Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 12.3s

Alternatives: 10

Speedup: 2.0×

• 42.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-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 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-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 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around inf 99.7%

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

   1. *-commutative 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in th around inf 99.6%

   \[\leadsto \color{blue}{\cos th \cdot \left(\sqrt{0.5} \cdot \left({a1}^{2} + {a2}^{2}\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 99.6%

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

    2. unpow2 99.6%

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

    3. +-commutative 99.6%

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

11. Simplified 99.6%

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

12. Final simplification 99.6%

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

Alternative 3: 56.7% accurate, 2.0× speedup

Math

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

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(a2 * Float64(a2 * Float64(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[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] / 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. +-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. Taylor expanded in a2 around inf 60.8%

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

   1. unpow2 60.8%

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

   2. associate-*r/ 60.7%

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

   3. associate-*r* 60.7%

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

6. Simplified 60.7%

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

7. Final simplification 60.7%

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

Alternative 4: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(a2 * a2), $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. 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 60.8%

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

   1. unpow2 60.8%

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

   2. *-commutative 60.8%

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

6. Simplified 60.8%

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

   1. div-inv 60.7%

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

   2. *-commutative 60.7%

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

   3. associate-*l* 60.7%

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

   4. *-commutative 60.7%

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

   5. add-sqr-sqrt 60.7%

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

   6. sqrt-unprod 60.7%

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

   7. frac-times 60.7%

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

   8. metadata-eval 60.7%

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

   9. add-sqr-sqrt 60.8%

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

   10. metadata-eval 60.8%

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

8. Applied egg-rr 60.8%

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

9. Final simplification 60.8%

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

Alternative 5: 56.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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. +-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. Taylor expanded in a2 around inf 60.8%

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

   1. unpow2 60.8%

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

   2. associate-*l/ 60.7%

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

6. Simplified 60.7%

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

   1. *-commutative 60.7%

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

   2. associate-/l* 60.8%

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

   3. associate-*r/ 60.8%

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

8. Applied egg-rr 60.8%

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

9. Final simplification 60.8%

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

Alternative 6: 65.1% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a2 \cdot a2 + a1 \cdot a1\\ \mathbf{if}\;a2 \cdot a2 \leq 10^{+213}:\\ \;\;\;\;\sqrt{0.5} \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a2 a2) (* a1 a1))))
   (if (<= (* a2 a2) 1e+213)
     (* (sqrt 0.5) t_1)
     (* t_1 (* (sqrt 0.5) (+ 1.0 (* -0.5 (* th th))))))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((a2 * a2) <= 1e+213) {
		tmp = sqrt(0.5) * t_1;
	} else {
		tmp = t_1 * (sqrt(0.5) * (1.0 + (-0.5 * (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) :: t_1
    real(8) :: tmp
    t_1 = (a2 * a2) + (a1 * a1)
    if ((a2 * a2) <= 1d+213) then
        tmp = sqrt(0.5d0) * t_1
    else
        tmp = t_1 * (sqrt(0.5d0) * (1.0d0 + ((-0.5d0) * (th * th))))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a2 * a2) + (a1 * a1);
	double tmp;
	if ((a2 * a2) <= 1e+213) {
		tmp = Math.sqrt(0.5) * t_1;
	} else {
		tmp = t_1 * (Math.sqrt(0.5) * (1.0 + (-0.5 * (th * th))));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a2 * a2) + (a1 * a1)
	tmp = 0
	if (a2 * a2) <= 1e+213:
		tmp = math.sqrt(0.5) * t_1
	else:
		tmp = t_1 * (math.sqrt(0.5) * (1.0 + (-0.5 * (th * th))))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a2 * a2) + Float64(a1 * a1))
	tmp = 0.0
	if (Float64(a2 * a2) <= 1e+213)
		tmp = Float64(sqrt(0.5) * t_1);
	else
		tmp = Float64(t_1 * Float64(sqrt(0.5) * Float64(1.0 + Float64(-0.5 * Float64(th * th)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a2 * a2) + (a1 * a1);
	tmp = 0.0;
	if ((a2 * a2) <= 1e+213)
		tmp = sqrt(0.5) * t_1;
	else
		tmp = t_1 * (sqrt(0.5) * (1.0 + (-0.5 * (th * th))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(a2 * a2), $MachinePrecision], 1e+213], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$1 * N[(N[Sqrt[0.5], $MachinePrecision] * N[(1.0 + N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 a2 a2) < 9.99999999999999984e212

