Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.1% → 99.7%

Time: 6.8s

Alternatives: 17

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.1% 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 95.0%

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

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

   4. lift-sin.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-sin.f64 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. unpow2 N/A

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

   10. lower-hypot.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 83.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1
         (*
          (/ (sin ky) (hypot (sin ky) (sin kx)))
          (* (fma (* th th) -0.16666666666666666 1.0) th)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.99)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 0.15)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_4 0.995)
           t_1
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), sin(kx))) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.99) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 0.15) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th))
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 0.15)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 84.3%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -0.98999999999999999 < (/.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))))) < -0.20000000000000001 or 0.149999999999999994 < (/.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 99.4%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.1

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

   7. Applied rewrites 53.1%

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

   if -0.20000000000000001 < (/.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))))) < 0.149999999999999994

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 96.2

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

   5. Applied rewrites 96.2%

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

   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)))))

   1. Initial program 93.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 98.4%

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

Alternative 3: 83.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -0.99:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_2}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (pow (sin ky) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_2 t_3)))))
   (if (<= t_4 -0.99)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.2)
       t_1
       (if (<= t_4 0.15)
         (* (/ (sin ky) (sqrt t_2)) (sin th))
         (if (<= t_4 0.995)
           t_1
           (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = pow(sin(ky), 2.0);
	double t_4 = sin(ky) / sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.99) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 0.15) {
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.pow(Math.sin(ky), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_2 + t_3));
	double tmp;
	if (t_4 <= -0.99) {
		tmp = (Math.sin(ky) / Math.sqrt(t_3)) * Math.sin(th);
	} else if (t_4 <= -0.2) {
		tmp = t_1;
	} else if (t_4 <= 0.15) {
		tmp = (Math.sin(ky) / Math.sqrt(t_2)) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_1;
	} else {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.pow(math.sin(ky), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_2 + t_3))
	tmp = 0
	if t_4 <= -0.99:
		tmp = (math.sin(ky) / math.sqrt(t_3)) * math.sin(th)
	elif t_4 <= -0.2:
		tmp = t_1
	elif t_4 <= 0.15:
		tmp = (math.sin(ky) / math.sqrt(t_2)) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = t_1
	else:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	t_2 = sin(kx) ^ 2.0
	t_3 = sin(ky) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_2 + t_3)))
	tmp = 0.0
	if (t_4 <= -0.99)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 0.15)
		tmp = Float64(Float64(sin(ky) / sqrt(t_2)) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = t_1;
	else
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_2 + t_3));
	tmp = 0.0;
	if (t_4 <= -0.99)
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	elseif (t_4 <= -0.2)
		tmp = t_1;
	elseif (t_4 <= 0.15)
		tmp = (sin(ky) / sqrt(t_2)) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = t_1;
	else
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.2], t$95$1, If[LessEqual[t$95$4, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$1, N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.98999999999999999

   1. Initial program 84.3%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if -0.98999999999999999 < (/.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))))) < -0.20000000000000001 or 0.149999999999999994 < (/.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 99.4%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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 52.8%

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

   if -0.20000000000000001 < (/.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))))) < 0.149999999999999994

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 96.2

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

   5. Applied rewrites 96.2%

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

   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)))))

