Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.3% → 99.7%

Time: 6.6s

Alternatives: 7

Speedup: N/A×

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: 94.3% 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, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in kx around inf

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

   1. unpow2 N/A

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

   2. unpow2 N/A

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

   3. lower-hypot.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-sin.f64 99.7

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in kx around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 3: 97.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right) \cdot -1\\ \left(\left(\cosh t\_1 + \sinh t\_1\right) \cdot \sin ky\right) \cdot \sin th \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (log (hypot (pow (sin ky) 1.0) (pow (sin kx) 1.0))) -1.0)))
   (* (* (+ (cosh t_1) (sinh t_1)) (sin ky)) (sin th))))

C

double code(double kx, double ky, double th) {
	double t_1 = log(hypot(pow(sin(ky), 1.0), pow(sin(kx), 1.0))) * -1.0;
	return ((cosh(t_1) + sinh(t_1)) * sin(ky)) * sin(th);
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.log(Math.hypot(Math.pow(Math.sin(ky), 1.0), Math.pow(Math.sin(kx), 1.0))) * -1.0;
	return ((Math.cosh(t_1) + Math.sinh(t_1)) * Math.sin(ky)) * Math.sin(th);
}

Python

def code(kx, ky, th):
	t_1 = math.log(math.hypot(math.pow(math.sin(ky), 1.0), math.pow(math.sin(kx), 1.0))) * -1.0
	return ((math.cosh(t_1) + math.sinh(t_1)) * math.sin(ky)) * math.sin(th)

Julia

function code(kx, ky, th)
	t_1 = Float64(log(hypot((sin(ky) ^ 1.0), (sin(kx) ^ 1.0))) * -1.0)
	return Float64(Float64(Float64(cosh(t_1) + sinh(t_1)) * sin(ky)) * sin(th))
end

MATLAB

function tmp = code(kx, ky, th)
	t_1 = log(hypot((sin(ky) ^ 1.0), (sin(kx) ^ 1.0))) * -1.0;
	tmp = ((cosh(t_1) + sinh(t_1)) * sin(ky)) * sin(th);
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Log[N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] * -1.0), $MachinePrecision]}, N[(N[(N[(N[Cosh[t$95$1], $MachinePrecision] + N[Sinh[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

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

3. Taylor expanded in kx around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. sqrt-div N/A

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

   4. metadata-eval N/A

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

   5. inv-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. unpow2 N/A

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

   8. unpow2 N/A

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

   9. lower-hypot.f64 N/A

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

   10. lift-sin.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. lift-sin.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-hypot.f64 N/A

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

   5. pow-to-exp N/A

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

   6. sinh-+-cosh-rev N/A

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

   7. lower-+.f64 N/A

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

7. Applied rewrites 97.9%

   \[\leadsto \left(\left(\cosh \left(\log \left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right) \cdot -1\right) + \sinh \left(\log \left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right) \cdot -1\right)\right) \cdot \sin \color{blue}{ky}\right) \cdot \sin th \]
8. Add Preprocessing

Alternative 4: 95.7% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 96.8%

      \[\leadsto \left({\left(\mathsf{fma}\left({\sin ky}^{1}, {\sin ky}^{1}, {\sin kx}^{2}\right)\right)}^{-0.5} \cdot \sin th\right) \cdot \color{blue}{\sin ky} \]

   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)))))

   1. Initial program 3.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

Alternative 5: 95.5% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 97.1%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 96.8%

      \[\leadsto \left({\left(\mathsf{fma}\left({\sin ky}^{1}, {\sin ky}^{1}, {\sin kx}^{2}\right)\right)}^{-0.5} \cdot \sin th\right) \cdot \color{blue}{\sin ky} \]

   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)))))

   1. Initial program 3.0%

      \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      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-hypot.f64 N/A

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

      5. sqr-pow N/A

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

      6. lower-*.f64 N/A

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

   7. Applied rewrites 98.4%

      \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5}\right) \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. sqr-pow N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-pow.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.f64 N/A

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

      10. lift-sin.f64 78.8

          \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{0.5} \cdot {\sin kx}^{0.5}\right)\right)}^{-0.5}\right) \cdot \left(\sin th \cdot \sin ky\right) \]

   9. Applied rewrites 78.8%

      \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{0.5} \cdot {\sin kx}^{0.5}\right)\right)}^{-0.5}\right) \cdot \left(\sin th \cdot \sin ky\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5}\\ \left(t\_1 \cdot t\_1\right) \cdot \left(\sin th \cdot \sin ky\right) \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (hypot (pow (sin ky) 1.0) (pow (sin kx) 1.0)) -0.5)))
   (* (* t_1 t_1) (* (sin th) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(hypot(pow(sin(ky), 1.0), pow(sin(kx), 1.0)), -0.5);
	return (t_1 * t_1) * (sin(th) * sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.hypot(Math.pow(Math.sin(ky), 1.0), Math.pow(Math.sin(kx), 1.0)), -0.5);
	return (t_1 * t_1) * (Math.sin(th) * Math.sin(ky));
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.hypot(math.pow(math.sin(ky), 1.0), math.pow(math.sin(kx), 1.0)), -0.5)
	return (t_1 * t_1) * (math.sin(th) * math.sin(ky))

