Toniolo and Linder, Equation (3a)

Percentage Accurate: 98.4% → 100.0%

Time: 12.4s

Alternatives: 8

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 98.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (*
   (/ 1.0 2.0)
   (+
    1.0
    (/
     1.0
     (sqrt
      (+
       1.0
       (*
        (pow (/ (* 2.0 l) Om) 2.0)
        (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + (pow(((2.0 * l) / Om), 2.0) * (pow(sin(kx), 2.0) + pow(sin(ky), 2.0)))))))));
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(((1.0d0 / 2.0d0) * (1.0d0 + (1.0d0 / sqrt((1.0d0 + ((((2.0d0 * l) / om) ** 2.0d0) * ((sin(kx) ** 2.0d0) + (sin(ky) ** 2.0d0)))))))))
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / Math.sqrt((1.0 + (Math.pow(((2.0 * l) / Om), 2.0) * (Math.pow(Math.sin(kx), 2.0) + Math.pow(Math.sin(ky), 2.0)))))))));
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(((1.0 / 2.0) * (1.0 + (1.0 / math.sqrt((1.0 + (math.pow(((2.0 * l) / Om), 2.0) * (math.pow(math.sin(kx), 2.0) + math.pow(math.sin(ky), 2.0)))))))))

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(1.0 / 2.0) * Float64(1.0 + Float64(1.0 / sqrt(Float64(1.0 + Float64((Float64(Float64(2.0 * l) / Om) ^ 2.0) * Float64((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))))
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(((1.0 / 2.0) * (1.0 + (1.0 / sqrt((1.0 + ((((2.0 * l) / Om) ^ 2.0) * ((sin(kx) ^ 2.0) + (sin(ky) ^ 2.0)))))))));
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(1.0 / 2.0), $MachinePrecision] * N[(1.0 + N[(1.0 / N[Sqrt[N[(1.0 + N[(N[Power[N[(N[(2.0 * l), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\ell \cdot \sin ky\right) \cdot \frac{2}{Om}\\ \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(t\_0, t\_0, {\left(\left(\sin kx \cdot \ell\right) \cdot \frac{2}{Om}\right)}^{2} + 1\right)}} + 1\right) \cdot \frac{1}{2}} \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (let* ((t_0 (* (* l (sin ky)) (/ 2.0 Om))))
   (sqrt
    (*
     (+
      (/
       1.0
       (sqrt (fma t_0 t_0 (+ (pow (* (* (sin kx) l) (/ 2.0 Om)) 2.0) 1.0))))
      1.0)
     (/ 1.0 2.0)))))

C

double code(double l, double Om, double kx, double ky) {
	double t_0 = (l * sin(ky)) * (2.0 / Om);
	return sqrt((((1.0 / sqrt(fma(t_0, t_0, (pow(((sin(kx) * l) * (2.0 / Om)), 2.0) + 1.0)))) + 1.0) * (1.0 / 2.0)));
}

Julia

function code(l, Om, kx, ky)
	t_0 = Float64(Float64(l * sin(ky)) * Float64(2.0 / Om))
	return sqrt(Float64(Float64(Float64(1.0 / sqrt(fma(t_0, t_0, Float64((Float64(Float64(sin(kx) * l) * Float64(2.0 / Om)) ^ 2.0) + 1.0)))) + 1.0) * Float64(1.0 / 2.0)))
end

Wolfram

code[l_, Om_, kx_, ky_] := Block[{t$95$0 = N[(N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision]}, N[Sqrt[N[(N[(N[(1.0 / N[Sqrt[N[(t$95$0 * t$95$0 + N[(N[Power[N[(N[(N[Sin[kx], $MachinePrecision] * l), $MachinePrecision] * N[(2.0 / Om), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.2%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. +-commutative N/A

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

   6. distribute-lft-in N/A

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

   7. *-commutative N/A

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

   8. associate-+l+ N/A

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

   9. lift-pow.f64 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow-prod-down N/A

