Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.8s

Alternatives: 10

Speedup: 1.9×

• 43.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    t_1 = cos(th) / sqrt(2.0d0)
    code = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	return (t_1 * (a1 * a1)) + (t_1 * (a2 * a2))

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(t_1 * Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = (t_1 * (a1 * a1)) + (t_1 * (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (/ (fma a1 a1 (* a2 a2)) (/ (sqrt 2.0) (cos th))))

C

double code(double a1, double a2, double th) {
	return fma(a1, a1, (a2 * a2)) / (sqrt(2.0) / cos(th));
}

Julia

function code(a1, a2, th)
	return Float64(fma(a1, a1, Float64(a2 * a2)) / Float64(sqrt(2.0) / cos(th)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}} \]

   3. clear-num N/A

      \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{2}}{\cos th}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)}{\frac{\sqrt{2}}{\cos th}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\frac{\sqrt{2}}{\cos th}}} \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\color{blue}{\sqrt{2}}}{\cos th}} \]

   10. cos-lowering-cos.f64 99.7

       \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\color{blue}{\cos th}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}}} \]
5. Add Preprocessing

Alternative 2: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot t\_1 \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* (* a2 a2) t_1)) -1e-123)
     (/ (* (* a2 a2) (fma th (* th -0.5) 1.0)) (sqrt 2.0))
     (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a1 * a1)) + ((a2 * a2) * t_1)) <= -1e-123) {
		tmp = ((a2 * a2) * fma(th, (th * -0.5), 1.0)) / sqrt(2.0);
	} else {
		tmp = fma(a1, a1, (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_1 * Float64(a1 * a1)) + Float64(Float64(a2 * a2) * t_1)) <= -1e-123)
		tmp = Float64(Float64(Float64(a2 * a2) * fma(th, Float64(th * -0.5), 1.0)) / sqrt(2.0));
	else
		tmp = Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -1e-123], N[(N[(N[(a2 * a2), $MachinePrecision] * N[(th * N[(th * -0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -1.0000000000000001e-123

   1. Initial program 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing

   3. Taylor expanded in a1 around 0

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \cos th}}{\sqrt{2}} \]

      3. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\cos th}}{\sqrt{2}} \]

      6. sqrt-lowering-sqrt.f64 58.5

         \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\color{blue}{\sqrt{2}}} \]

   5. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}} \]

   6. Taylor expanded in th around 0

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \left({a2}^{2} \cdot {th}^{2}\right) + {a2}^{2}}}{\sqrt{2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({a2}^{2} \cdot {th}^{2}\right) \cdot \frac{-1}{2}} + {a2}^{2}}{\sqrt{2}} \]

      2. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \left({th}^{2} \cdot \frac{-1}{2}\right)} + {a2}^{2}}{\sqrt{2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{{a2}^{2} \cdot \color{blue}{\left(\frac{-1}{2} \cdot {th}^{2}\right)} + {a2}^{2}}{\sqrt{2}} \]

      4. *-rgt-identity N/A

         \[\leadsto \frac{{a2}^{2} \cdot \left(\frac{-1}{2} \cdot {th}^{2}\right) + \color{blue}{{a2}^{2} \cdot 1}}{\sqrt{2}} \]

      5. distribute-lft-out N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \left(\frac{-1}{2} \cdot {th}^{2} + 1\right)}}{\sqrt{2}} \]

      6. +-commutative N/A

         \[\leadsto \frac{{a2}^{2} \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}}{\sqrt{2}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}}{\sqrt{2}} \]

      8. unpow2 N/A

         \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}{\sqrt{2}} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \left(1 + \frac{-1}{2} \cdot {th}^{2}\right)}{\sqrt{2}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\left(\frac{-1}{2} \cdot {th}^{2} + 1\right)}}{\sqrt{2}} \]

      11. unpow2 N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\left(th \cdot th\right)} + 1\right)}{\sqrt{2}} \]

      12. associate-*r* N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \left(\color{blue}{\left(\frac{-1}{2} \cdot th\right) \cdot th} + 1\right)}{\sqrt{2}} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \left(\color{blue}{th \cdot \left(\frac{-1}{2} \cdot th\right)} + 1\right)}{\sqrt{2}} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\mathsf{fma}\left(th, \frac{-1}{2} \cdot th, 1\right)}}{\sqrt{2}} \]