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. distribute-lft-out 99.4%

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

   3. Simplified 99.4%

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

      1. clear-num 99.4%

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

      2. associate-/r/ 99.3%

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

      3. pow1/2 99.3%

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

      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 64.0%

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

   if 9.99999999999999984e212 < (*.f64 a2 a2)

   1. Initial program 99.8%

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

      1. +-commutative 99.8%

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

      2. distribute-lft-out 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.8%

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

      3. pow1/2 99.8%

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

      4. pow-flip 99.9%

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

      5. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in th around 0 73.3%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} + -0.5 \cdot \left({th}^{2} \cdot \sqrt{0.5}\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 73.3%

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

      2. associate-*r* 73.3%

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

      3. distribute-rgt-out 73.3%

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

      4. unpow2 73.3%

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

   8. Simplified 73.3%

      \[\leadsto \color{blue}{\left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \cdot a2 \leq 10^{+213}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a2 \cdot a2 + a1 \cdot a1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \left(\sqrt{0.5} \cdot \left(1 + -0.5 \cdot \left(th \cdot th\right)\right)\right)\\ \end{array} \]

Alternative 7: 65.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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. +-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 99.7%

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

   5. metadata-eval 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in th around 0 64.0%

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

7. Final simplification 64.0%

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

Alternative 8: 39.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ a1 \cdot \frac{a1}{\sqrt{2}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * N[(a1 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. Taylor expanded in th around 0 63.9%

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

5. Taylor expanded in a2 around 0 36.3%

   \[\leadsto \frac{1}{\sqrt{2}} \cdot \color{blue}{{a1}^{2}} \]
6. Step-by-step derivation

   1. unpow2 36.3%

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

7. Simplified 36.3%

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

8. Taylor expanded in a1 around 0 36.3%

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

   1. unpow2 36.3%

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

   2. associate-*r/ 36.3%

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

10. Simplified 36.3%

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

11. Final simplification 36.3%

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

Alternative 9: 39.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
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. Taylor expanded in th around 0 63.9%

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

5. Taylor expanded in a2 around inf 41.7%

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

   1. unpow2 41.7%

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

   2. associate-*r/ 41.7%

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

7. Simplified 41.7%

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

   1. expm1-log1p-u 29.0%

      \[\leadsto a2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)\right)} \]

   2. expm1-udef 24.7%

      \[\leadsto a2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{a2}{\sqrt{2}}\right)} - 1\right)} \]

   3. div-inv 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{a2 \cdot \frac{1}{\sqrt{2}}}\right)} - 1\right) \]

   4. add-sqr-sqrt 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \color{blue}{\left(\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}\right)}\right)} - 1\right) \]

   5. sqrt-unprod 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \]

   6. frac-times 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \]

   7. metadata-eval 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \]

   8. add-sqr-sqrt 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \]

   9. metadata-eval 24.7%

      \[\leadsto a2 \cdot \left(e^{\mathsf{log1p}\left(a2 \cdot \sqrt{\color{blue}{0.5}}\right)} - 1\right) \]

9. Applied egg-rr 24.7%

   \[\leadsto a2 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(a2 \cdot \sqrt{0.5}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 28.9%

       \[\leadsto a2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a2 \cdot \sqrt{0.5}\right)\right)} \]

    2. expm1-log1p 41.8%

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

    3. *-commutative 41.8%

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

11. Simplified 41.8%

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

12. Final simplification 41.8%

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

Alternative 10: 39.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Sqrt[0.5], $MachinePrecision] * N[(a2 * a2), $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. 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 60.8%

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

   1. unpow2 60.8%

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

   2. *-commutative 60.8%

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

6. Simplified 60.8%

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

   1. div-inv 60.7%

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

   2. *-commutative 60.7%

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

   3. associate-*l* 60.7%

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

   4. *-commutative 60.7%

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

   5. add-sqr-sqrt 60.7%

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

   6. sqrt-unprod 60.7%

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

   7. frac-times 60.7%

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

   8. metadata-eval 60.7%

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

   9. add-sqr-sqrt 60.8%

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

   10. metadata-eval 60.8%

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

8. Applied egg-rr 60.8%

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

9. Taylor expanded in th around 0 41.8%

   \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{0.5}} \]
10. Step-by-step derivation

    1. unpow2 41.8%

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

    2. *-commutative 41.8%

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

11. Simplified 41.8%

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

12. Final simplification 41.8%

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

Reproduce

herbie shell --seed 2023278 
(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))))