Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 13.0s

Alternatives: 10

Speedup: 2.0×

• 42.5% 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, 1.2× speedup

Math

\[\begin{array}{l} \\ \left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot {2}^{-0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (fma a2 a2 (* a1 a1))) (pow 2.0 -0.5)))

C

double code(double a1, double a2, double th) {
	return (cos(th) * fma(a2, a2, (a1 * a1))) * pow(2.0, -0.5);
}

Julia

function code(a1, a2, th)
	return Float64(Float64(cos(th) * fma(a2, a2, Float64(a1 * a1))) * (2.0 ^ -0.5))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Final simplification 99.7%

   \[\leadsto \left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot {2}^{-0.5} \]
6. Add Preprocessing

Alternative 2: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ t_2 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_2 \cdot \left(a2 \cdot a2\right) + t\_2 \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\left(\left(t\_1 \cdot th\right) \cdot th\right) \cdot \left(\sqrt{0.5} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (fma a1 a1 (* a2 a2))) (t_2 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_2 (* a2 a2)) (* t_2 (* a1 a1))) -2e-80)
     (* (* (* t_1 th) th) (* (sqrt 0.5) -0.5))
     (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = fma(a1, a1, (a2 * a2));
	double t_2 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_2 * (a2 * a2)) + (t_2 * (a1 * a1))) <= -2e-80) {
		tmp = ((t_1 * th) * th) * (sqrt(0.5) * -0.5);
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	return tmp;
}

Julia

function code(a1, a2, th)
	t_1 = fma(a1, a1, Float64(a2 * a2))
	t_2 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(t_2 * Float64(a2 * a2)) + Float64(t_2 * Float64(a1 * a1))) <= -2e-80)
		tmp = Float64(Float64(Float64(t_1 * th) * th) * Float64(sqrt(0.5) * -0.5));
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-80], N[(N[(N[(t$95$1 * th), $MachinePrecision] * th), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $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))) < -1.99999999999999992e-80

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.6

          \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({th}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)\right) + \sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {th}^{2}, 1\right)} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{th \cdot th}, 1\right) \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{th \cdot th}, 1\right) \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \color{blue}{\left(\left({a1}^{2} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \color{blue}{\left(\left({a1}^{2} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\left(\color{blue}{a1 \cdot a1} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      15. lower-sqrt.f64 60.7

          \[\leadsto \mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}}\right) \]

   7. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right)} \]

   8. Taylor expanded in th around inf

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left({th}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

   10. Applied rewrites 60.7%

       \[\leadsto \left(-0.5 \cdot \sqrt{0.5}\right) \cdot \color{blue}{\left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot th\right) \cdot th\right)} \]

   if -1.99999999999999992e-80 < (+.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. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.7

          \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

      7. lower-sqrt.f64 86.1

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot th\right) \cdot th\right) \cdot \left(\sqrt{0.5} \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a2 \cdot a2\right) + t\_1 \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a2 a2)) (* t_1 (* a1 a1))) -2e-80)
     (* (* (fma (* th th) -0.5 1.0) (sqrt 0.5)) (* a2 a2))
     (* (sqrt 0.5) (fma a1 a1 (* a2 a2))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if (((t_1 * (a2 * a2)) + (t_1 * (a1 * a1))) <= -2e-80) {
		tmp = (fma((th * th), -0.5, 1.0) * sqrt(0.5)) * (a2 * a2);
	} else {
		tmp = sqrt(0.5) * fma(a1, a1, (a2 * a2));
	}
	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(a2 * a2)) + Float64(t_1 * Float64(a1 * a1))) <= -2e-80)
		tmp = Float64(Float64(fma(Float64(th * th), -0.5, 1.0) * sqrt(0.5)) * Float64(a2 * a2));
	else
		tmp = Float64(sqrt(0.5) * fma(a1, a1, Float64(a2 * a2)));
	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[(a2 * a2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-80], N[(N[(N[(N[(th * th), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $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))) < -1.99999999999999992e-80

