Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 9.8s

Alternatives: 16

Speedup: 1.9×

• 43.6% 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: 99.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (- (cos th)) -1.0) (/ (fma a2 a2 (* a1 a1)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	return (-cos(th) / -1.0) * (fma(a2, a2, (a1 * a1)) / sqrt(2.0));
}

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(-cos(th)) / -1.0) * Float64(fma(a2, a2, Float64(a1 * a1)) / sqrt(2.0)))
end

Wolfram

code[a1_, a2_, th_] := N[(N[((-N[Cos[th], $MachinePrecision]) / -1.0), $MachinePrecision] * N[(N[(a2 * a2 + N[(a1 * a1), $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. 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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 74.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \leq -5 \cdot 10^{-105}:\\ \;\;\;\;\left(\left(-0.5 \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\right) \cdot th\right) \cdot th\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(a1, a1, a2 \cdot a2\right)}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2))) -5e-105)
     (* (* (* -0.5 (* a2 (/ a2 (sqrt 2.0)))) th) th)
     (/ (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)) + (t_1 * (a2 * a2))) <= -5e-105) {
		tmp = ((-0.5 * (a2 * (a2 / sqrt(2.0)))) * th) * th;
	} 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(t_1 * Float64(a2 * a2))) <= -5e-105)
		tmp = Float64(Float64(Float64(-0.5 * Float64(a2 * Float64(a2 / sqrt(2.0)))) * th) * th);
	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[(t$95$1 * N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-105], N[(N[(N[(-0.5 * N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * th), $MachinePrecision] * th), $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))) < -4.99999999999999963e-105

   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. *-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in th around 0

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

   8. Applied rewrites 6.8%

      \[\leadsto \frac{\mathsf{fma}\left(\left(-0.5 \cdot a2\right) \cdot a2, th \cdot th, a2 \cdot a2\right)}{\color{blue}{\sqrt{2}}} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.0%

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

   11. Taylor expanded in th around inf

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

   13. Applied rewrites 30.6%

       \[\leadsto \left(\left(-0.5 \cdot \left(a2 \cdot \frac{a2}{\sqrt{2}}\right)\right) \cdot th\right) \cdot th \]

   if -4.99999999999999963e-105 < (+.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 99.6%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. div-add-rev N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sqrt.f64 80.8

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

   5. Applied rewrites 80.8%

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

   7. Applied rewrites 80.8%

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

Alternative 3: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   3. lift-*.f64 N/A

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

   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. associate-*l/ N/A

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

   7. distribute-lft-in N/A

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

   8. lower-/.f64 N/A

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

   9. distribute-lft-in N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 4: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   6. lower-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lower-fma.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 5: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{-\cos th}{-1} \cdot \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. metadata-eval N/A

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

   10. /-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \cos th\right) \cdot 0.5} \]
9. Add Preprocessing

Alternative 6: 99.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{-\cos th}{-1} \cdot \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. metadata-eval N/A

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

   10. /-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \cos th\right) \cdot 0.5} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.6

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

   8. lift-*.f64 N/A

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

   9. lift-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-*.f64 99.6

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

10. Applied rewrites 99.6%

    \[\leadsto \color{blue}{\left(\left(\cos th \cdot \mathsf{fma}\left(a2, a2, a1 \cdot a1\right)\right) \cdot \sqrt{2}\right)} \cdot 0.5 \]
11. Add Preprocessing

Alternative 7: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return ((a2 / sqrt(2.0)) * cos(th)) * 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 = ((a2 / sqrt(2.0d0)) * cos(th)) * a2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $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. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 58.6

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

5. Applied rewrites 58.6%

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

7. Applied rewrites 58.7%

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

Alternative 8: 57.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return (cos(th) * 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 = (cos(th) * a2) * (a2 / sqrt(2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] * 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 a1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 58.6

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

5. Applied rewrites 58.6%

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

Alternative 9: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{2} \cdot \left(a2 \cdot a2\right)\right) \cdot \cos th\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (* (sqrt 2.0) (* a2 a2)) (cos th)) 0.5))

C

double code(double a1, double a2, double th) {
	return ((sqrt(2.0) * (a2 * a2)) * cos(th)) * 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 = ((sqrt(2.0d0) * (a2 * a2)) * cos(th)) * 0.5d0
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.sqrt(2.0) * (a2 * a2)) * Math.cos(th)) * 0.5;
}

Python

def code(a1, a2, th):
	return ((math.sqrt(2.0) * (a2 * a2)) * math.cos(th)) * 0.5

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(sqrt(2.0) * Float64(a2 * a2)) * cos(th)) * 0.5)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((sqrt(2.0) * (a2 * a2)) * cos(th)) * 0.5;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{-\cos th}{-1} \cdot \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. metadata-eval N/A

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

   10. /-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in a1 around 0

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

    1. unpow2 N/A

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

    2. lower-*.f64 58.7

       \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \cos th\right) \cdot 0.5 \]

11. Applied rewrites 58.7%

    \[\leadsto \left(\left(\sqrt{2} \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot \cos th\right) \cdot 0.5 \]
12. Add Preprocessing

Alternative 10: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right)\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (* (sqrt 2.0) (cos th)) (* a2 a2)) 0.5))

C

double code(double a1, double a2, double th) {
	return ((sqrt(2.0) * cos(th)) * (a2 * a2)) * 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 = ((sqrt(2.0d0) * cos(th)) * (a2 * a2)) * 0.5d0
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.sqrt(2.0) * Math.cos(th)) * (a2 * a2)) * 0.5;
}

