Toniolo and Linder, Equation (3b), real

Percentage Accurate: 94.2% → 99.7%

Time: 4.4s

Alternatives: 12

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: 94.2% 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 94.2%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. 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

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

3. Applied rewrites 99.7%

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

Alternative 2: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, ky \cdot ky, 1\right) \cdot ky, ky, t\_2\right)}}\\ \mathbf{elif}\;t\_4 \leq 0.99999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin kx) (sin ky)))))
        (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 -1.0)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.01)
       t_1
       (if (<= t_4 1e-6)
         (*
          (sin ky)
          (/
           (sin th)
           (sqrt (fma (* (fma -0.3333333333333333 (* ky ky) 1.0) ky) ky t_2))))
         (if (<= t_4 0.99999998) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	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 <= -1.0) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_1;
	} else if (t_4 <= 1e-6) {
		tmp = sin(ky) * (sin(th) / sqrt(fma((fma(-0.3333333333333333, (ky * ky), 1.0) * ky), ky, t_2)));
	} else if (t_4 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))))
	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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.01)
		tmp = t_1;
	elseif (t_4 <= 1e-6)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(fma(Float64(fma(-0.3333333333333333, Float64(ky * ky), 1.0) * ky), ky, t_2))));
	elseif (t_4 <= 0.99999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $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, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.01], t$95$1, If[LessEqual[t$95$4, 1e-6], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[N[(N[(N[(-0.3333333333333333 * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] * ky), $MachinePrecision] * ky + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.99999998], t$95$1, N[Sin[th], $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))))) < -1

   1. Initial program 86.2%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   3. 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 86.2

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

   4. Applied rewrites 86.2%

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

   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))))) < -0.0100000000000000002 or 9.99999999999999955e-7 < (/.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))))) < 0.999999980000000011

   1. Initial program 99.1%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 50.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 50.3

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

   6. Applied rewrites 50.3%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{3} \cdot {ky}^{2}, \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{3} \cdot {ky}^{2} + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \frac{-1}{3} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. lift-pow.f64 99.0

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   4. Applied rewrites 99.0%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\sin ky \cdot \sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \]

      6. lower-/.f64 99.0

         \[\leadsto \sin ky \cdot \color{blue}{\frac{\sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \]

      7. lift-fma.f64 N/A

         \[\leadsto \sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right) \cdot \left(ky \cdot ky\right) + \color{blue}{{\sin kx}^{2}}}} \]

   6. Applied rewrites 99.0%

      \[\leadsto \color{blue}{\sin ky \cdot \frac{\sin th}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.3333333333333333, ky \cdot ky, 1\right) \cdot ky, ky, {\sin kx}^{2}\right)}}} \]

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

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

Alternative 3: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 0.1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(ky, ky, t\_2\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq 0.99999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin kx) (sin ky)))))
        (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 -1.0)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.01)
       t_1
       (if (<= t_4 0.1)
         (* (/ (sin ky) (sqrt (fma ky ky t_2))) (sin th))
         (if (<= t_4 0.99999998) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	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 <= -1.0) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_1;
	} else if (t_4 <= 0.1) {
		tmp = (sin(ky) / sqrt(fma(ky, ky, t_2))) * sin(th);
	} else if (t_4 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))))
	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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.01)
		tmp = t_1;
	elseif (t_4 <= 0.1)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(ky, ky, t_2))) * sin(th));
	elseif (t_4 <= 0.99999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $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, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.01], t$95$1, If[LessEqual[t$95$4, 0.1], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(ky * ky + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.99999998], t$95$1, N[Sin[th], $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))))) < -1

   1. Initial program 86.2%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   3. 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 86.2

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

   4. Applied rewrites 86.2%

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

   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))))) < -0.0100000000000000002 or 0.10000000000000001 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.999999980000000011

   1. Initial program 99.1%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 50.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 50.2