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.6

          \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left({th}^{2} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)\right) + \sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. distribute-lft1-in N/A

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

      3. +-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {th}^{2}, 1\right)} \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{th \cdot th}, 1\right) \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{th \cdot th}, 1\right) \cdot \left(\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \color{blue}{\left(\left({a1}^{2} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \color{blue}{\left(\left({a1}^{2} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right)} \]

      11. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\left(\color{blue}{a1 \cdot a1} + {a2}^{2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}}\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}}\right) \]

      15. lower-sqrt.f64 60.7

          \[\leadsto \mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}}\right) \]

   7. Applied rewrites 60.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, th \cdot th, 1\right) \cdot \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right)} \]

   8. Taylor expanded in a1 around 0

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

   10. Applied rewrites 38.1%

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

   if -1.99999999999999992e-80 < (+.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. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft-out N/A

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

      5. lift-/.f64 N/A

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

      6. clear-num N/A

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

      7. associate-/r/ N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. pow1/2 N/A

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

      12. pow-flip N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 99.7

          \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

   5. Taylor expanded in th around 0

      \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

      7. lower-sqrt.f64 86.1

         \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

   7. Applied rewrites 86.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\left(\mathsf{fma}\left(th \cdot th, -0.5, 1\right) \cdot \sqrt{0.5}\right) \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. div-inv N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. associate-*l/ N/A

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

   11. *-lft-identity N/A

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

   12. lower-/.f64 N/A

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 5: 99.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \cos th \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (sqrt 0.5) (fma a1 a1 (* a2 a2))) (cos th)))

C

double code(double a1, double a2, double th) {
	return (sqrt(0.5) * fma(a1, a1, (a2 * a2))) * cos(th);
}

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(0.5) * fma(a1, a1, Float64(a2 * a2))) * cos(th))
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{{a1}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right) + {a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-cos.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \cdot \color{blue}{\cos th} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \cdot \cos th} \]

8. Final simplification 99.6%

   \[\leadsto \left(\sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \cos th \]
9. Add Preprocessing

Alternative 6: 58.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{0.5} \cdot a2\right) \cdot a2\right) \cdot \cos th \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* (* (sqrt 0.5) a2) a2) (cos th)))

C

double code(double a1, double a2, double th) {
	return ((sqrt(0.5) * a2) * a2) * cos(th);
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((sqrt(0.5d0) * a2) * a2) * cos(th)
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.sqrt(0.5) * a2) * a2) * Math.cos(th);
}

Python

def code(a1, a2, th):
	return ((math.sqrt(0.5) * a2) * a2) * math.cos(th)

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(sqrt(0.5) * a2) * a2) * cos(th))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((sqrt(0.5) * a2) * a2) * cos(th);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision] * a2), $MachinePrecision] * N[Cos[th], $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in a1 around 0

   \[\leadsto \color{blue}{{a1}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right) + {a2}^{2} \cdot \left(\cos th \cdot \sqrt{\frac{1}{2}}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-out N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-cos.f64 99.6

       \[\leadsto \left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \cdot \color{blue}{\cos th} \]

7. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}\right) \cdot \cos th} \]

8. Taylor expanded in a1 around 0

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

10. Applied rewrites 59.7%

    \[\leadsto \left(\left(\sqrt{0.5} \cdot a2\right) \cdot a2\right) \cdot \cos \color{blue}{th} \]
11. Add Preprocessing

Alternative 7: 66.3% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (sqrt 0.5) (fma a1 a1 (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return sqrt(0.5) * fma(a1, a1, (a2 * a2));
}

Julia

function code(a1, a2, th)
	return Float64(sqrt(0.5) * fma(a1, a1, Float64(a2 * a2)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   7. lower-sqrt.f64 70.0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

7. Applied rewrites 70.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]

8. Final simplification 70.0%

   \[\leadsto \sqrt{0.5} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \]
9. Add Preprocessing

