Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 10.9s

Alternatives: 10

Speedup: 2.0×

• 43.2% 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, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2 \cdot a2 + a1 \cdot a1}{\frac{\sqrt{2}}{\cos th}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. unpow2 99.6%

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

   4. unpow2 99.6%

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

   5. fma-def 99.6%

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

6. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

8. Applied egg-rr 99.6%

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

9. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 99.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cos th}{\frac{\sqrt{2}}{a2 \cdot a2 + a1 \cdot a1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. unpow2 99.6%

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

   4. unpow2 99.6%

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

   5. fma-def 99.6%

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

6. Simplified 99.6%

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

   1. fma-udef 99.6%

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

   2. +-commutative 99.6%

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

8. Applied egg-rr 99.6%

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

9. Taylor expanded in th around inf 99.6%

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

    1. associate-/l* 99.6%

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

    2. unpow2 99.6%

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

    3. +-commutative 99.6%

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

    4. unpow2 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 4: 56.9% 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 99.6%

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

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

   3. associate-*l/ 99.6%

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

   4. div-inv 99.5%

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

   5. associate-*l* 99.5%

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

   6. add-sqr-sqrt 99.5%

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

   7. pow2 99.5%

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

   8. fma-udef 99.5%

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

   9. +-commutative 99.5%

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

   10. hypot-def 99.5%

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

   11. pow1/2 99.5%

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

   12. pow-flip 99.6%

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

   13. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

4. Taylor expanded in a2 around inf 55.1%

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

   1. unpow2 55.1%

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

   2. associate-*l* 55.0%

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

6. Simplified 55.0%

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

7. Final simplification 55.0%

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

Alternative 5: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * N[Sqrt[0.5], $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. cos-neg 99.6%

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.0%

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

   1. unpow2 55.0%

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

   2. *-commutative 55.0%

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

6. Simplified 55.0%

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

   1. div-inv 55.0%

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

   2. associate-*r* 55.0%

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

   3. associate-*l* 55.0%

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

   4. *-commutative 55.0%

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

   5. add-sqr-sqrt 55.0%

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

   6. sqrt-unprod 55.0%

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

   7. frac-times 55.0%

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

   8. metadata-eval 55.0%

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

   9. add-sqr-sqrt 55.1%

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

   10. metadata-eval 55.1%

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

8. Applied egg-rr 55.1%

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

9. Final simplification 55.1%

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

Alternative 6: 56.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. unpow2 99.6%

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

   4. unpow2 99.6%

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

   5. fma-def 99.6%

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

6. Simplified 99.6%

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

7. Taylor expanded in a1 around 0 55.0%

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

   1. unpow2 55.0%

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

   2. associate-*l* 55.0%

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

9. Simplified 55.0%

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

10. Final simplification 55.0%

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

Alternative 7: 66.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 1.5 \cdot 10^{+201}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot \sqrt{0.5}\right) \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 1.5e+201)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (* (* a2 (sqrt 0.5)) (+ a2 (* -0.5 (* a2 (* th th)))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.5e+201) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else {
		tmp = (a2 * sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	}
	return tmp;
}

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) :: tmp
    if (a2 <= 1.5d+201) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else
        tmp = (a2 * sqrt(0.5d0)) * (a2 + ((-0.5d0) * (a2 * (th * th))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 1.5e+201) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else {
		tmp = (a2 * Math.sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 1.5e+201:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	else:
		tmp = (a2 * math.sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 1.5e+201)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	else
		tmp = Float64(Float64(a2 * sqrt(0.5)) * Float64(a2 + Float64(-0.5 * Float64(a2 * Float64(th * th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 1.5e+201)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	else
		tmp = (a2 * sqrt(0.5)) * (a2 + (-0.5 * (a2 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 1.5e+201], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], N[(N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] * N[(a2 + N[(-0.5 * N[(a2 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a2 < 1.50000000000000012e201

   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. +-commutative 99.5%

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

      2. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. associate-/r/ 99.5%

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

      3. pow1/2 99.5%

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

      4. pow-flip 99.5%

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

      5. metadata-eval 99.5%

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

   5. Applied egg-rr 99.5%

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

   6. Taylor expanded in th around 0 66.5%

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

   if 1.50000000000000012e201 < a2

   1. Initial program 100.0%

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

      1. distribute-lft-out 100.0%

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

      2. cos-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. cos-neg 100.0%

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

      5. fma-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in a1 around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   6. Simplified 100.0%

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

      1. div-inv 100.0%

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

      2. associate-*r* 100.0%

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

      3. associate-*l* 100.0%

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

      4. *-commutative 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. frac-times 100.0%

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

      8. metadata-eval 100.0%

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

      9. add-sqr-sqrt 100.0%

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

      10. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in th around 0 86.4%

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

       1. unpow2 86.4%

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

   11. Simplified 86.4%

       \[\leadsto \color{blue}{\left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)} \cdot \left(a2 \cdot \sqrt{0.5}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 1.5 \cdot 10^{+201}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot \sqrt{0.5}\right) \cdot \left(a2 + -0.5 \cdot \left(a2 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 8: 66.0% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.5%

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

   3. pow1/2 99.5%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 67.0%

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

7. Final simplification 67.0%

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

Alternative 9: 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 99.6%

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

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

   5. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 55.0%

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

   1. unpow2 55.0%

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

   2. *-commutative 55.0%

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

6. Simplified 55.0%

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

   1. div-inv 55.0%

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

   2. associate-*r* 55.0%

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

   3. associate-*l* 55.0%

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

   4. *-commutative 55.0%

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

   5. add-sqr-sqrt 55.0%

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

   6. sqrt-unprod 55.0%

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

   7. frac-times 55.0%

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

   8. metadata-eval 55.0%

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

   9. add-sqr-sqrt 55.1%

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

   10. metadata-eval 55.1%

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

8. Applied egg-rr 55.1%

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

9. Taylor expanded in th around 0 38.7%

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

10. Final simplification 38.7%

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

Alternative 10: 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 99.6%

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

   1. +-commutative 99.6%

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

   2. distribute-lft-out 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in th around 0 66.9%

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

5. Taylor expanded in a2 around inf 38.7%

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

   1. unpow2 38.7%

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

7. Simplified 38.7%

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

   1. associate-/l* 38.7%

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

   2. associate-/r/ 38.7%

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

9. Applied egg-rr 38.7%

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

10. Final simplification 38.7%

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

Reproduce

herbie shell --seed 2023271 
(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))))