      15. *-commutative N/A

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, \color{blue}{th \cdot \frac{-1}{2}}, 1\right)}{\sqrt{2}} \]

      16. *-lowering-*.f64 46.4

          \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, \color{blue}{th \cdot -0.5}, 1\right)}{\sqrt{2}} \]

   8. Simplified 46.4%

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}}{\sqrt{2}} \]

   if -1.0000000000000001e-123 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-out N/A

         \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}} \]

      3. clear-num N/A

         \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \]

      4. un-div-inv N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{2}}{\cos th}} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)}{\frac{\sqrt{2}}{\cos th}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\frac{\sqrt{2}}{\cos th}}} \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\color{blue}{\sqrt{2}}}{\cos th}} \]

      10. cos-lowering-cos.f64 99.7

          \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\color{blue}{\cos th}}} \]

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}}} \]

   5. Taylor expanded in th around 0

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f64 87.1

         \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

   7. Simplified 87.1%

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\frac{\left(a2 \cdot a2\right) \cdot \mathsf{fma}\left(th, th \cdot -0.5, 1\right)}{\sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return cos(th) * (fma(a1, a1, (a2 * a2)) / sqrt(2.0));
}

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\cos th \cdot \frac{1}{\sqrt{2}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\cos th \cdot \left(\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right)} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \cdot \cos th} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\right) \cdot \cos th} \]

   6. associate-*l/ N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)}{\sqrt{2}}} \cdot \cos th \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a1 + a2 \cdot a2}}{\sqrt{2}} \cdot \cos th \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}} \cdot \cos th \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\sqrt{2}} \cdot \cos th \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)}{\sqrt{2}} \cdot \cos th \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \cdot \cos th \]

   12. cos-lowering-cos.f64 99.6

       \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \cdot \color{blue}{\cos th} \]

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \cdot \cos th} \]

5. Final simplification 99.6%

   \[\leadsto \cos th \cdot \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \]
6. Add Preprocessing

Alternative 4: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (* a2 (* a2 (cos th))) (sqrt 2.0)))

C

double code(double a1, double a2, double th) {
	return (a2 * (a2 * cos(th))) / sqrt(2.0);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * (a2 * cos(th))) / sqrt(2.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * (a2 * Math.cos(th))) / Math.sqrt(2.0);
}

Python

def code(a1, a2, th):
	return (a2 * (a2 * math.cos(th))) / math.sqrt(2.0)

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * Float64(a2 * cos(th))) / sqrt(2.0))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * (a2 * cos(th))) / sqrt(2.0);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \cos th}}{\sqrt{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\cos th}}{\sqrt{2}} \]

   6. sqrt-lowering-sqrt.f64 58.0

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\color{blue}{\sqrt{2}}} \]

5. Simplified 58.0%

   \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{a2 \cdot \left(a2 \cdot \cos th\right)}}{\sqrt{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot \cos th\right) \cdot a2}}{\sqrt{2}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot \cos th\right) \cdot a2}}{\sqrt{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot \cos th\right)} \cdot a2}{\sqrt{2}} \]

   5. cos-lowering-cos.f64 58.0

      \[\leadsto \frac{\left(a2 \cdot \color{blue}{\cos th}\right) \cdot a2}{\sqrt{2}} \]

7. Applied egg-rr 58.0%

   \[\leadsto \frac{\color{blue}{\left(a2 \cdot \cos th\right) \cdot a2}}{\sqrt{2}} \]

8. Final simplification 58.0%

   \[\leadsto \frac{a2 \cdot \left(a2 \cdot \cos th\right)}{\sqrt{2}} \]
9. Add Preprocessing

Alternative 5: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (* (* a2 a2) (cos th)) (sqrt 2.0)))

C

double code(double a1, double a2, double th) {
	return ((a2 * a2) * cos(th)) / sqrt(2.0);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a2 * a2) * cos(th)) / sqrt(2.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return ((a2 * a2) * Math.cos(th)) / Math.sqrt(2.0);
}

Python

def code(a1, a2, th):
	return ((a2 * a2) * math.cos(th)) / math.sqrt(2.0)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(a2 * a2) * cos(th)) / sqrt(2.0))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((a2 * a2) * cos(th)) / sqrt(2.0);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \cos th}}{\sqrt{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\cos th}}{\sqrt{2}} \]

   6. sqrt-lowering-sqrt.f64 58.0

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\color{blue}{\sqrt{2}}} \]