   1. Initial program 93.2%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 98.4%

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

Alternative 4: 86.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ t_3 := {\sin kx}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_3 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_4 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.995:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th))
        (t_3 (pow (sin kx) 2.0))
        (t_4 (/ (sin ky) (sqrt (+ t_3 (pow (sin ky) 2.0))))))
   (if (<= t_4 -1.0)
     t_1
     (if (<= t_4 -0.2)
       t_2
       (if (<= t_4 0.15)
         (* (/ (sin ky) (sqrt t_3)) (sin th))
         (if (<= t_4 0.995) t_2 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	double t_3 = pow(sin(kx), 2.0);
	double t_4 = sin(ky) / sqrt((t_3 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.2) {
		tmp = t_2;
	} else if (t_4 <= 0.15) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	double t_3 = Math.pow(Math.sin(kx), 2.0);
	double t_4 = Math.sin(ky) / Math.sqrt((t_3 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_4 <= -1.0) {
		tmp = t_1;
	} else if (t_4 <= -0.2) {
		tmp = t_2;
	} else if (t_4 <= 0.15) {
		tmp = (Math.sin(ky) / Math.sqrt(t_3)) * Math.sin(th);
	} else if (t_4 <= 0.995) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	t_3 = math.pow(math.sin(kx), 2.0)
	t_4 = math.sin(ky) / math.sqrt((t_3 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_4 <= -1.0:
		tmp = t_1
	elif t_4 <= -0.2:
		tmp = t_2
	elif t_4 <= 0.15:
		tmp = (math.sin(ky) / math.sqrt(t_3)) * math.sin(th)
	elif t_4 <= 0.995:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	t_3 = sin(kx) ^ 2.0
	t_4 = Float64(sin(ky) / sqrt(Float64(t_3 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.2)
		tmp = t_2;
	elseif (t_4 <= 0.15)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= 0.995)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	t_3 = sin(kx) ^ 2.0;
	t_4 = sin(ky) / sqrt((t_3 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -1.0)
		tmp = t_1;
	elseif (t_4 <= -0.2)
		tmp = t_2;
	elseif (t_4 <= 0.15)
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	elseif (t_4 <= 0.995)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$3 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -1.0], t$95$1, If[LessEqual[t$95$4, -0.2], t$95$2, If[LessEqual[t$95$4, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.995], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.1%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 99.1%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.149999999999999994 < (/.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 99.4%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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 52.0%

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

   if -0.20000000000000001 < (/.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))))) < 0.149999999999999994

   1. Initial program 99.5%

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

   3. Taylor expanded in ky around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-pow.f64 96.2

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

   5. Applied rewrites 96.2%

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

Alternative 5: 86.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ t_3 := \frac{\sin ky}{\mathsf{hypot}\left(\sin ky, \sin kx\right)} \cdot th\\ \mathbf{if}\;t\_2 \leq -1:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -0.2:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.15:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.995:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (/ (sin ky) (hypot (sin ky) kx)) (sin th)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))))
        (t_3 (* (/ (sin ky) (hypot (sin ky) (sin kx))) th)))
   (if (<= t_2 -1.0)
     t_1
     (if (<= t_2 -0.2)
       t_3
       (if (<= t_2 0.15)
         (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
         (if (<= t_2 0.995) t_3 t_1))))))

C

double code(double kx, double ky, double th) {
	double t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double t_3 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.15) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	double t_2 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double t_3 = (Math.sin(ky) / Math.hypot(Math.sin(ky), Math.sin(kx))) * th;
	double tmp;
	if (t_2 <= -1.0) {
		tmp = t_1;
	} else if (t_2 <= -0.2) {
		tmp = t_3;
	} else if (t_2 <= 0.15) {
		tmp = (Math.sin(ky) / Math.hypot(ky, Math.sin(kx))) * Math.sin(th);
	} else if (t_2 <= 0.995) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	t_2 = math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))
	t_3 = (math.sin(ky) / math.hypot(math.sin(ky), math.sin(kx))) * th
	tmp = 0
	if t_2 <= -1.0:
		tmp = t_1
	elif t_2 <= -0.2:
		tmp = t_3
	elif t_2 <= 0.15:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	elif t_2 <= 0.995:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	t_3 = Float64(Float64(sin(ky) / hypot(sin(ky), sin(kx))) * th)
	tmp = 0.0
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.15)
		tmp = Float64(Float64(sin(ky) / hypot(ky, sin(kx))) * sin(th));
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	t_2 = sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)));
	t_3 = (sin(ky) / hypot(sin(ky), sin(kx))) * th;
	tmp = 0.0;
	if (t_2 <= -1.0)
		tmp = t_1;
	elseif (t_2 <= -0.2)
		tmp = t_3;
	elseif (t_2 <= 0.15)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	elseif (t_2 <= 0.995)
		tmp = t_3;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision]}, If[LessEqual[t$95$2, -1.0], t$95$1, If[LessEqual[t$95$2, -0.2], t$95$3, If[LessEqual[t$95$2, 0.15], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.995], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 89.1%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 100.0

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

   4. Applied rewrites 100.0%

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

   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 99.1%

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

   if -1 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < -0.20000000000000001 or 0.149999999999999994 < (/.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 99.4%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.3