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

   12. pow2 N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \sqrt{\left(\frac{1}{\sqrt{\mathsf{fma}\left(\left(\ell \cdot \sin ky\right) \cdot \frac{2}{Om}, \left(\ell \cdot \sin ky\right) \cdot \frac{2}{Om}, {\left(\left(\sin kx \cdot \ell\right) \cdot \frac{2}{Om}\right)}^{2} + 1\right)}} + 1\right) \cdot \frac{1}{2}} \]
6. Add Preprocessing

Alternative 2: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.9998:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right)}{Om} \cdot \left(ky \cdot ky\right)\right), 4, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (/
       1.0
       (sqrt
        (+
         (*
          (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
          (pow (/ (* l 2.0) Om) 2.0))
         1.0)))
      0.9998)
   (sqrt
    (fma
     (sqrt
      (/
       1.0
       (fma
        (*
         (* (/ l Om) l)
         (*
          (/
           (fma
            (fma 0.044444444444444446 (* ky ky) -0.3333333333333333)
            (* ky ky)
            1.0)
           Om)
          (* ky ky)))
        4.0
        1.0)))
     0.5
     0.5))
   (sqrt 1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.9998) {
		tmp = sqrt(fma(sqrt((1.0 / fma((((l / Om) * l) * ((fma(fma(0.044444444444444446, (ky * ky), -0.3333333333333333), (ky * ky), 1.0) / Om) * (ky * ky))), 4.0, 1.0))), 0.5, 0.5));
	} else {
		tmp = sqrt(1.0);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.9998)
		tmp = sqrt(fma(sqrt(Float64(1.0 / fma(Float64(Float64(Float64(l / Om) * l) * Float64(Float64(fma(fma(0.044444444444444446, Float64(ky * ky), -0.3333333333333333), Float64(ky * ky), 1.0) / Om) * Float64(ky * ky))), 4.0, 1.0))), 0.5, 0.5));
	else
		tmp = sqrt(1.0);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.9998], N[Sqrt[N[(N[Sqrt[N[(1.0 / N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * N[(N[(N[(N[(0.044444444444444446 * N[(ky * ky), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(ky * ky), $MachinePrecision] + 1.0), $MachinePrecision] / Om), $MachinePrecision] * N[(ky * ky), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 4.0 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))) < 0.99980000000000002

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

   8. Taylor expanded in ky around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left({ky}^{2} \cdot \left({ky}^{2} \cdot \left(\frac{2}{45} \cdot \frac{{ky}^{2}}{Om} - \frac{1}{3} \cdot \frac{1}{Om}\right) + \frac{1}{Om}\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, \frac{ky \cdot ky}{Om}, \frac{-0.3333333333333333}{Om}\right), ky \cdot ky, \frac{1}{Om}\right) \cdot \left(ky \cdot ky\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

   11. Taylor expanded in Om around 0

       \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\frac{1 + {ky}^{2} \cdot \left(\frac{2}{45} \cdot {ky}^{2} - \frac{1}{3}\right)}{Om} \cdot \left(ky \cdot ky\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 73.8%

       \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right)}{Om} \cdot \left(ky \cdot ky\right)\right) \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

   if 0.99980000000000002 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.9998:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.044444444444444446, ky \cdot ky, -0.3333333333333333\right), ky \cdot ky, 1\right)}{Om} \cdot \left(ky \cdot ky\right)\right), 4, 1\right)}}, 0.5, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.9998:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, \frac{ky \cdot ky}{Om}, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (/
       1.0
       (sqrt
        (+
         (*
          (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
          (pow (/ (* l 2.0) Om) 2.0))
         1.0)))
      0.9998)
   (sqrt
    (+ (/ 0.5 (sqrt (fma (* (* (/ l Om) l) 4.0) (/ (* ky ky) Om) 1.0))) 0.5))
   (sqrt 1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.9998) {
		tmp = sqrt(((0.5 / sqrt(fma((((l / Om) * l) * 4.0), ((ky * ky) / Om), 1.0))) + 0.5));
	} else {
		tmp = sqrt(1.0);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.9998)
		tmp = sqrt(Float64(Float64(0.5 / sqrt(fma(Float64(Float64(Float64(l / Om) * l) * 4.0), Float64(Float64(ky * ky) / Om), 1.0))) + 0.5));
	else
		tmp = sqrt(1.0);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.9998], N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(N[(N[(N[(l / Om), $MachinePrecision] * l), $MachinePrecision] * 4.0), $MachinePrecision] * N[(N[(ky * ky), $MachinePrecision] / Om), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))) < 0.99980000000000002

