Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.5s

Alternatives: 7

Speedup: 2.0×

• 43.9% 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, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

      \[\leadsto \left(\frac{1}{\color{blue}{{2}^{0.5}}} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. pow-flip 99.7%

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

   5. metadata-eval 99.7%

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

6. Applied egg-rr 99.7%

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

7. Taylor expanded in th around inf 99.7%

   \[\leadsto \color{blue}{\left(\cos th \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. *-commutative 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

    \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\sqrt{0.5} \cdot \cos th\right) \]
11. Add Preprocessing

Alternative 2: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * 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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 51.8%

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

   1. pow2 51.8%

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

   2. associate-/l* 51.7%

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

   3. *-commutative 51.7%

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

   4. associate-*r* 51.7%

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

   5. div-inv 51.7%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 51.7%

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

   8. frac-times 51.7%

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

   9. metadata-eval 51.7%

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

   10. rem-square-sqrt 51.8%

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

   11. metadata-eval 51.8%

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

7. Applied egg-rr 51.8%

   \[\leadsto \color{blue}{\left(\left(\cos th \cdot \sqrt{0.5}\right) \cdot a2\right) \cdot a2} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 41.6%

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

   2. pow2 41.6%

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

   3. associate-*l* 41.6%

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

   4. *-commutative 41.6%

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

   5. sqrt-prod 39.1%

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

   6. sqrt-prod 18.8%

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

   7. add-sqr-sqrt 39.1%

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

9. Applied egg-rr 39.1%

   \[\leadsto \color{blue}{{\left(\sqrt{\sqrt{0.5} \cdot \cos th} \cdot a2\right)}^{2}} \]
10. Step-by-step derivation

    1. unpow2 39.1%

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

    2. swap-sqr 39.0%

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

    3. add-sqr-sqrt 51.8%

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

    4. associate-*l* 51.8%

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

    5. *-commutative 51.8%

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

    6. associate-*l* 51.8%

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

11. Applied egg-rr 51.8%

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

12. Final simplification 51.8%

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

Alternative 3: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * 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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 51.8%

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

   1. pow2 51.8%

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

   2. associate-/l* 51.7%

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

   3. *-commutative 51.7%

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

   4. associate-*r* 51.7%

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

   5. div-inv 51.7%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 51.7%

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

   8. frac-times 51.7%

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

   9. metadata-eval 51.7%

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

   10. rem-square-sqrt 51.8%

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

   11. metadata-eval 51.8%

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

7. Applied egg-rr 51.8%

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

8. Taylor expanded in th around inf 51.8%

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

   1. *-commutative 51.8%

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

   2. *-commutative 51.8%

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

   3. associate-*r* 51.8%

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

10. Simplified 51.8%

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

11. Final simplification 51.8%

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

Alternative 4: 57.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 51.8%

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

   1. pow2 51.8%

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

   2. associate-/l* 51.7%

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

   3. *-commutative 51.7%

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

   4. associate-*r* 51.7%

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

   5. div-inv 51.7%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 51.7%

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

   8. frac-times 51.7%

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

   9. metadata-eval 51.7%

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

   10. rem-square-sqrt 51.8%

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

   11. metadata-eval 51.8%

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

7. Applied egg-rr 51.8%

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

8. Final simplification 51.8%

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

Alternative 5: 39.5% accurate, 4.0× 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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 68.5%

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

6. Taylor expanded in a1 around inf 42.1%

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

   1. pow2 42.1%

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

   2. associate-/l* 42.1%

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

8. Applied egg-rr 42.1%

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

9. Final simplification 42.1%

   \[\leadsto a1 \cdot \frac{a1}{\sqrt{2}} \]
10. Add Preprocessing

Alternative 6: 39.8% accurate, 4.0× 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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 68.5%

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

6. Taylor expanded in a1 around 0 39.0%

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

   1. pow2 39.0%

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

   2. associate-/l* 39.0%

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

8. Applied egg-rr 39.0%

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

9. Final simplification 39.0%

   \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
10. Add Preprocessing

Alternative 7: 39.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Sqrt[0.5], $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. Step-by-step derivation

   1. distribute-lft-out 99.6%

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

   2. cos-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.6%

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

   5. cos-neg 99.6%

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

   6. +-commutative 99.6%

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

   7. fma-define 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in a2 around inf 51.8%

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

   1. pow2 51.8%

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

   2. associate-/l* 51.7%

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

   3. *-commutative 51.7%

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

   4. associate-*r* 51.7%

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

   5. div-inv 51.7%

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

   6. add-sqr-sqrt 51.7%

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

   7. sqrt-unprod 51.7%

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

   8. frac-times 51.7%

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

   9. metadata-eval 51.7%

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

   10. rem-square-sqrt 51.8%

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

   11. metadata-eval 51.8%

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

7. Applied egg-rr 51.8%

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

8. Taylor expanded in th around 0 39.1%

   \[\leadsto \color{blue}{\left(a2 \cdot \sqrt{0.5}\right)} \cdot a2 \]
9. Step-by-step derivation

   1. *-commutative 39.1%

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

10. Simplified 39.1%

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

11. Final simplification 39.1%

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

Reproduce

herbie shell --seed 2024100 
(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))))