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

   4. Applied rewrites 99.3%

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

   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 52.0%

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

   if -0.20000000000000001 < (/.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))))) < 0.149999999999999994

   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-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 95.1%

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

Alternative 6: 53.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -0.99:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot {\left(\mathsf{hypot}\left(kx, \sin ky\right)\right)}^{-1}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -0.99)
     (* (* (sin ky) th) (pow (hypot kx (sin ky)) -1.0))
     (if (<= t_1 5e-14) (* (/ (sin th) (sin kx)) (sin ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (sin(ky) * th) * pow(hypot(kx, sin(ky)), -1.0);
	} else if (t_1 <= 5e-14) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -0.99) {
		tmp = (Math.sin(ky) * th) * Math.pow(Math.hypot(kx, Math.sin(ky)), -1.0);
	} else if (t_1 <= 5e-14) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} 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 <= -0.99:
		tmp = (math.sin(ky) * th) * math.pow(math.hypot(kx, math.sin(ky)), -1.0)
	elif t_1 <= 5e-14:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	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 <= -0.99)
		tmp = Float64(Float64(sin(ky) * th) * (hypot(kx, sin(ky)) ^ -1.0));
	elseif (t_1 <= 5e-14)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	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 <= -0.99)
		tmp = (sin(ky) * th) * (hypot(kx, sin(ky)) ^ -1.0);
	elseif (t_1 <= 5e-14)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	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, -0.99], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Power[N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-14], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $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))))) < -0.98999999999999999

   1. Initial program 84.3%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 41.1

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 41.1%

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

   if -0.98999999999999999 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-14

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 49.1

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

   10. Applied rewrites 49.1%

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

   if 5.0000000000000002e-14 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.9

         \[\leadsto \sin th \]

   5. Applied rewrites 68.9%

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

Alternative 7: 54.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_1 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \left(\mathsf{fma}\left(th \cdot th, -0.16666666666666666, 1\right) \cdot th\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

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 -1.0)
     (*
      (/ (sin ky) (hypot (sin ky) kx))
      (* (fma (* th th) -0.16666666666666666 1.0) th))
     (if (<= t_1 5e-14) (* (/ (sin th) (sin kx)) (sin ky)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_1 <= -1.0) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * (fma((th * th), -0.16666666666666666, 1.0) * th);
	} else if (t_1 <= 5e-14) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} else {
		tmp = 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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * Float64(fma(Float64(th * th), -0.16666666666666666, 1.0) * th));
	elseif (t_1 <= 5e-14)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * th), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-14], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $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))))) < -1

   1. Initial program 84.1%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in th around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 53.4%

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

   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))))) < 5.0000000000000002e-14

   1. Initial program 99.5%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 97.7

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

   5. Applied rewrites 97.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 48.8

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

   10. Applied rewrites 48.8%

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

   if 5.0000000000000002e-14 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.9

         \[\leadsto \sin th \]

   5. Applied rewrites 68.9%

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

Alternative 8: 45.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;\frac{\sin th}{\sin kx} \cdot \sin ky\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-14)
   (* (/ (sin th) (sin kx)) (sin ky))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 5e-14) {
		tmp = (sin(th) / sin(kx)) * sin(ky);
	} 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)))) <= 5d-14) then
        tmp = (sin(th) / sin(kx)) * sin(ky)
    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)))) <= 5e-14) {
		tmp = (Math.sin(th) / Math.sin(kx)) * Math.sin(ky);
	} 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)))) <= 5e-14:
		tmp = (math.sin(th) / math.sin(kx)) * math.sin(ky)
	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)))) <= 5e-14)
		tmp = Float64(Float64(sin(th) / sin(kx)) * sin(ky));
	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)))) <= 5e-14)
		tmp = (sin(th) / sin(kx)) * sin(ky);
	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], 5e-14], N[(N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 5.0000000000000002e-14

   1. Initial program 94.8%

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

   3. Taylor expanded in kx around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. inv-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-sin.f64 94.9

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

   5. Applied rewrites 94.9%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-sin.f64 36.5

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

   10. Applied rewrites 36.5%

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

   if 5.0000000000000002e-14 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.9

         \[\leadsto \sin th \]