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

   5. Applied rewrites 70.8%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in ky around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, \frac{1}{2}, \frac{1}{2}\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 70.2%

      \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{ky \cdot ky}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 73.4%

       \[\leadsto \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, \frac{ky \cdot ky}{Om}, 1\right)}} + \color{blue}{0.5}} \]

   if 0.99980000000000002 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 99.2%

      \[\leadsto \sqrt{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.9998:\\ \;\;\;\;\sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(\left(\frac{\ell}{Om} \cdot \ell\right) \cdot 4, \frac{ky \cdot ky}{Om}, 1\right)}} + 0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.5:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\ell \cdot ky}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (/
       1.0
       (sqrt
        (+
         (*
          (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
          (pow (/ (* l 2.0) Om) 2.0))
         1.0)))
      0.5)
   (sqrt (fma (/ Om (* l ky)) 0.25 0.5))
   (sqrt 1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.5) {
		tmp = sqrt(fma((Om / (l * ky)), 0.25, 0.5));
	} else {
		tmp = sqrt(1.0);
	}
	return tmp;
}

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.5)
		tmp = sqrt(fma(Float64(Om / Float64(l * ky)), 0.25, 0.5));
	else
		tmp = sqrt(1.0);
	end
	return tmp
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.5], N[Sqrt[N[(N[(Om / N[(l * ky), $MachinePrecision]), $MachinePrecision] * 0.25 + 0.5), $MachinePrecision]], $MachinePrecision], N[Sqrt[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))) < 0.5

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in kx around 0

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

      1. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} + 1\right)}} \]

      2. distribute-rgt-in N/A

         \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

   5. Applied rewrites 71.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]

   6. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\frac{1}{2} + \color{blue}{\frac{1}{4} \cdot \frac{Om}{\ell \cdot \sin ky}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.4%

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{\sin ky \cdot \ell}, \color{blue}{0.25}, 0.5\right)} \]

   9. Taylor expanded in ky around 0

      \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, \frac{1}{4}, \frac{1}{2}\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 80.4%

       \[\leadsto \sqrt{\mathsf{fma}\left(\frac{Om}{ky \cdot \ell}, 0.25, 0.5\right)} \]

   if 0.5 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \sqrt{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.5:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\frac{Om}{\ell \cdot ky}, 0.25, 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.46:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (if (<=
      (/
       1.0
       (sqrt
        (+
         (*
          (+ (pow (sin ky) 2.0) (pow (sin kx) 2.0))
          (pow (/ (* l 2.0) Om) 2.0))
         1.0)))
      0.46)
   (sqrt 0.5)
   (sqrt 1.0)))

C

double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / sqrt((((pow(sin(ky), 2.0) + pow(sin(kx), 2.0)) * pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.46) {
		tmp = sqrt(0.5);
	} else {
		tmp = sqrt(1.0);
	}
	return tmp;
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    real(8) :: tmp
    if ((1.0d0 / sqrt(((((sin(ky) ** 2.0d0) + (sin(kx) ** 2.0d0)) * (((l * 2.0d0) / om) ** 2.0d0)) + 1.0d0))) <= 0.46d0) then
        tmp = sqrt(0.5d0)
    else
        tmp = sqrt(1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	double tmp;
	if ((1.0 / Math.sqrt((((Math.pow(Math.sin(ky), 2.0) + Math.pow(Math.sin(kx), 2.0)) * Math.pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.46) {
		tmp = Math.sqrt(0.5);
	} else {
		tmp = Math.sqrt(1.0);
	}
	return tmp;
}