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 50.4

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

   6. Applied rewrites 50.4%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.10000000000000001

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

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

      1. unpow2 N/A

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

      2. lower-fma.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

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

Alternative 4: 82.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := {\sin kx}^{2}\\ t_3 := {\sin ky}^{2}\\ t_4 := \frac{\sin ky}{\sqrt{t\_2 + t\_3}}\\ \mathbf{if}\;t\_4 \leq -1:\\ \;\;\;\;\frac{\sin ky}{\sqrt{t\_3}} \cdot \sin th\\ \mathbf{elif}\;t\_4 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_4 \leq 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_4 \leq 0.99999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin kx) (sin ky)))))
        (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 -1.0)
     (* (/ (sin ky) (sqrt t_3)) (sin th))
     (if (<= t_4 -0.01)
       t_1
       (if (<= t_4 1e-6)
         (* (sin ky) (/ (sin th) (sqrt t_2)))
         (if (<= t_4 0.99999998) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	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 <= -1.0) {
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_1;
	} else if (t_4 <= 1e-6) {
		tmp = sin(ky) * (sin(th) / sqrt(t_2));
	} else if (t_4 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.sin(ky) * (th / Math.hypot(Math.sin(kx), Math.sin(ky)));
	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 <= -1.0) {
		tmp = (Math.sin(ky) / Math.sqrt(t_3)) * Math.sin(th);
	} else if (t_4 <= -0.01) {
		tmp = t_1;
	} else if (t_4 <= 1e-6) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sqrt(t_2));
	} else if (t_4 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.sin(ky) * (th / math.hypot(math.sin(kx), math.sin(ky)))
	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 <= -1.0:
		tmp = (math.sin(ky) / math.sqrt(t_3)) * math.sin(th)
	elif t_4 <= -0.01:
		tmp = t_1
	elif t_4 <= 1e-6:
		tmp = math.sin(ky) * (math.sin(th) / math.sqrt(t_2))
	elif t_4 <= 0.99999998:
		tmp = t_1
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))))
	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 <= -1.0)
		tmp = Float64(Float64(sin(ky) / sqrt(t_3)) * sin(th));
	elseif (t_4 <= -0.01)
		tmp = t_1;
	elseif (t_4 <= 1e-6)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_2)));
	elseif (t_4 <= 0.99999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	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 <= -1.0)
		tmp = (sin(ky) / sqrt(t_3)) * sin(th);
	elseif (t_4 <= -0.01)
		tmp = t_1;
	elseif (t_4 <= 1e-6)
		tmp = sin(ky) * (sin(th) / sqrt(t_2));
	elseif (t_4 <= 0.99999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $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, -1.0], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -0.01], t$95$1, If[LessEqual[t$95$4, 1e-6], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.99999998], t$95$1, N[Sin[th], $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))))) < -1

   1. Initial program 86.2%

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

   2. Taylor expanded in kx around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin ky}^{2}}}} \cdot \sin th \]
   3. 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 86.2

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

   4. Applied rewrites 86.2%

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

   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))))) < -0.0100000000000000002 or 9.99999999999999955e-7 < (/.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))))) < 0.999999980000000011

   1. Initial program 99.1%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 50.1

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 50.3

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

   6. Applied rewrites 50.3%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   3. 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 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.8

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

   6. Applied rewrites 98.8%

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

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

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

Alternative 5: 65.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.99999998:\\ \;\;\;\;\sin ky \cdot \frac{th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.01)
     (* (/ (sin ky) t_1) th)
     (if (<= t_2 4e-186)
       (*
        (/
         (sin ky)
         (sqrt
          (fma
           (fma (* ky ky) -0.3333333333333333 1.0)
           (* ky ky)
           (- 0.5 (* (cos (* 2.0 kx)) 0.5)))))
        (sin th))
       (if (<= t_2 1e-6)
         (* (/ ky (sin kx)) (sin th))
         (if (<= t_2 0.99999998) (* (sin ky) (/ th t_1)) (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.01) {
		tmp = (sin(ky) / t_1) * th;
	} else if (t_2 <= 4e-186) {
		tmp = (sin(ky) / sqrt(fma(fma((ky * ky), -0.3333333333333333, 1.0), (ky * ky), (0.5 - (cos((2.0 * kx)) * 0.5))))) * sin(th);
	} else if (t_2 <= 1e-6) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.99999998) {
		tmp = sin(ky) * (th / t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (t_2 <= 4e-186)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(fma(Float64(ky * ky), -0.3333333333333333, 1.0), Float64(ky * ky), Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5))))) * sin(th));
	elseif (t_2 <= 1e-6)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.99999998)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $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]}, If[LessEqual[t$95$2, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$2, 4e-186], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-6], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99999998], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 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))))) < -0.0100000000000000002

