5)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(ky) <= -0.06)
		tmp = (sin(ky) / sqrt(((1.0 - cos((2.0 * ky))) * 0.5))) * th;
	elseif (sin(ky) <= 1e-5)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.06], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-5], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.059999999999999998

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 52.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 27.7%

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

   if -0.059999999999999998 < (sin.f64 ky) < 1.00000000000000008e-5

   1. Initial program 86.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 86.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 97.4%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 97.6%

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

   if 1.00000000000000008e-5 < (sin.f64 ky)

   1. Initial program 99.8%

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

   3. Taylor expanded in kx around 0

      \[\leadsto \color{blue}{\sin th} \]
   4. Step-by-step derivation

   5. Applied rewrites 68.8%

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

Alternative 13: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
2. Add Preprocessing
3. 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. associate-*l/ N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 92.3

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

   7. lift-sqrt.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. +-commutative N/A

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

   10. lift-pow.f64 N/A

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

   11. unpow2 N/A

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

   12. lift-pow.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-hypot.f64 99.5

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

4. Applied rewrites 99.5%

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

Alternative 14: 79.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;kx \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\left(0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot ky\right)\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= kx 8e-5)
   (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
   (*
    (/
     (sin ky)
     (sqrt
      (+ (- 0.5 (* (cos (* 2.0 kx)) 0.5)) (- 0.5 (* 0.5 (cos (* 2.0 ky)))))))
    (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e-5) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else {
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (0.5 - (0.5 * cos((2.0 * ky))))))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (kx <= 8e-5) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sqrt(((0.5 - (Math.cos((2.0 * kx)) * 0.5)) + (0.5 - (0.5 * Math.cos((2.0 * ky))))))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if kx <= 8e-5:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sqrt(((0.5 - (math.cos((2.0 * kx)) * 0.5)) + (0.5 - (0.5 * math.cos((2.0 * ky))))))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (kx <= 8e-5)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * ky))))))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (kx <= 8e-5)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	else
		tmp = (sin(ky) / sqrt(((0.5 - (cos((2.0 * kx)) * 0.5)) + (0.5 - (0.5 * cos((2.0 * ky))))))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[kx, 8e-5], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if kx < 8.00000000000000065e-5

   1. Initial program 89.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 73.8%

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

   if 8.00000000000000065e-5 < kx

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. cos-2 N/A

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

      10. cos-sum N/A

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

      11. lower-cos.f64 N/A

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

      12. count-2-rev N/A

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

      13. lower-*.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 15: 66.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0044:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0044)
   (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0044) {
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0044) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0044:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0044)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th);
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0044)
		tmp = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0044], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00440000000000000027

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 64.5%

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

   if 0.00440000000000000027 < th

   1. Initial program 88.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 68.0%

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

Alternative 16: 66.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 0.0044:\\ \;\;\;\;\sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= th 0.0044)
   (* (sin ky) (/ th (hypot (sin ky) (sin kx))))
   (* ky (/ (sin th) (hypot ky (sin kx))))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0044) {
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	} else {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (th <= 0.0044) {
		tmp = Math.sin(ky) * (th / Math.hypot(Math.sin(ky), Math.sin(kx)));
	} else {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if th <= 0.0044:
		tmp = math.sin(ky) * (th / math.hypot(math.sin(ky), math.sin(kx)))
	else:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (th <= 0.0044)
		tmp = Float64(sin(ky) * Float64(th / hypot(sin(ky), sin(kx))));
	else
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (th <= 0.0044)
		tmp = sin(ky) * (th / hypot(sin(ky), sin(kx)));
	else
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[th, 0.0044], N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 0.00440000000000000027

   1. Initial program 93.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in th around 0

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

   7. Applied rewrites 64.4%

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

   if 0.00440000000000000027 < th

   1. Initial program 88.9%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 88.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 55.2%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 68.0%

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

Alternative 17: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.7:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.7)
   (* ky (/ (sin th) (hypot ky (sin kx))))
   (* (/ (sin ky) (sqrt (* 0.5 (- 1.0 (cos (* -2.0 ky)))))) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.7) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((-2.0 * ky)))))) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.7) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 * (1.0 - Math.cos((-2.0 * ky)))))) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.7:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = (math.sin(ky) / math.sqrt((0.5 * (1.0 - math.cos((-2.0 * ky)))))) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.7)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 * Float64(1.0 - cos(Float64(-2.0 * ky)))))) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.7)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = (sin(ky) / sqrt((0.5 * (1.0 - cos((-2.0 * ky)))))) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.7], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 * N[(1.0 - N[Cos[N[(-2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.7000000000000002

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 76.7%

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

   if 2.7000000000000002 < ky

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 54.5%

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

Alternative 18: 70.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ky \leq 2.7:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{hypot}\left(ky, \sin kx\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{2}{1 - \cos \left(2 \cdot ky\right)}} \cdot \sin ky\right) \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= ky 2.7)
   (* ky (/ (sin th) (hypot ky (sin kx))))
   (* (* (sqrt (/ 2.0 (- 1.0 (cos (* 2.0 ky))))) (sin ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.7) {
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	} else {
		tmp = (sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * sin(ky)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (ky <= 2.7) {
		tmp = ky * (Math.sin(th) / Math.hypot(ky, Math.sin(kx)));
	} else {
		tmp = (Math.sqrt((2.0 / (1.0 - Math.cos((2.0 * ky))))) * Math.sin(ky)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ky <= 2.7:
		tmp = ky * (math.sin(th) / math.hypot(ky, math.sin(kx)))
	else:
		tmp = (math.sqrt((2.0 / (1.0 - math.cos((2.0 * ky))))) * math.sin(ky)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (ky <= 2.7)
		tmp = Float64(ky * Float64(sin(th) / hypot(ky, sin(kx))));
	else
		tmp = Float64(Float64(sqrt(Float64(2.0 / Float64(1.0 - cos(Float64(2.0 * ky))))) * sin(ky)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (ky <= 2.7)
		tmp = ky * (sin(th) / hypot(ky, sin(kx)));
	else
		tmp = (sqrt((2.0 / (1.0 - cos((2.0 * ky))))) * sin(ky)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[ky, 2.7], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(2.0 / N[(1.0 - N[Cos[N[(2.0 * ky), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ky < 2.7000000000000002

   1. Initial program 90.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   2. Add Preprocessing
   3. 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. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.0

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. unpow2 N/A

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

      12. lift-pow.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-hypot.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in ky around 0

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

   10. Applied rewrites 76.7%

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

   if 2.7000000000000002 < ky

   1. Initial program 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in kx around 0

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

   7. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{0.5 \cdot \left(1 - \cos \left(-2 \cdot ky\right)\right)}} \cdot \left(\sin th \cdot \sin ky\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 54.3%

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

Alternative 19: 13.8% accurate, 632.0× speedup

Math

\[\begin{array}{l} \\ th \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(kx, ky, th)
	return th
end

MATLAB

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

Wolfram

code[kx_, ky_, th_] := th

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in kx around 0

   \[\leadsto \color{blue}{\sin th} \]
4. Step-by-step derivation

5. Applied rewrites 23.0%

   \[\leadsto \color{blue}{\sin th} \]

6. Taylor expanded in th around 0

   \[\leadsto th \]
7. Step-by-step derivation

8. Applied rewrites 12.8%

   \[\leadsto th \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025019 
(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)))