Python

def code(l, Om, kx, ky):
	tmp = 0
	if (1.0 / math.sqrt((((math.pow(math.sin(ky), 2.0) + math.pow(math.sin(kx), 2.0)) * math.pow(((l * 2.0) / Om), 2.0)) + 1.0))) <= 0.46:
		tmp = math.sqrt(0.5)
	else:
		tmp = math.sqrt(1.0)
	return tmp

Julia

function code(l, Om, kx, ky)
	tmp = 0.0
	if (Float64(1.0 / sqrt(Float64(Float64(Float64((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (Float64(Float64(l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.46)
		tmp = sqrt(0.5);
	else
		tmp = sqrt(1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(l, Om, kx, ky)
	tmp = 0.0;
	if ((1.0 / sqrt(((((sin(ky) ^ 2.0) + (sin(kx) ^ 2.0)) * (((l * 2.0) / Om) ^ 2.0)) + 1.0))) <= 0.46)
		tmp = sqrt(0.5);
	else
		tmp = sqrt(1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[l_, Om_, kx_, ky_] := If[LessEqual[N[(1.0 / N[Sqrt[N[(N[(N[(N[Power[N[Sin[ky], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[kx], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(l * 2.0), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.46], N[Sqrt[0.5], $MachinePrecision], N[Sqrt[1.0], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64))))))) < 0.46000000000000002

   1. Initial program 100.0%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around 0

      \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.7%

      \[\leadsto \sqrt{\color{blue}{0.5}} \]

   if 0.46000000000000002 < (/.f64 #s(literal 1 binary64) (sqrt.f64 (+.f64 #s(literal 1 binary64) (*.f64 (pow.f64 (/.f64 (*.f64 #s(literal 2 binary64) l) Om) #s(literal 2 binary64)) (+.f64 (pow.f64 (sin.f64 kx) #s(literal 2 binary64)) (pow.f64 (sin.f64 ky) #s(literal 2 binary64)))))))

   1. Initial program 98.6%

      \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in Om around inf

      \[\leadsto \sqrt{\color{blue}{1}} \]
   4. Step-by-step derivation

   5. Applied rewrites 98.8%

      \[\leadsto \sqrt{\color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{\left({\sin ky}^{2} + {\sin kx}^{2}\right) \cdot {\left(\frac{\ell \cdot 2}{Om}\right)}^{2} + 1}} \leq 0.46:\\ \;\;\;\;\sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(e^{-0.5 \cdot \mathsf{log1p}\left(\frac{4}{{\left(\frac{Om}{\ell \cdot \sin ky}\right)}^{2}}\right)}, 0.5, 0.5\right)} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt
  (fma
   (exp (* -0.5 (log1p (/ 4.0 (pow (/ Om (* l (sin ky))) 2.0)))))
   0.5
   0.5)))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(fma(exp((-0.5 * log1p((4.0 / pow((Om / (l * sin(ky))), 2.0))))), 0.5, 0.5));
}

Julia

function code(l, Om, kx, ky)
	return sqrt(fma(exp(Float64(-0.5 * log1p(Float64(4.0 / (Float64(Om / Float64(l * sin(ky))) ^ 2.0))))), 0.5, 0.5))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[Exp[N[(-0.5 * N[Log[1 + N[(4.0 / N[Power[N[(Om / N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5 + 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

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

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} + 1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

5. Applied rewrites 81.3%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 80.2%

   \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\left(\ell \cdot \sin ky\right)}^{2}}{Om \cdot Om}, 4, 1\right)}}, 0.5, 0.5\right)} \]

9. Applied rewrites 93.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(e^{\mathsf{log1p}\left(\frac{4}{{\left(\frac{Om}{\ell \cdot \sin ky}\right)}^{2}}\right) \cdot -0.5}, 0.5, 0.5\right)} \]

10. Final simplification 93.0%

    \[\leadsto \sqrt{\mathsf{fma}\left(e^{-0.5 \cdot \mathsf{log1p}\left(\frac{4}{{\left(\frac{Om}{\ell \cdot \sin ky}\right)}^{2}}\right)}, 0.5, 0.5\right)} \]
11. Add Preprocessing