   1. Initial program 91.7%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 51.4%

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

   if -0.0100000000000000002 < (/.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))))) < 3.9999999999999996e-186

   1. Initial program 99.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{3} \cdot {ky}^{2}, \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{3} \cdot {ky}^{2} + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \frac{-1}{3} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. lift-pow.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 75.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 75.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th \]

   if 3.9999999999999996e-186 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.2

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

   4. Applied rewrites 55.2%

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

   if 9.99999999999999955e-7 < (/.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))))) < 0.999999980000000011

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 49.0

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

   6. Applied rewrites 49.0%

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

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\sin th} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 65.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin ky \cdot \frac{th}{\mathsf{hypot}\left(\sin kx, \sin ky\right)}\\ t_2 := \frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_2 \leq -0.01:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 10^{-6}:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{elif}\;t\_2 \leq 0.99999998:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (* (sin ky) (/ th (hypot (sin kx) (sin ky)))))
        (t_2 (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))
   (if (<= t_2 -0.01)
     t_1
     (if (<= t_2 4e-186)
       (*
        (/
         (sin ky)
         (sqrt
          (fma
           (fma (* ky ky) -0.3333333333333333 1.0)
           (* ky ky)
           (- 0.5 (* (cos (* 2.0 kx)) 0.5)))))
        (sin th))
       (if (<= t_2 1e-6)
         (* (/ ky (sin kx)) (sin th))
         (if (<= t_2 0.99999998) t_1 (sin th)))))))

C

double code(double kx, double ky, double th) {
	double t_1 = sin(ky) * (th / hypot(sin(kx), sin(ky)));
	double t_2 = sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)));
	double tmp;
	if (t_2 <= -0.01) {
		tmp = t_1;
	} else if (t_2 <= 4e-186) {
		tmp = (sin(ky) / sqrt(fma(fma((ky * ky), -0.3333333333333333, 1.0), (ky * ky), (0.5 - (cos((2.0 * kx)) * 0.5))))) * sin(th);
	} else if (t_2 <= 1e-6) {
		tmp = (ky / sin(kx)) * sin(th);
	} else if (t_2 <= 0.99999998) {
		tmp = t_1;
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Julia

function code(kx, ky, th)
	t_1 = Float64(sin(ky) * Float64(th / hypot(sin(kx), sin(ky))))
	t_2 = Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = t_1;
	elseif (t_2 <= 4e-186)
		tmp = Float64(Float64(sin(ky) / sqrt(fma(fma(Float64(ky * ky), -0.3333333333333333, 1.0), Float64(ky * ky), Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5))))) * sin(th));
	elseif (t_2 <= 1e-6)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	elseif (t_2 <= 0.99999998)
		tmp = t_1;
	else
		tmp = sin(th);
	end
	return tmp
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[(N[Sin[ky], $MachinePrecision] * N[(th / N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]), $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]}, If[LessEqual[t$95$2, -0.01], t$95$1, If[LessEqual[t$95$2, 4e-186], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[(N[(ky * ky), $MachinePrecision] * -0.3333333333333333 + 1.0), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-6], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.99999998], t$95$1, N[Sin[th], $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))))) < -0.0100000000000000002 or 9.99999999999999955e-7 < (/.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))))) < 0.999999980000000011

   1. Initial program 93.7%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 47.7

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 50.7

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

   6. Applied rewrites 50.7%

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

   if -0.0100000000000000002 < (/.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))))) < 3.9999999999999996e-186

