Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.5s

Alternatives: 5

Speedup: 2.0×

• 43.1% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision] * N[(N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision] / N[Sqrt[2.0], $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 99.5%

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

   2. cos-neg 99.5%

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

   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. Step-by-step derivation

   1. add-sqr-sqrt 99.6%

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

   2. associate-/l* 99.6%

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

   3. fma-undefine 99.6%

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

   4. hypot-define 99.6%

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

   5. fma-undefine 99.6%

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

   6. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. inv-pow 99.5%

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

6. Applied egg-rr 99.5%

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

   1. unpow-1 99.5%

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

   2. associate-/r/ 99.5%

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

8. Simplified 99.5%

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

   1. *-un-lft-identity 61.0%

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

   2. add-sqr-sqrt 61.0%

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

   3. sqrt-unprod 61.0%

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

   4. frac-times 61.0%

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

   5. metadata-eval 61.0%

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

   6. rem-square-sqrt 61.0%

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

   7. metadata-eval 61.0%

      \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

10. Applied egg-rr 99.6%

    \[\leadsto \left(\color{blue}{\left(1 \cdot \sqrt{0.5}\right)} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
11. Step-by-step derivation

    1. *-lft-identity 61.0%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

Alternative 3: 57.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $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 99.5%

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

   2. cos-neg 99.5%

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

   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 53.0%

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

   1. *-commutative 53.0%

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

   2. pow2 53.0%

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

   3. div-inv 52.9%

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

   4. associate-*r* 52.9%

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

   5. *-commutative 52.9%

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

   6. associate-*r/ 52.9%

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

   7. pow1/2 52.9%

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

   8. metadata-eval 52.9%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \frac{\cos th \cdot 1}{{2}^{\color{blue}{\left(0.25 + 0.25\right)}}} \]

   9. pow-prod-up 52.9%

      \[\leadsto \left(a2 \cdot a2\right) \cdot \frac{\cos th \cdot 1}{\color{blue}{{2}^{0.25} \cdot {2}^{0.25}}} \]

   10. frac-times 52.8%

       \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\left(\frac{\cos th}{{2}^{0.25}} \cdot \frac{1}{{2}^{0.25}}\right)} \]

   11. div-inv 52.9%

       \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \]

   12. clear-num 52.9%

       \[\leadsto \left(a2 \cdot a2\right) \cdot \color{blue}{\frac{1}{\frac{{2}^{0.25}}{\frac{\cos th}{{2}^{0.25}}}}} \]

   13. un-div-inv 52.9%

       \[\leadsto \color{blue}{\frac{a2 \cdot a2}{\frac{{2}^{0.25}}{\frac{\cos th}{{2}^{0.25}}}}} \]

   14. pow2 52.9%

       \[\leadsto \frac{\color{blue}{{a2}^{2}}}{\frac{{2}^{0.25}}{\frac{\cos th}{{2}^{0.25}}}} \]

   15. clear-num 52.9%

       \[\leadsto \frac{{a2}^{2}}{\color{blue}{\frac{1}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}}}} \]

   16. associate-/l/ 52.9%

       \[\leadsto \frac{{a2}^{2}}{\frac{1}{\color{blue}{\frac{\cos th}{{2}^{0.25} \cdot {2}^{0.25}}}}} \]

   17. pow-prod-up 52.9%

       \[\leadsto \frac{{a2}^{2}}{\frac{1}{\frac{\cos th}{\color{blue}{{2}^{\left(0.25 + 0.25\right)}}}}} \]

   18. metadata-eval 52.9%

       \[\leadsto \frac{{a2}^{2}}{\frac{1}{\frac{\cos th}{{2}^{\color{blue}{0.5}}}}} \]

   19. pow1/2 52.9%

       \[\leadsto \frac{{a2}^{2}}{\frac{1}{\frac{\cos th}{\color{blue}{\sqrt{2}}}}} \]

   20. div-inv 52.9%

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

7. Applied egg-rr 52.9%

   \[\leadsto \color{blue}{\frac{{a2}^{2}}{\frac{1}{\cos th \cdot \sqrt{0.5}}}} \]
8. Step-by-step derivation

   1. associate-/r/ 52.9%

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

   2. /-rgt-identity 52.9%

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

   3. unpow2 52.9%

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

   4. associate-*l* 52.9%

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

9. Applied egg-rr 52.9%

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

10. Final simplification 52.9%

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

Alternative 4: 66.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in th around 0 61.0%