Julia

function code(kx, ky, th)
	t_1 = hypot((sin(ky) ^ 1.0), (sin(kx) ^ 1.0)) ^ -0.5
	return Float64(Float64(t_1 * t_1) * Float64(sin(th) * sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	t_1 = hypot((sin(ky) ^ 1.0), (sin(kx) ^ 1.0)) ^ -0.5;
	tmp = (t_1 * t_1) * (sin(th) * sin(ky));
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sqrt[N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision] ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision], -0.5], $MachinePrecision]}, N[(N[(t$95$1 * t$95$1), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 95.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 95.6

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

5. Applied rewrites 95.6%

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

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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-hypot.f64 N/A

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

   5. sqr-pow N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 95.2%

   \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5}\right) \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
8. Add Preprocessing

Alternative 7: 47.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\sin kx}^{0.5}\\ t_2 := {\sin ky}^{1}\\ \left({\left(\mathsf{hypot}\left(t\_2, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left(t\_2, t\_1 \cdot t\_1\right)\right)}^{-0.5}\right) \cdot \left(\sin th \cdot \sin ky\right) \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (pow (sin kx) 0.5)) (t_2 (pow (sin ky) 1.0)))
   (*
    (*
     (pow (hypot t_2 (pow (sin kx) 1.0)) -0.5)
     (pow (hypot t_2 (* t_1 t_1)) -0.5))
    (* (sin th) (sin ky)))))

C

double code(double kx, double ky, double th) {
	double t_1 = pow(sin(kx), 0.5);
	double t_2 = pow(sin(ky), 1.0);
	return (pow(hypot(t_2, pow(sin(kx), 1.0)), -0.5) * pow(hypot(t_2, (t_1 * t_1)), -0.5)) * (sin(th) * sin(ky));
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.pow(Math.sin(kx), 0.5);
	double t_2 = Math.pow(Math.sin(ky), 1.0);
	return (Math.pow(Math.hypot(t_2, Math.pow(Math.sin(kx), 1.0)), -0.5) * Math.pow(Math.hypot(t_2, (t_1 * t_1)), -0.5)) * (Math.sin(th) * Math.sin(ky));
}

Python

def code(kx, ky, th):
	t_1 = math.pow(math.sin(kx), 0.5)
	t_2 = math.pow(math.sin(ky), 1.0)
	return (math.pow(math.hypot(t_2, math.pow(math.sin(kx), 1.0)), -0.5) * math.pow(math.hypot(t_2, (t_1 * t_1)), -0.5)) * (math.sin(th) * math.sin(ky))

Julia

function code(kx, ky, th)
	t_1 = sin(kx) ^ 0.5
	t_2 = sin(ky) ^ 1.0
	return Float64(Float64((hypot(t_2, (sin(kx) ^ 1.0)) ^ -0.5) * (hypot(t_2, Float64(t_1 * t_1)) ^ -0.5)) * Float64(sin(th) * sin(ky)))
end

MATLAB

function tmp = code(kx, ky, th)
	t_1 = sin(kx) ^ 0.5;
	t_2 = sin(ky) ^ 1.0;
	tmp = ((hypot(t_2, (sin(kx) ^ 1.0)) ^ -0.5) * (hypot(t_2, (t_1 * t_1)) ^ -0.5)) * (sin(th) * sin(ky));
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Power[N[Sin[kx], $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[ky], $MachinePrecision], 1.0], $MachinePrecision]}, N[(N[(N[Power[N[Sqrt[t$95$2 ^ 2 + N[Power[N[Sin[kx], $MachinePrecision], 1.0], $MachinePrecision] ^ 2], $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[Sqrt[t$95$2 ^ 2 + N[(t$95$1 * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 95.2%

   \[\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \cdot \sin th \]
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 95.6

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

5. Applied rewrites 95.6%

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

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   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-hypot.f64 N/A

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

   5. sqr-pow N/A

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

   6. lower-*.f64 N/A

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

7. Applied rewrites 95.2%

   \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5}\right) \cdot \left(\color{blue}{\sin th} \cdot \sin ky\right) \]
8. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. sqr-pow N/A

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

   4. lower-*.f64 N/A

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

   5. metadata-eval N/A

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

   6. lower-pow.f64 N/A

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

   7. lift-sin.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-sin.f64 50.1

       \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{0.5} \cdot {\sin kx}^{0.5}\right)\right)}^{-0.5}\right) \cdot \left(\sin th \cdot \sin ky\right) \]

9. Applied rewrites 50.1%

   \[\leadsto \left({\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{1}\right)\right)}^{-0.5} \cdot {\left(\mathsf{hypot}\left({\sin ky}^{1}, {\sin kx}^{0.5} \cdot {\sin kx}^{0.5}\right)\right)}^{-0.5}\right) \cdot \left(\sin th \cdot \sin ky\right) \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025065 
(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)))