5. Simplified 58.0%

   \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}} \]
6. Add Preprocessing

Alternative 6: 57.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ (* a2 (cos th)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return a2 * ((a2 * cos(th)) / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * ((a2 * cos(th)) / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * ((a2 * Math.cos(th)) / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return a2 * ((a2 * math.cos(th)) / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(Float64(a2 * cos(th)) / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * ((a2 * cos(th)) / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{a2}^{2} \cdot \cos th}{\sqrt{2}}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{{a2}^{2} \cdot \cos th}}{\sqrt{2}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(a2 \cdot a2\right)} \cdot \cos th}{\sqrt{2}} \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \color{blue}{\cos th}}{\sqrt{2}} \]

   6. sqrt-lowering-sqrt.f64 58.0

      \[\leadsto \frac{\left(a2 \cdot a2\right) \cdot \cos th}{\color{blue}{\sqrt{2}}} \]

5. Simplified 58.0%

   \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \cos th}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{a2 \cdot \left(a2 \cdot \cos th\right)}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto a2 \cdot \color{blue}{\frac{a2 \cdot \cos th}{\sqrt{2}}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto a2 \cdot \frac{\color{blue}{a2 \cdot \cos th}}{\sqrt{2}} \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto a2 \cdot \frac{a2 \cdot \color{blue}{\cos th}}{\sqrt{2}} \]

   7. sqrt-lowering-sqrt.f64 58.0

      \[\leadsto a2 \cdot \frac{a2 \cdot \cos th}{\color{blue}{\sqrt{2}}} \]

7. Applied egg-rr 58.0%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2 \cdot \cos th}{\sqrt{2}}} \]
8. Add Preprocessing

Alternative 7: 66.2% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (fma a1 a1 (* a2 a2)) (sqrt 2.0)))

C

double code(double a1, double a2, double th) {
	return fma(a1, a1, (a2 * a2)) / sqrt(2.0);
}

Julia

function code(a1, a2, th)
	return Float64(fma(a1, a1, Float64(a2 * a2)) / sqrt(2.0))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-lft-out N/A

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}}} \]

   3. clear-num N/A

      \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \color{blue}{\frac{1}{\frac{\sqrt{2}}{\cos th}}} \]

   4. un-div-inv N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a1 \cdot a1 + a2 \cdot a2}{\frac{\sqrt{2}}{\cos th}}} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}}{\frac{\sqrt{2}}{\cos th}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right)}{\frac{\sqrt{2}}{\cos th}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\frac{\sqrt{2}}{\cos th}}} \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\color{blue}{\sqrt{2}}}{\cos th}} \]

   10. cos-lowering-cos.f64 99.7

       \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\color{blue}{\cos th}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\frac{\sqrt{2}}{\cos th}}} \]

5. Taylor expanded in th around 0

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 65.7

      \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]

7. Simplified 65.7%

   \[\leadsto \frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]
8. Add Preprocessing

Alternative 8: 40.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \frac{a2 \cdot a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (/ (* a2 a2) (sqrt 2.0)))

C

double code(double a1, double a2, double th) {
	return (a2 * a2) / sqrt(2.0);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (a2 * a2) / sqrt(2.0d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * a2) / Math.sqrt(2.0);
}

Python

def code(a1, a2, th):
	return (a2 * a2) / math.sqrt(2.0)

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * a2) / sqrt(2.0))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * a2) / sqrt(2.0);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{a1}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   9. sqrt-lowering-sqrt.f64 65.7

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\color{blue}{\sqrt{2}}}\right) \]

5. Simplified 65.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)} \]

6. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto a2 \cdot \color{blue}{\frac{a2}{\sqrt{2}}} \]

   5. sqrt-lowering-sqrt.f64 38.6

      \[\leadsto a2 \cdot \frac{a2}{\color{blue}{\sqrt{2}}} \]

8. Simplified 38.6%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
9. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   4. sqrt-lowering-sqrt.f64 38.6

      \[\leadsto \frac{a2 \cdot a2}{\color{blue}{\sqrt{2}}} \]

10. Applied egg-rr 38.6%

    \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}}} \]
11. Add Preprocessing

Alternative 9: 40.0% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \frac{a2}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (/ a2 (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a2 * (a2 / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return a2 * (a2 / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{a1}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   9. sqrt-lowering-sqrt.f64 65.7