   5. Applied rewrites 68.9%

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

Alternative 9: 44.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-14)
   (* ky (/ (sin th) (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)))) <= 5e-14) {
		tmp = 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) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-14) then
        tmp = 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 tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-14) {
		tmp = ky * (Math.sin(th) / 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)))) <= 5e-14:
		tmp = ky * (math.sin(th) / 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)))) <= 5e-14)
		tmp = Float64(ky * Float64(sin(th) / 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)))) <= 5e-14)
		tmp = ky * (sin(th) / 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], 5e-14], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $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))))) < 5.0000000000000002e-14

   1. Initial program 94.8%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 34.1

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

   5. Applied rewrites 34.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 34.6

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

   7. Applied rewrites 34.6%

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

   if 5.0000000000000002e-14 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.9

         \[\leadsto \sin th \]

   5. Applied rewrites 68.9%

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

Alternative 10: 35.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-25}:\\ \;\;\;\;ky \cdot \frac{\sin th}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-25)
   (*
    ky
    (/
     (sin th)
     (*
      (fma
       (-
        (*
         (fma -0.0001984126984126984 (* kx kx) 0.008333333333333333)
         (* kx kx))
        0.16666666666666666)
       (* kx kx)
       1.0)
      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)))) <= 1e-25) {
		tmp = ky * (sin(th) / (fma(((fma(-0.0001984126984126984, (kx * kx), 0.008333333333333333) * (kx * kx)) - 0.16666666666666666), (kx * kx), 1.0) * kx));
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-25)
		tmp = Float64(ky * Float64(sin(th) / Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(kx * kx), 0.008333333333333333) * Float64(kx * kx)) - 0.16666666666666666), Float64(kx * kx), 1.0) * kx)));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-25], N[(ky * N[(N[Sin[th], $MachinePrecision] / N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(kx * kx), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(kx * kx), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(kx * kx), $MachinePrecision] + 1.0), $MachinePrecision] * kx), $MachinePrecision]), $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))))) < 1.00000000000000004e-25

   1. Initial program 94.8%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 34.3

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

   5. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in kx around 0

      \[\leadsto ky \cdot \frac{\sin th}{kx \cdot \color{blue}{\left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto ky \cdot \frac{\sin th}{\left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot kx} \]

      2. lower-*.f64 N/A

         \[\leadsto ky \cdot \frac{\sin th}{\left(1 + {kx}^{2} \cdot \left({kx}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {kx}^{2}\right) - \frac{1}{6}\right)\right) \cdot kx} \]

   10. Applied rewrites 21.9%

       \[\leadsto ky \cdot \frac{\sin th}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, kx \cdot kx, 0.008333333333333333\right) \cdot \left(kx \cdot kx\right) - 0.16666666666666666, kx \cdot kx, 1\right) \cdot \color{blue}{kx}} \]

   if 1.00000000000000004e-25 < (/.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 95.5%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.3

         \[\leadsto \sin th \]

   5. Applied rewrites 68.3%

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

Alternative 11: 15.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \leq 10^{-308}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<=
      (*
       (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))
       (sin th))
      1e-308)
   (* (* (* th th) -0.16666666666666666) th)
   th))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) * sin(th)) <= 1e-308) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if (((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) * sin(th)) <= 1d-308) then
        tmp = ((th * th) * (-0.16666666666666666d0)) * th
    else
        tmp = th
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) * Math.sin(th)) <= 1e-308) {
		tmp = ((th * th) * -0.16666666666666666) * th;
	} else {
		tmp = th;
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if ((math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) * math.sin(th)) <= 1e-308:
		tmp = ((th * th) * -0.16666666666666666) * th
	else:
		tmp = th
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-308)
		tmp = Float64(Float64(Float64(th * th) * -0.16666666666666666) * th);
	else
		tmp = th;
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) * sin(th)) <= 1e-308)
		tmp = ((th * th) * -0.16666666666666666) * th;
	else
		tmp = th;
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], 1e-308], N[(N[(N[(th * th), $MachinePrecision] * -0.16666666666666666), $MachinePrecision] * th), $MachinePrecision], th]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) (sin.f64 th)) < 9.9999999999999991e-309

   1. Initial program 95.0%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 24.4

         \[\leadsto \sin th \]