Alternative 8: 39.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \left(a2 \cdot a2\right) \cdot \sqrt{0.5} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* a2 a2) (sqrt 0.5)))

C

double code(double a1, double a2, double th) {
	return (a2 * a2) * sqrt(0.5);
}

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(0.5d0)
end function

Java

public static double code(double a1, double a2, double th) {
	return (a2 * a2) * Math.sqrt(0.5);
}

Python

def code(a1, a2, th):
	return (a2 * a2) * math.sqrt(0.5)

Julia

function code(a1, a2, th)
	return Float64(Float64(a2 * a2) * sqrt(0.5))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (a2 * a2) * sqrt(0.5);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] * N[Sqrt[0.5], $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   7. lower-sqrt.f64 70.0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

7. Applied rewrites 70.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]

8. Taylor expanded in a1 around 0

   \[\leadsto {a2}^{2} \cdot \sqrt{\color{blue}{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 45.4%

    \[\leadsto \left(a2 \cdot a2\right) \cdot \sqrt{\color{blue}{0.5}} \]
11. Add Preprocessing

Alternative 9: 39.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{0.5} \cdot a2\right) \cdot a2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* (sqrt 0.5) a2) a2))

C

double code(double a1, double a2, double th) {
	return (sqrt(0.5) * 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
    code = (sqrt(0.5d0) * a2) * a2
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.sqrt(0.5) * a2) * a2;
}

Python

def code(a1, a2, th):
	return (math.sqrt(0.5) * a2) * a2

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(0.5) * a2) * a2)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (sqrt(0.5) * a2) * a2;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * a2), $MachinePrecision] * a2), $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   7. lower-sqrt.f64 70.0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

7. Applied rewrites 70.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]

8. Taylor expanded in a1 around 0

   \[\leadsto {a2}^{2} \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 45.4%

    \[\leadsto \left(\sqrt{0.5} \cdot a2\right) \cdot \color{blue}{a2} \]
11. Add Preprocessing

Alternative 10: 40.1% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt{0.5} \cdot a1\right) \cdot a1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (* (sqrt 0.5) a1) a1))

C

double code(double a1, double a2, double th) {
	return (sqrt(0.5) * a1) * a1;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (sqrt(0.5d0) * a1) * a1
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.sqrt(0.5) * a1) * a1;
}

Python

def code(a1, a2, th):
	return (math.sqrt(0.5) * a1) * a1

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(0.5) * a1) * a1)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (sqrt(0.5) * a1) * a1;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * a1), $MachinePrecision] * a1), $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. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-lft-out N/A

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

   5. lift-/.f64 N/A

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

   6. clear-num N/A

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

   7. associate-/r/ N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-sqrt.f64 N/A

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

   11. pow1/2 N/A

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

   12. pow-flip N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto {2}^{-0.5} \cdot \left(\color{blue}{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)} \cdot \cos th\right) \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{{2}^{-0.5} \cdot \left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \cos th\right)} \]

5. Taylor expanded in th around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{2}} \cdot \left({a1}^{2} + {a2}^{2}\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, {a2}^{2}\right)} \cdot \sqrt{\frac{1}{2}} \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a1, a1, \color{blue}{a2 \cdot a2}\right) \cdot \sqrt{\frac{1}{2}} \]

   7. lower-sqrt.f64 70.0

      \[\leadsto \mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \color{blue}{\sqrt{0.5}} \]

7. Applied rewrites 70.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right) \cdot \sqrt{0.5}} \]

8. Taylor expanded in a1 around inf

   \[\leadsto {a1}^{2} \cdot \color{blue}{\sqrt{\frac{1}{2}}} \]
9. Step-by-step derivation

10. Applied rewrites 38.6%

    \[\leadsto \left(\sqrt{0.5} \cdot a1\right) \cdot \color{blue}{a1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024242 
(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))))