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\color{blue}{\sqrt{2}}}\right) \]

5. Simplified 65.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)} \]

6. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto a2 \cdot \color{blue}{\frac{a2}{\sqrt{2}}} \]

   5. sqrt-lowering-sqrt.f64 38.6

      \[\leadsto a2 \cdot \frac{a2}{\color{blue}{\sqrt{2}}} \]

8. Simplified 38.6%

   \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
9. Add Preprocessing

Alternative 10: 39.8% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ a1 \cdot \frac{a1}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a1 (/ a1 (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return a1 * (a1 / sqrt(2.0));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = a1 * (a1 / sqrt(2.0d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a1 * (a1 / Math.sqrt(2.0));
}

Python

def code(a1, a2, th):
	return a1 * (a1 / math.sqrt(2.0))

Julia

function code(a1, a2, th)
	return Float64(a1 * Float64(a1 / sqrt(2.0)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a1 * (a1 / sqrt(2.0));
end

Wolfram

code[a1_, a2_, th_] := N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. Add Preprocessing

3. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} + \frac{{a2}^{2}}{\sqrt{2}} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{{a2}^{2}}{\sqrt{2}}\right)} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \color{blue}{\frac{a1}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\color{blue}{\sqrt{2}}}, \frac{{a2}^{2}}{\sqrt{2}}\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \color{blue}{\frac{{a2}^{2}}{\sqrt{2}}}\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}}\right) \]

   9. sqrt-lowering-sqrt.f64 65.7

      \[\leadsto \mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\color{blue}{\sqrt{2}}}\right) \]

5. Simplified 65.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, \frac{a1}{\sqrt{2}}, \frac{a2 \cdot a2}{\sqrt{2}}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\sqrt{2}} + a1 \cdot \frac{a1}{\sqrt{2}}} \]

   2. clear-num N/A

      \[\leadsto \frac{a2 \cdot a2}{\sqrt{2}} + a1 \cdot \color{blue}{\frac{1}{\frac{\sqrt{2}}{a1}}} \]

   3. un-div-inv N/A

      \[\leadsto \frac{a2 \cdot a2}{\sqrt{2}} + \color{blue}{\frac{a1}{\frac{\sqrt{2}}{a1}}} \]

   4. frac-add N/A

      \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \frac{\sqrt{2}}{a1} + \sqrt{2} \cdot a1}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(a2 \cdot a2\right) \cdot \frac{\sqrt{2}}{a1} + \sqrt{2} \cdot a1}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}}} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{a2 \cdot a2}, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   8. /-lowering-/.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \color{blue}{\frac{\sqrt{2}}{a1}}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   9. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\color{blue}{\sqrt{2}}}{a1}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \color{blue}{\sqrt{2} \cdot a1}\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   11. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \color{blue}{\sqrt{2}} \cdot a1\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}} \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\color{blue}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}}} \]

   13. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\color{blue}{\sqrt{2}} \cdot \frac{\sqrt{2}}{a1}} \]

   14. /-lowering-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \color{blue}{\frac{\sqrt{2}}{a1}}} \]

   15. sqrt-lowering-sqrt.f64 62.6

       \[\leadsto \frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \frac{\color{blue}{\sqrt{2}}}{a1}} \]

7. Applied egg-rr 62.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \frac{\sqrt{2}}{a1}, \sqrt{2} \cdot a1\right)}{\sqrt{2} \cdot \frac{\sqrt{2}}{a1}}} \]

8. Taylor expanded in a2 around 0

   \[\leadsto \color{blue}{\frac{{a1}^{2}}{\sqrt{2}}} \]
9. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{\color{blue}{a1 \cdot a1}}{\sqrt{2}} \]

   2. associate-/l* N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto a1 \cdot \color{blue}{\frac{a1}{\sqrt{2}}} \]

   5. sqrt-lowering-sqrt.f64 39.5

      \[\leadsto a1 \cdot \frac{a1}{\color{blue}{\sqrt{2}}} \]

10. Simplified 39.5%

    \[\leadsto \color{blue}{a1 \cdot \frac{a1}{\sqrt{2}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(FPCore (a1 a2 th)
  :name "Migdal et al, Equation (64)"
  :precision binary64
  (+ (* (/ (cos th) (sqrt 2.0)) (* a1 a1)) (* (/ (cos th) (sqrt 2.0)) (* a2 a2))))