Alternative 7: 93.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{\ell \cdot \sin ky}{Om}\right)}^{2}, 1\right)}} + 0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky)
 :precision binary64
 (sqrt (+ (/ 0.5 (sqrt (fma 4.0 (pow (/ (* l (sin ky)) Om) 2.0) 1.0))) 0.5)))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(((0.5 / sqrt(fma(4.0, pow(((l * sin(ky)) / Om), 2.0), 1.0))) + 0.5));
}

Julia

function code(l, Om, kx, ky)
	return sqrt(Float64(Float64(0.5 / sqrt(fma(4.0, (Float64(Float64(l * sin(ky)) / Om) ^ 2.0), 1.0))) + 0.5))
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[N[(N[(0.5 / N[Sqrt[N[(4.0 * N[Power[N[(N[(l * N[Sin[ky], $MachinePrecision]), $MachinePrecision] / Om), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in kx around 0

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

   1. +-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2} \cdot \color{blue}{\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} + 1\right)}} \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\color{blue}{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2}}} \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}} \cdot \frac{1}{2} + \color{blue}{\frac{1}{2}}} \]

   4. lower-fma.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{1 + 4 \cdot \frac{{\ell}^{2} \cdot {\sin ky}^{2}}{{Om}^{2}}}}, \frac{1}{2}, \frac{1}{2}\right)}} \]

5. Applied rewrites 81.3%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \frac{\ell \cdot \ell}{Om}, 4, 1\right)}}, 0.5, 0.5\right)}} \]
6. Step-by-step derivation

7. Applied rewrites 87.1%

   \[\leadsto \sqrt{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(\frac{{\sin ky}^{2}}{Om} \cdot \left(\frac{\ell}{Om} \cdot \ell\right), 4, 1\right)}}, 0.5, 0.5\right)} \]

9. Applied rewrites 93.0%

   \[\leadsto \sqrt{\frac{0.5}{\sqrt{\mathsf{fma}\left(4, {\left(\frac{\ell \cdot \sin ky}{Om}\right)}^{2}, 1\right)}} + \color{blue}{0.5}} \]
10. Add Preprocessing

Alternative 8: 55.8% accurate, 52.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \end{array} \]

FPCore

(FPCore (l Om kx ky) :precision binary64 (sqrt 0.5))

C

double code(double l, double Om, double kx, double ky) {
	return sqrt(0.5);
}

Fortran

real(8) function code(l, om, kx, ky)
    real(8), intent (in) :: l
    real(8), intent (in) :: om
    real(8), intent (in) :: kx
    real(8), intent (in) :: ky
    code = sqrt(0.5d0)
end function

Java

public static double code(double l, double Om, double kx, double ky) {
	return Math.sqrt(0.5);
}

Python

def code(l, Om, kx, ky):
	return math.sqrt(0.5)

Julia

function code(l, Om, kx, ky)
	return sqrt(0.5)
end

MATLAB

function tmp = code(l, Om, kx, ky)
	tmp = sqrt(0.5);
end

Wolfram

code[l_, Om_, kx_, ky_] := N[Sqrt[0.5], $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\sqrt{\frac{1}{2} \cdot \left(1 + \frac{1}{\sqrt{1 + {\left(\frac{2 \cdot \ell}{Om}\right)}^{2} \cdot \left({\sin kx}^{2} + {\sin ky}^{2}\right)}}\right)} \]
2. Add Preprocessing

3. Taylor expanded in Om around 0

   \[\leadsto \sqrt{\color{blue}{\frac{1}{2}}} \]
4. Step-by-step derivation

5. Applied rewrites 54.3%

   \[\leadsto \sqrt{\color{blue}{0.5}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(FPCore (l Om kx ky)
  :name "Toniolo and Linder, Equation (3a)"
  :precision binary64
  (sqrt (* (/ 1.0 2.0) (+ 1.0 (/ 1.0 (sqrt (+ 1.0 (* (pow (/ (* 2.0 l) Om) 2.0) (+ (pow (sin kx) 2.0) (pow (sin ky) 2.0))))))))))