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

   1. *-un-lft-identity 61.0%

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

   2. add-sqr-sqrt 61.0%

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

   3. sqrt-unprod 61.0%

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

   4. frac-times 61.0%

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

   5. metadata-eval 61.0%

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

   6. rem-square-sqrt 61.0%

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

   7. metadata-eval 61.0%

      \[\leadsto \left(1 \cdot \sqrt{\color{blue}{0.5}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{\left(1 \cdot \sqrt{0.5}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
8. Step-by-step derivation

   1. *-lft-identity 61.0%

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

9. Simplified 61.0%

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

10. Final simplification 61.0%

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

Alternative 5: 40.3% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ a2 \cdot \left(a2 \cdot {0.25}^{0.25}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* a2 (* a2 (pow 0.25 0.25))))

C

double code(double a1, double a2, double th) {
	return a2 * (a2 * pow(0.25, 0.25));
}

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 * (0.25d0 ** 0.25d0))
end function

Java

public static double code(double a1, double a2, double th) {
	return a2 * (a2 * Math.pow(0.25, 0.25));
}

Python

def code(a1, a2, th):
	return a2 * (a2 * math.pow(0.25, 0.25))

Julia

function code(a1, a2, th)
	return Float64(a2 * Float64(a2 * (0.25 ^ 0.25)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = a2 * (a2 * (0.25 ^ 0.25));
end

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[Power[0.25, 0.25], $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 99.5%

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

3. Simplified 99.5%

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

   1. *-un-lft-identity 99.5%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.2%

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

   4. pow1/2 99.2%

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

   5. sqrt-pow1 99.2%

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

   6. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.2%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

9. Taylor expanded in a1 around 0 52.9%

   \[\leadsto \color{blue}{{0.25}^{0.25} \cdot \left({a2}^{2} \cdot \cos th\right)} \]

10. Taylor expanded in th around 0 32.4%

    \[\leadsto \color{blue}{{0.25}^{0.25} \cdot {a2}^{2}} \]
11. Step-by-step derivation

    1. metadata-eval 32.4%

       \[\leadsto {0.25}^{\color{blue}{\left(2 \cdot 0.125\right)}} \cdot {a2}^{2} \]

    2. pow-sqr 32.4%

       \[\leadsto \color{blue}{\left({0.25}^{0.125} \cdot {0.25}^{0.125}\right)} \cdot {a2}^{2} \]

    3. unpow2 32.4%

       \[\leadsto \left({0.25}^{0.125} \cdot {0.25}^{0.125}\right) \cdot \color{blue}{\left(a2 \cdot a2\right)} \]

    4. swap-sqr 32.4%

       \[\leadsto \color{blue}{\left({0.25}^{0.125} \cdot a2\right) \cdot \left({0.25}^{0.125} \cdot a2\right)} \]

    5. unpow2 32.4%

       \[\leadsto \color{blue}{{\left({0.25}^{0.125} \cdot a2\right)}^{2}} \]

    6. *-commutative 32.4%

       \[\leadsto {\color{blue}{\left(a2 \cdot {0.25}^{0.125}\right)}}^{2} \]

12. Simplified 32.4%

    \[\leadsto \color{blue}{{\left(a2 \cdot {0.25}^{0.125}\right)}^{2}} \]
13. Step-by-step derivation

    1. unpow-prod-down 32.4%

       \[\leadsto \color{blue}{{a2}^{2} \cdot {\left({0.25}^{0.125}\right)}^{2}} \]

    2. unpow2 32.4%

       \[\leadsto \color{blue}{\left(a2 \cdot a2\right)} \cdot {\left({0.25}^{0.125}\right)}^{2} \]

    3. associate-*l* 32.4%

       \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot {\left({0.25}^{0.125}\right)}^{2}\right)} \]

    4. pow-pow 32.4%

       \[\leadsto a2 \cdot \left(a2 \cdot \color{blue}{{0.25}^{\left(0.125 \cdot 2\right)}}\right) \]

    5. metadata-eval 32.4%

       \[\leadsto a2 \cdot \left(a2 \cdot {0.25}^{\color{blue}{0.25}}\right) \]

14. Applied egg-rr 32.4%

    \[\leadsto \color{blue}{a2 \cdot \left(a2 \cdot {0.25}^{0.25}\right)} \]

15. Final simplification 32.4%

    \[\leadsto a2 \cdot \left(a2 \cdot {0.25}^{0.25}\right) \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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))))