   1. Initial program 99.2%

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

   2. Taylor expanded in ky around 0

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

      1. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\left(1 + \frac{-1}{3} \cdot {ky}^{2}\right) \cdot {ky}^{2} + {\color{blue}{\sin kx}}^{2}}} \cdot \sin th \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{3} \cdot {ky}^{2}, \color{blue}{{ky}^{2}}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      3. +-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\frac{-1}{3} \cdot {ky}^{2} + 1, {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left({ky}^{2} \cdot \frac{-1}{3} + 1, {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left({ky}^{2}, \frac{-1}{3}, 1\right), {\color{blue}{ky}}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      6. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), {ky}^{2}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      8. unpow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot \color{blue}{ky}, {\sin kx}^{2}\right)}} \cdot \sin th \]

      10. lift-sin.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      11. lift-pow.f64 98.9

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

   4. Applied rewrites 98.9%

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}}} \cdot \sin th \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, {\sin kx}^{2}\right)}} \cdot \sin th \]

      2. pow2 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      3. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      4. lift-sin.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \sin kx \cdot \sin kx\right)}} \cdot \sin th \]

      5. sqr-sin-a N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot kx\right)\right)}} \cdot \sin th \]

      7. *-commutative N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      9. count-2-rev N/A

         \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      10. lower-cos.f64 N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(kx + kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      11. count-2-rev N/A

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, \frac{-1}{3}, 1\right), ky \cdot ky, \frac{1}{2} - \cos \left(2 \cdot kx\right) \cdot \frac{1}{2}\right)}} \cdot \sin th \]

      12. lower-*.f64 75.7

          \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th \]

   6. Applied rewrites 75.7%

      \[\leadsto \frac{\sin ky}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ky \cdot ky, -0.3333333333333333, 1\right), ky \cdot ky, 0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5\right)}} \cdot \sin th \]

   if 3.9999999999999996e-186 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.2

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

   4. Applied rewrites 55.2%

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

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

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

Alternative 7: 75.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{hypot}\left(\sin kx, \sin ky\right)\\ t_2 := {\sin kx}^{2}\\ t_3 := \frac{\sin ky}{\sqrt{t\_2 + {\sin ky}^{2}}}\\ \mathbf{if}\;t\_3 \leq -0.01:\\ \;\;\;\;\frac{\sin ky}{t\_1} \cdot th\\ \mathbf{elif}\;t\_3 \leq 10^{-6}:\\ \;\;\;\;\sin ky \cdot \frac{\sin th}{\sqrt{t\_2}}\\ \mathbf{elif}\;t\_3 \leq 0.99999998:\\ \;\;\;\;\sin ky \cdot \frac{th}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (let* ((t_1 (hypot (sin kx) (sin ky)))
        (t_2 (pow (sin kx) 2.0))
        (t_3 (/ (sin ky) (sqrt (+ t_2 (pow (sin ky) 2.0))))))
   (if (<= t_3 -0.01)
     (* (/ (sin ky) t_1) th)
     (if (<= t_3 1e-6)
       (* (sin ky) (/ (sin th) (sqrt t_2)))
       (if (<= t_3 0.99999998) (* (sin ky) (/ th t_1)) (sin th))))))

C

double code(double kx, double ky, double th) {
	double t_1 = hypot(sin(kx), sin(ky));
	double t_2 = pow(sin(kx), 2.0);
	double t_3 = sin(ky) / sqrt((t_2 + pow(sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.01) {
		tmp = (sin(ky) / t_1) * th;
	} else if (t_3 <= 1e-6) {
		tmp = sin(ky) * (sin(th) / sqrt(t_2));
	} else if (t_3 <= 0.99999998) {
		tmp = sin(ky) * (th / t_1);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Java