   5. Applied rewrites 24.4%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 13.9

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

   8. Applied rewrites 13.9%

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

   9. Taylor expanded in th around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 16.7

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

   11. Applied rewrites 16.7%

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

   if 9.9999999999999991e-309 < (*.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 95.1%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 43.5

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 18.8%

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

Alternative 12: 35.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-25}:\\ \;\;\;\;ky \cdot \frac{\sin th}{kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-25)
   (* ky (/ (sin th) 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)))) <= 1e-25) {
		tmp = ky * (sin(th) / 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)))) <= 1d-25) then
        tmp = ky * (sin(th) / 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)))) <= 1e-25) {
		tmp = ky * (Math.sin(th) / 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)))) <= 1e-25:
		tmp = ky * (math.sin(th) / 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)))) <= 1e-25)
		tmp = Float64(ky * Float64(sin(th) / 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)))) <= 1e-25)
		tmp = ky * (sin(th) / 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], 1e-25], N[(ky * N[(N[Sin[th], $MachinePrecision] / 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))))) < 1.00000000000000004e-25

   1. Initial program 94.8%

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

   3. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 34.3

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

   5. Applied rewrites 34.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-sin.f64 34.8

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

   7. Applied rewrites 34.8%

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

   8. Taylor expanded in kx around 0

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

   10. Applied rewrites 22.0%

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

   if 1.00000000000000004e-25 < (/.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 95.5%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.3

         \[\leadsto \sin th \]

   5. Applied rewrites 68.3%

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

Alternative 13: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 5 \cdot 10^{-14}:\\ \;\;\;\;ky \cdot \frac{th}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 5e-14)
   (* ky (/ th (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)))) <= 5e-14) {
		tmp = ky * (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) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 5d-14) then
        tmp = ky * (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 tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 5e-14) {
		tmp = ky * (th / 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)))) <= 5e-14:
		tmp = ky * (th / 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)))) <= 5e-14)
		tmp = Float64(ky * Float64(th / 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)))) <= 5e-14)
		tmp = ky * (th / 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], 5e-14], N[(ky * N[(th / N[Sin[kx], $MachinePrecision]), $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))))) < 5.0000000000000002e-14

   1. Initial program 94.8%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in ky around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sin.f64 20.3

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

   8. Applied rewrites 20.3%

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

   if 5.0000000000000002e-14 < (/.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 95.4%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 68.9

         \[\leadsto \sin th \]

   5. Applied rewrites 68.9%

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

Alternative 14: 30.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 3.9 \cdot 10^{-81}:\\ \;\;\;\;\left(\left(th \cdot th\right) \cdot -0.16666666666666666\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 3.9e-81)
   (* (* (* 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)))) <= 3.9e-81) {
		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)))) <= 3.9d-81) 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)))) <= 3.9e-81) {
		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)))) <= 3.9e-81:
		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)))) <= 3.9e-81)
		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)))) <= 3.9e-81)
		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], 3.9e-81], 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))))) < 3.89999999999999985e-81

   1. Initial program 94.6%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 3.3

         \[\leadsto \sin th \]

   5. Applied rewrites 3.3%

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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 3.2

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

   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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 14.9

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

   11. Applied rewrites 14.9%

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

   if 3.89999999999999985e-81 < (/.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 95.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 63.5

         \[\leadsto \sin th \]

   5. Applied rewrites 63.5%

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

Alternative 15: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(\sin ky, kx\right)} \cdot \sin th\\ \mathbf{elif}\;\sin ky \leq 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.01)
   (* (/ (sin ky) (hypot (sin ky) kx)) (sin th))
   (if (<= (sin ky) 1e-14)
     (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.01) {
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	} else if (sin(ky) <= 1e-14) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.01) {
		tmp = (Math.sin(ky) / Math.hypot(Math.sin(ky), kx)) * Math.sin(th);
	} else if (Math.sin(ky) <= 1e-14) {
		tmp = (Math.sin(ky) / Math.hypot(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) <= -0.01:
		tmp = (math.sin(ky) / math.hypot(math.sin(ky), kx)) * math.sin(th)
	elif math.sin(ky) <= 1e-14:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.01)
		tmp = Float64(Float64(sin(ky) / hypot(sin(ky), kx)) * sin(th));
	elseif (sin(ky) <= 1e-14)
		tmp = Float64(Float64(sin(ky) / hypot(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) <= -0.01)
		tmp = (sin(ky) / hypot(sin(ky), kx)) * sin(th);
	elseif (sin(ky) <= 1e-14)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[Sin[ky], $MachinePrecision] ^ 2 + kx ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-14], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 ky) < -0.0100000000000000002

