Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 11.6s

Alternatives: 8

Speedup: 2.0×

• 44.1% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. *-commutative 99.6%

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

   5. associate-*l/ 99.6%

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

   6. fma-def 99.6%

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

   7. cos-neg 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.5%

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

   2. fma-def 99.5%

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

   3. add-sqr-sqrt 99.5%

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

   4. pow2 99.5%

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

   5. hypot-def 99.5%

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

   6. pow1/2 99.5%

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

   7. pow-flip 99.7%

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

   8. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Final simplification 99.7%

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

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 99.5%

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

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. *-commutative 99.6%

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

   5. associate-*l/ 99.6%

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

   6. fma-def 99.6%

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

   7. cos-neg 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.7%

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

   1. pow2 39.3%

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

   2. *-un-lft-identity 39.3%

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

   3. associate-*l/ 39.3%

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

   4. associate-*r* 39.3%

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

   5. add-sqr-sqrt 39.3%

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

   6. sqrt-unprod 39.3%

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

   7. frac-times 39.3%

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

   8. metadata-eval 39.3%

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

   9. rem-square-sqrt 39.3%

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

   10. metadata-eval 39.3%

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

6. Applied egg-rr 56.7%

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

7. Final simplification 56.7%

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

Alternative 4: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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-/r/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.7%

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

   1. associate-/r/ 56.7%

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

   2. associate-*l/ 56.7%

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

   3. associate-*r/ 56.7%

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

   4. unpow2 56.7%

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

   5. associate-*l/ 56.7%

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

   6. associate-*r* 56.7%

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

   7. *-commutative 56.7%

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

6. Applied egg-rr 56.7%

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

7. Final simplification 56.7%

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

Alternative 5: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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-/r/ 99.6%

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

   4. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.7%

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

   1. associate-/r/ 56.7%

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

   2. associate-*l/ 56.7%

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

   3. associate-*r/ 56.7%

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

   4. unpow2 56.7%

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

   5. associate-*r/ 56.7%

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

   6. /-rgt-identity 56.7%

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

   7. *-commutative 56.7%

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

   8. /-rgt-identity 56.7%

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

   9. *-commutative 56.7%

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

   10. associate-*l* 56.7%

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

6. Applied egg-rr 56.7%

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

7. Final simplification 56.7%

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

Alternative 6: 65.8% 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 99.5%

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

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around 0 65.2%

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

7. Final simplification 65.2%

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

Alternative 7: 39.3% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
2. 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. Taylor expanded in th around 0 65.1%

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

5. Taylor expanded in a1 around 0 39.3%

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

   1. pow2 39.3%

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

   2. *-un-lft-identity 39.3%

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

   3. associate-*l/ 39.3%

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

   4. associate-*r* 39.3%

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

   5. add-sqr-sqrt 39.3%

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

   6. sqrt-unprod 39.3%

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

   7. frac-times 39.3%

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

   8. metadata-eval 39.3%

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

   9. rem-square-sqrt 39.3%

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

   10. metadata-eval 39.3%

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

7. Applied egg-rr 39.3%

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

8. Final simplification 39.3%

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

Alternative 8: 39.3% 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 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. Taylor expanded in th around 0 65.1%

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

5. Taylor expanded in a1 around 0 39.3%

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

   1. pow2 39.3%

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

   2. add-sqr-sqrt 39.2%

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

   3. pow2 39.2%

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

   4. div-inv 39.2%

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

   5. sqrt-prod 39.3%

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

   6. sqrt-prod 20.1%

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

   7. add-sqr-sqrt 39.3%

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

   8. inv-pow 39.3%

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

   9. sqrt-pow2 39.2%

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

   10. metadata-eval 39.2%

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

   11. sqrt-pow1 39.3%

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

   12. metadata-eval 39.3%

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

7. Applied egg-rr 39.3%

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

   1. unpow2 39.3%

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

   2. swap-sqr 39.3%

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

   3. unpow2 39.3%

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

   4. pow-prod-up 39.3%

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

   5. metadata-eval 39.3%

      \[\leadsto {a2}^{2} \cdot {2}^{\color{blue}{-0.5}} \]

   6. metadata-eval 39.3%

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

   7. pow-flip 39.3%

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

   8. pow1/2 39.3%

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

   9. div-inv 39.3%

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

   10. unpow2 39.3%

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

   11. associate-*l/ 39.3%

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

9. Applied egg-rr 39.3%

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

10. Final simplification 39.3%

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

Reproduce

herbie shell --seed 2023299 
(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))))