public static double code(double kx, double ky, double th) {
	double t_1 = Math.hypot(Math.sin(kx), Math.sin(ky));
	double t_2 = Math.pow(Math.sin(kx), 2.0);
	double t_3 = Math.sin(ky) / Math.sqrt((t_2 + Math.pow(Math.sin(ky), 2.0)));
	double tmp;
	if (t_3 <= -0.01) {
		tmp = (Math.sin(ky) / t_1) * th;
	} else if (t_3 <= 1e-6) {
		tmp = Math.sin(ky) * (Math.sin(th) / Math.sqrt(t_2));
	} else if (t_3 <= 0.99999998) {
		tmp = Math.sin(ky) * (th / t_1);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	t_1 = math.hypot(math.sin(kx), math.sin(ky))
	t_2 = math.pow(math.sin(kx), 2.0)
	t_3 = math.sin(ky) / math.sqrt((t_2 + math.pow(math.sin(ky), 2.0)))
	tmp = 0
	if t_3 <= -0.01:
		tmp = (math.sin(ky) / t_1) * th
	elif t_3 <= 1e-6:
		tmp = math.sin(ky) * (math.sin(th) / math.sqrt(t_2))
	elif t_3 <= 0.99999998:
		tmp = math.sin(ky) * (th / t_1)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky))
	t_2 = sin(kx) ^ 2.0
	t_3 = Float64(sin(ky) / sqrt(Float64(t_2 + (sin(ky) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= -0.01)
		tmp = Float64(Float64(sin(ky) / t_1) * th);
	elseif (t_3 <= 1e-6)
		tmp = Float64(sin(ky) * Float64(sin(th) / sqrt(t_2)));
	elseif (t_3 <= 0.99999998)
		tmp = Float64(sin(ky) * Float64(th / t_1));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	t_1 = hypot(sin(kx), sin(ky));
	t_2 = sin(kx) ^ 2.0;
	t_3 = sin(ky) / sqrt((t_2 + (sin(ky) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -0.01)
		tmp = (sin(ky) / t_1) * th;
	elseif (t_3 <= 1e-6)
		tmp = sin(ky) * (sin(th) / sqrt(t_2));
	elseif (t_3 <= 0.99999998)
		tmp = sin(ky) * (th / t_1);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := Block[{t$95$1 = N[Sqrt[N[Sin[kx], $MachinePrecision] ^ 2 + N[Sin[ky], $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(t$95$2 + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -0.01], N[(N[(N[Sin[ky], $MachinePrecision] / t$95$1), $MachinePrecision] * th), $MachinePrecision], If[LessEqual[t$95$3, 1e-6], N[(N[Sin[ky], $MachinePrecision] * N[(N[Sin[th], $MachinePrecision] / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.99999998], N[(N[Sin[ky], $MachinePrecision] * N[(th / t$95$1), $MachinePrecision]), $MachinePrecision], N[Sin[th], $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))))) < -0.0100000000000000002

   1. Initial program 91.7%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 47.3%

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

   6. Applied rewrites 51.4%

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

   if -0.0100000000000000002 < (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 99.1%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   3. 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 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.8

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

   6. Applied rewrites 98.8%

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

   if 9.99999999999999955e-7 < (/.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))))) < 0.999999980000000011

   1. Initial program 99.0%

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

   2. Taylor expanded in th around 0

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

   4. Applied rewrites 48.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 48.8

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

      7. lift-sqrt.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 49.0

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

   6. Applied rewrites 49.0%

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

   if 0.999999980000000011 < (/.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 86.9%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 93.9

         \[\leadsto \sin th \]

   4. Applied rewrites 93.9%

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

Alternative 8: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.12:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.12)
   (* (/ (sin ky) (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.12) {
		tmp = (sin(ky) / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.12d0) then
        tmp = (sin(ky) / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.12) {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.12:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.12)
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.12)
		tmp = (sin(ky) / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.12], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.12

   1. Initial program 95.5%

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

   2. Taylor expanded in ky around 0

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

      1. lift-sin.f64 35.2

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

   4. Applied rewrites 35.2%

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

   if 0.12 < (/.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 91.5%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 66.9

         \[\leadsto \sin th \]

   4. Applied rewrites 66.9%

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

Alternative 9: 44.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 0.04:\\ \;\;\;\;\frac{ky}{\sin kx} \cdot \sin th\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 0.04)
   (* (/ ky (sin kx)) (sin th))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 0.04) {
		tmp = (ky / sin(kx)) * sin(th);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 0.04d0) then
        tmp = (ky / sin(kx)) * sin(th)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 0.04) {
		tmp = (ky / Math.sin(kx)) * Math.sin(th);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 0.04:
		tmp = (ky / math.sin(kx)) * math.sin(th)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.04)
		tmp = Float64(Float64(ky / sin(kx)) * sin(th));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 0.04)
		tmp = (ky / sin(kx)) * sin(th);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.04], N[(N[(ky / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 0.0400000000000000008