Python

def code(a1, a2, th):
	return ((math.sqrt(2.0) * math.cos(th)) * (a2 * a2)) * 0.5

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(sqrt(2.0) * cos(th)) * Float64(a2 * a2)) * 0.5)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((sqrt(2.0) * cos(th)) * (a2 * a2)) * 0.5;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{-\cos th}{-1} \cdot \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. metadata-eval N/A

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

   10. /-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in a1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-sqrt.f64 N/A

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

    6. lower-cos.f64 N/A

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

    7. unpow2 N/A

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

    8. lower-*.f64 58.6

       \[\leadsto \left(\left(\sqrt{2} \cdot \cos th\right) \cdot \color{blue}{\left(a2 \cdot a2\right)}\right) \cdot 0.5 \]

11. Applied rewrites 58.6%

    \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \cos th\right) \cdot \left(a2 \cdot a2\right)\right)} \cdot 0.5 \]
12. Add Preprocessing

Alternative 11: 57.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot \cos th\right) \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot a2\right) \end{array} \]

FPCore

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

C

double code(double a1, double a2, double th) {
	return (0.5 * cos(th)) * ((sqrt(2.0) * 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 = (0.5d0 * cos(th)) * ((sqrt(2.0d0) * a2) * a2)
end function

Java

public static double code(double a1, double a2, double th) {
	return (0.5 * Math.cos(th)) * ((Math.sqrt(2.0) * a2) * a2);
}

Python

def code(a1, a2, th):
	return (0.5 * math.cos(th)) * ((math.sqrt(2.0) * a2) * a2)

Julia

function code(a1, a2, th)
	return Float64(Float64(0.5 * cos(th)) * Float64(Float64(sqrt(2.0) * a2) * a2))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (0.5 * cos(th)) * ((sqrt(2.0) * a2) * a2);
end

Wolfram

code[a1_, a2_, th_] := N[(N[(0.5 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] * a2), $MachinePrecision] * a2), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

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

7. Taylor expanded in a1 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. associate-*r* N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. *-commutative N/A

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

   10. unpow2 N/A

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

   11. associate-*r* N/A

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

   12. lower-*.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 58.6

       \[\leadsto \left(0.5 \cdot \cos th\right) \cdot \left(\left(\color{blue}{\sqrt{2}} \cdot a2\right) \cdot a2\right) \]

9. Applied rewrites 58.6%

   \[\leadsto \color{blue}{\left(0.5 \cdot \cos th\right) \cdot \left(\left(\sqrt{2} \cdot a2\right) \cdot a2\right)} \]
10. Add Preprocessing

Alternative 12: 65.9% 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. 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. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 63.5

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

5. Applied rewrites 63.5%

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

7. Applied rewrites 63.5%

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

Alternative 13: 65.9% accurate, 8.3× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (fma a2 a2 (* a1 a1)) (sqrt 2.0)) 0.5))

C

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

Julia

function code(a1, a2, th)
	return Float64(Float64(fma(a2, a2, Float64(a1 * a1)) * sqrt(2.0)) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2 + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $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. frac-2neg N/A

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

   7. associate-*l/ N/A

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

   8. neg-mul-1 N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-neg.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-*.f64 N/A

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

   16. lower-fma.f64 99.7

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{-\cos th}{-1} \cdot \frac{\mathsf{fma}\left(a2, a2, a1 \cdot a1\right)}{\sqrt{2}}} \]
5. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. div-add N/A

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

   4. frac-add N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-/.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 99.6

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

6. Applied rewrites 99.6%

   \[\leadsto \frac{-\cos th}{-1} \cdot \color{blue}{\frac{\mathsf{fma}\left(a2 \cdot a2, \sqrt{2}, \sqrt{2} \cdot \left(a1 \cdot a1\right)\right)}{2}} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. div-inv N/A

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

   5. lift-/.f64 N/A

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

   6. lift-neg.f64 N/A

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

   7. distribute-frac-neg N/A

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

   8. distribute-neg-frac2 N/A

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

   9. metadata-eval N/A

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

   10. /-rgt-identity N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in th around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. unpow2 N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. lower-sqrt.f64 63.5

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

11. Applied rewrites 63.5%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(a2, a2, a1 \cdot a1\right) \cdot \sqrt{2}\right)} \cdot 0.5 \]
12. Add Preprocessing

Alternative 14: 65.9% accurate, 8.3× speedup

Math

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

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (sqrt 2.0) (fma a1 a1 (* a2 a2))) 0.5))

C

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

Julia

function code(a1, a2, th)
	return Float64(Float64(sqrt(2.0) * fma(a1, a1, Float64(a2 * a2))) * 0.5)
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $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. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 63.5

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

5. Applied rewrites 63.5%

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

7. Applied rewrites 62.0%

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

9. Applied rewrites 63.5%

   \[\leadsto \left(\sqrt{2} \cdot \mathsf{fma}\left(a1, a1, a2 \cdot a2\right)\right) \cdot \color{blue}{0.5} \]
10. Add Preprocessing

Alternative 15: 39.4% 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. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 63.5

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

5. Applied rewrites 63.5%

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

6. Taylor expanded in a1 around 0

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

8. Applied rewrites 38.8%

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

Alternative 16: 39.7% 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. div-add-rev N/A

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

   2. lower-/.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sqrt.f64 63.5

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

5. Applied rewrites 63.5%

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

6. Taylor expanded in a1 around inf

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

8. Applied rewrites 41.2%

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

Reproduce

herbie shell --seed 2024305 
(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))))