   1. Initial program 99.7%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.7

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

   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 52.3%

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

   if -0.0100000000000000002 < (sin.f64 ky) < 9.99999999999999999e-15

   1. Initial program 89.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 99.5%

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

   if 9.99999999999999999e-15 < (sin.f64 ky)

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 64.3

         \[\leadsto \sin th \]

   5. Applied rewrites 64.3%

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

Alternative 16: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin ky \leq -0.02:\\ \;\;\;\;\left(\sin ky \cdot th\right) \cdot {\left(\mathsf{hypot}\left(kx, \sin ky\right)\right)}^{-1}\\ \mathbf{elif}\;\sin ky \leq 10^{-14}:\\ \;\;\;\;\frac{\sin ky}{\mathsf{hypot}\left(ky, \sin kx\right)} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin ky) -0.02)
   (* (* (sin ky) th) (pow (hypot kx (sin ky)) -1.0))
   (if (<= (sin ky) 1e-14)
     (* (/ (sin ky) (hypot ky (sin kx))) (sin th))
     (sin th))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(ky) <= -0.02) {
		tmp = (sin(ky) * th) * pow(hypot(kx, sin(ky)), -1.0);
	} else if (sin(ky) <= 1e-14) {
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(ky) <= -0.02) {
		tmp = (Math.sin(ky) * th) * Math.pow(Math.hypot(kx, Math.sin(ky)), -1.0);
	} else if (Math.sin(ky) <= 1e-14) {
		tmp = (Math.sin(ky) / Math.hypot(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) <= -0.02:
		tmp = (math.sin(ky) * th) * math.pow(math.hypot(kx, math.sin(ky)), -1.0)
	elif math.sin(ky) <= 1e-14:
		tmp = (math.sin(ky) / math.hypot(ky, math.sin(kx))) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(ky) <= -0.02)
		tmp = Float64(Float64(sin(ky) * th) * (hypot(kx, sin(ky)) ^ -1.0));
	elseif (sin(ky) <= 1e-14)
		tmp = Float64(Float64(sin(ky) / hypot(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) <= -0.02)
		tmp = (sin(ky) * th) * (hypot(kx, sin(ky)) ^ -1.0);
	elseif (sin(ky) <= 1e-14)
		tmp = (sin(ky) / hypot(ky, sin(kx))) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[ky], $MachinePrecision], -0.02], N[(N[(N[Sin[ky], $MachinePrecision] * th), $MachinePrecision] * N[Power[N[Sqrt[kx ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[ky], $MachinePrecision], 1e-14], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[ky ^ 2 + N[Sin[kx], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.7%

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

   3. Taylor expanded in th around 0

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqrt-div N/A

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

      6. metadata-eval N/A

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

      7. inv-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in kx around 0

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

   8. Applied rewrites 25.5%

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

   if -0.0200000000000000004 < (sin.f64 ky) < 9.99999999999999999e-15

   1. Initial program 89.6%

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

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. unpow2 N/A

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

      10. lower-hypot.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ky around 0

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

   7. Applied rewrites 98.7%

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

   if 9.99999999999999999e-15 < (sin.f64 ky)

   1. Initial program 99.7%

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

   3. Taylor expanded in kx around 0

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

      1. lift-sin.f64 64.3

         \[\leadsto \sin th \]

   5. Applied rewrites 64.3%

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

Alternative 17: 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 95.0%

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

3. Taylor expanded in th around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. sqrt-div N/A

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

   6. metadata-eval N/A

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

   7. inv-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. unpow2 N/A

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

   10. unpow2 N/A

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

   11. lower-hypot.f64 N/A

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

   12. lift-sin.f64 N/A

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

   13. lift-sin.f64 49.0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in kx around 0

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

8. Applied rewrites 16.4%

   \[\leadsto th \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025060 
(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)))