   1. Initial program 95.5%

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

   2. Taylor expanded in ky around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 33.4

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

   4. Applied rewrites 33.4%

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

   if 0.0400000000000000008 < (/.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 91.6%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 66.3

         \[\leadsto \sin th \]

   4. Applied rewrites 66.3%

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

Alternative 10: 43.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sin ky}{\sqrt{{\sin kx}^{2} + {\sin ky}^{2}}} \leq 10^{-6}:\\ \;\;\;\;\frac{\sin th \cdot ky}{\sin kx}\\ \mathbf{else}:\\ \;\;\;\;\sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (/ (sin ky) (sqrt (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0)))) 1e-6)
   (/ (* (sin th) ky) (sin kx))
   (sin th)))

C

double code(double kx, double ky, double th) {
	double tmp;
	if ((sin(ky) / sqrt((pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))) <= 1e-6) {
		tmp = (sin(th) * ky) / sin(kx);
	} else {
		tmp = sin(th);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(kx, ky, th)
use fmin_fmax_functions
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8), intent (in) :: th
    real(8) :: tmp
    if ((sin(ky) / sqrt(((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))) <= 1d-6) then
        tmp = (sin(th) * ky) / sin(kx)
    else
        tmp = sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if ((Math.sin(ky) / Math.sqrt((Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))) <= 1e-6) {
		tmp = (Math.sin(th) * ky) / Math.sin(kx);
	} else {
		tmp = Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if (math.sin(ky) / math.sqrt((math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))) <= 1e-6:
		tmp = (math.sin(th) * ky) / math.sin(kx)
	else:
		tmp = math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (Float64(sin(ky) / sqrt(Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-6)
		tmp = Float64(Float64(sin(th) * ky) / sin(kx));
	else
		tmp = sin(th);
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if ((sin(ky) / sqrt(((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))) <= 1e-6)
		tmp = (sin(th) * ky) / sin(kx);
	else
		tmp = sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1e-6], N[(N[(N[Sin[th], $MachinePrecision] * ky), $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision], N[Sin[th], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sin.f64 ky) (sqrt.f64 (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))) < 9.99999999999999955e-7

   1. Initial program 95.5%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \color{blue}{\frac{ky \cdot \sin th}{\sin kx}} \]
   3. 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 32.6

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

   4. Applied rewrites 32.6%

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

   if 9.99999999999999955e-7 < (/.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 91.7%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 65.6

         \[\leadsto \sin th \]

   4. Applied rewrites 65.6%

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

Alternative 11: 51.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sin kx \leq -0.005:\\ \;\;\;\;\frac{\sin ky}{\sqrt{0.5 - \cos \left(2 \cdot kx\right) \cdot 0.5}} \cdot \sin th\\ \mathbf{elif}\;\sin kx \leq 1.4 \cdot 10^{-89}:\\ \;\;\;\;\sin th\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin ky}{\sin kx} \cdot \sin th\\ \end{array} \end{array} \]

FPCore

(FPCore (kx ky th)
 :precision binary64
 (if (<= (sin kx) -0.005)
   (* (/ (sin ky) (sqrt (- 0.5 (* (cos (* 2.0 kx)) 0.5)))) (sin th))
   (if (<= (sin kx) 1.4e-89) (sin th) (* (/ (sin ky) (sin kx)) (sin th)))))

C

double code(double kx, double ky, double th) {
	double tmp;
	if (sin(kx) <= -0.005) {
		tmp = (sin(ky) / sqrt((0.5 - (cos((2.0 * kx)) * 0.5)))) * sin(th);
	} else if (sin(kx) <= 1.4e-89) {
		tmp = sin(th);
	} else {
		tmp = (sin(ky) / sin(kx)) * 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(kx) <= (-0.005d0)) then
        tmp = (sin(ky) / sqrt((0.5d0 - (cos((2.0d0 * kx)) * 0.5d0)))) * sin(th)
    else if (sin(kx) <= 1.4d-89) then
        tmp = sin(th)
    else
        tmp = (sin(ky) / sin(kx)) * sin(th)
    end if
    code = tmp
end function

