Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.7s

Alternatives: 9

Speedup: 2.0×

• 44.4% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

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%

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

   1. *-commutative 99.7%

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

   2. unpow2 99.7%

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

   3. unpow2 99.7%

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

   4. +-commutative 99.7%

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

8. Simplified 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. +-commutative 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 3: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

   1. fma-def 99.6%

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

   2. associate-*r/ 99.6%

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

   3. associate-*l/ 99.5%

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

   4. distribute-lft-in 99.5%

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

   5. associate-*l/ 99.5%

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

   6. associate-*l/ 99.6%

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

   7. frac-add 99.3%

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

   8. fma-def 99.3%

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

   9. *-commutative 99.3%

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

   10. add-sqr-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. *-commutative 99.7%

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

   5. distribute-lft-in 99.7%

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

   6. +-commutative 99.7%

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

   7. unpow2 99.7%

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

   8. unpow2 99.7%

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

   9. *-commutative 99.7%

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

   10. unpow2 99.7%

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

   11. unpow2 99.7%

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

   12. fma-def 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in a2 around inf 57.0%

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

   1. unpow2 57.0%

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

10. Simplified 57.0%

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

11. Final simplification 57.0%

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

Alternative 4: 57.0% 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. distribute-lft-out 99.5%

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l* 56.9%

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

6. Simplified 56.9%

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

7. Final simplification 56.9%

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

Alternative 5: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l* 56.9%

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

6. Simplified 56.9%

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

   1. expm1-log1p-u 37.0%

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

   2. expm1-udef 30.8%

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

   3. div-inv 30.8%

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

   4. add-sqr-sqrt 30.8%

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

   5. sqrt-unprod 30.8%

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

   6. frac-times 30.8%

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

   7. metadata-eval 30.8%

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

   8. add-sqr-sqrt 30.8%

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

   9. metadata-eval 30.8%

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

8. Applied egg-rr 30.8%

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

   1. expm1-def 37.0%

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

   2. expm1-log1p 57.0%

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

   3. associate-*r* 57.0%

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

   4. *-commutative 57.0%

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

10. Simplified 57.0%

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

11. Final simplification 57.0%

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

Alternative 6: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l* 56.9%

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

6. Simplified 56.9%

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

   1. associate-*r* 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l/ 56.9%

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

   4. *-commutative 56.9%

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

   5. expm1-log1p-u 47.7%

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

   6. expm1-udef 41.0%

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

8. Applied egg-rr 41.0%

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

   1. expm1-def 47.7%

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

   2. expm1-log1p 57.0%

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

   3. associate-*r* 57.0%

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

   4. *-commutative 57.0%

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

   5. associate-*l* 57.0%

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

10. Simplified 57.0%

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

11. Final simplification 57.0%

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

Alternative 7: 57.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 * a2), $MachinePrecision] * N[(N[Sqrt[0.5], $MachinePrecision] * 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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l* 56.9%

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

6. Simplified 56.9%

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

7. Taylor expanded in a2 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r* 56.9%

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

   3. associate-/l* 56.9%

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

   4. associate-/r/ 57.0%

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

   5. *-commutative 57.0%

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

9. Simplified 57.0%

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

    1. expm1-log1p-u 47.7%

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

    2. expm1-udef 41.0%

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

11. Applied egg-rr 41.0%

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

    1. expm1-def 47.7%

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

    2. expm1-log1p 57.0%

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

    3. associate-*r* 57.0%

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

    4. *-commutative 57.0%

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

13. Simplified 57.0%

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

14. Final simplification 57.0%

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

Alternative 8: 66.7% 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. Taylor expanded in th around 0 70.5%

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

   1. expm1-log1p-u 70.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. expm1-udef 70.5%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\sqrt{2}}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   3. add-sqr-sqrt 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}}} \cdot \sqrt{\frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   4. sqrt-unprod 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   5. frac-times 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\frac{1 \cdot 1}{\sqrt{2} \cdot \sqrt{2}}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   6. metadata-eval 70.5%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{\color{blue}{1}}{\sqrt{2} \cdot \sqrt{2}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. add-sqr-sqrt 70.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\color{blue}{2}}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. metadata-eval 70.3%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{0.5}}\right)} - 1\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

6. Applied egg-rr 70.3%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{0.5}\right)} - 1\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
7. Step-by-step derivation

   1. expm1-def 70.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{0.5}\right)\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. expm1-log1p 70.6%

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

8. Simplified 70.6%

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

9. Final simplification 70.6%

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

Alternative 9: 39.9% 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. associate-*l/ 99.6%

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

   3. associate-*r/ 99.6%

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

   4. fma-def 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in a1 around 0 56.9%

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

   1. unpow2 56.9%

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

   2. associate-*r/ 56.9%

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

   3. associate-*l* 56.9%

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

6. Simplified 56.9%

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

7. Taylor expanded in th around 0 42.7%

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

   1. expm1-log1p-u 41.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(a2 \cdot \left(a2 \cdot \frac{1}{\sqrt{2}}\right)\right)\right)} \]

   2. expm1-udef 38.5%

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

   3. add-sqr-sqrt 38.5%

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

   4. sqrt-unprod 38.5%

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

   5. frac-times 38.5%

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

   6. metadata-eval 38.5%

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

   7. add-sqr-sqrt 38.5%

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

   8. metadata-eval 38.5%

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

9. Applied egg-rr 38.5%

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

    1. expm1-def 41.9%

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

    2. expm1-log1p 42.7%

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

    3. associate-*r* 42.7%

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

11. Simplified 42.7%

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

12. Final simplification 42.7%

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

Reproduce

herbie shell --seed 2023217 
(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))))