Java

public static double code(double kx, double ky, double th) {
	double tmp;
	if (Math.sin(kx) <= -0.005) {
		tmp = (Math.sin(ky) / Math.sqrt((0.5 - (Math.cos((2.0 * kx)) * 0.5)))) * Math.sin(th);
	} else if (Math.sin(kx) <= 1.4e-89) {
		tmp = Math.sin(th);
	} else {
		tmp = (Math.sin(ky) / Math.sin(kx)) * Math.sin(th);
	}
	return tmp;
}

Python

def code(kx, ky, th):
	tmp = 0
	if math.sin(kx) <= -0.005:
		tmp = (math.sin(ky) / math.sqrt((0.5 - (math.cos((2.0 * kx)) * 0.5)))) * math.sin(th)
	elif math.sin(kx) <= 1.4e-89:
		tmp = math.sin(th)
	else:
		tmp = (math.sin(ky) / math.sin(kx)) * math.sin(th)
	return tmp

Julia

function code(kx, ky, th)
	tmp = 0.0
	if (sin(kx) <= -0.005)
		tmp = Float64(Float64(sin(ky) / sqrt(Float64(0.5 - Float64(cos(Float64(2.0 * kx)) * 0.5)))) * sin(th));
	elseif (sin(kx) <= 1.4e-89)
		tmp = sin(th);
	else
		tmp = Float64(Float64(sin(ky) / sin(kx)) * sin(th));
	end
	return tmp
end

MATLAB

function tmp_2 = code(kx, ky, th)
	tmp = 0.0;
	if (sin(kx) <= -0.005)
		tmp = (sin(ky) / sqrt((0.5 - (cos((2.0 * kx)) * 0.5)))) * sin(th);
	elseif (sin(kx) <= 1.4e-89)
		tmp = sin(th);
	else
		tmp = (sin(ky) / sin(kx)) * sin(th);
	end
	tmp_2 = tmp;
end

Wolfram

code[kx_, ky_, th_] := If[LessEqual[N[Sin[kx], $MachinePrecision], -0.005], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sqrt[N[(0.5 - N[(N[Cos[N[(2.0 * kx), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Sin[kx], $MachinePrecision], 1.4e-89], N[Sin[th], $MachinePrecision], N[(N[(N[Sin[ky], $MachinePrecision] / N[Sin[kx], $MachinePrecision]), $MachinePrecision] * N[Sin[th], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 kx) < -0.0050000000000000001

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0

      \[\leadsto \frac{\sin ky}{\sqrt{\color{blue}{{\sin kx}^{2}}}} \cdot \sin th \]
   3. 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 60.2

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

   4. Applied rewrites 60.2%

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

      1. lift-pow.f64 N/A

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

      2. pow2 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. sqr-sin-a N/A

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

      6. lower--.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. count-2-rev N/A

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

      10. lower-cos.f64 N/A

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

      11. count-2-rev N/A

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

      12. lower-*.f64 59.9

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

   6. Applied rewrites 59.9%

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

   if -0.0050000000000000001 < (sin.f64 kx) < 1.3999999999999999e-89

   1. Initial program 87.1%

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

   2. Taylor expanded in kx around 0

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

      1. lift-sin.f64 41.4

         \[\leadsto \sin th \]

   4. Applied rewrites 41.4%

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

   if 1.3999999999999999e-89 < (sin.f64 kx)

   1. Initial program 99.5%

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

   2. Taylor expanded in ky around 0

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

      1. lift-sin.f64 56.4

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

   4. Applied rewrites 56.4%

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

Alternative 12: 23.9% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[kx_, ky_, th_] := N[Sin[th], $MachinePrecision]

Derivation

1. Initial program 94.2%

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

2. Taylor expanded in kx around 0

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

   1. lift-sin.f64 23.9

      \[\leadsto \sin th \]

4. Applied rewrites 23.9%

   \[\leadsto \color{blue}{\sin th} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025104 
(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)))