Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.0s

Alternatives: 21

Speedup: 2.0×

• 43.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1 + N[(a2 * a2), $MachinePrecision]), $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. distribute-lft-out 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * 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 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.6% accurate, 2.0× speedup

Math

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

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(Float64(cos(th) / 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[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $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 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 4: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. associate-/l* 60.6%

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

   3. associate-/r/ 60.6%

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

6. Simplified 60.6%

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

7. Final simplification 60.6%

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

Alternative 5: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $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. distribute-lft-out 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

6. Simplified 60.6%

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

7. Final simplification 60.6%

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

Alternative 6: 66.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a2 \leq 8 \cdot 10^{+171}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{+261} \lor \neg \left(a2 \leq 2 \cdot 10^{+279}\right) \land a2 \leq 10^{+297}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= a2 8e+171)
   (* (+ (* a2 a2) (* a1 a1)) (sqrt 0.5))
   (if (or (<= a2 7.6e+261) (and (not (<= a2 2e+279)) (<= a2 1e+297)))
     (* (sqrt 0.5) (* (* a2 a2) (+ (* -0.5 (* th th)) 1.0)))
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8e+171) {
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	} else if ((a2 <= 7.6e+261) || (!(a2 <= 2e+279) && (a2 <= 1e+297))) {
		tmp = sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	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 <= 8d+171) then
        tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5d0)
    else if ((a2 <= 7.6d+261) .or. (.not. (a2 <= 2d+279)) .and. (a2 <= 1d+297)) then
        tmp = sqrt(0.5d0) * ((a2 * a2) * (((-0.5d0) * (th * th)) + 1.0d0))
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (a2 <= 8e+171) {
		tmp = ((a2 * a2) + (a1 * a1)) * Math.sqrt(0.5);
	} else if ((a2 <= 7.6e+261) || (!(a2 <= 2e+279) && (a2 <= 1e+297))) {
		tmp = Math.sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if a2 <= 8e+171:
		tmp = ((a2 * a2) + (a1 * a1)) * math.sqrt(0.5)
	elif (a2 <= 7.6e+261) or (not (a2 <= 2e+279) and (a2 <= 1e+297)):
		tmp = math.sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (a2 <= 8e+171)
		tmp = Float64(Float64(Float64(a2 * a2) + Float64(a1 * a1)) * sqrt(0.5));
	elseif ((a2 <= 7.6e+261) || (!(a2 <= 2e+279) && (a2 <= 1e+297)))
		tmp = Float64(sqrt(0.5) * Float64(Float64(a2 * a2) * Float64(Float64(-0.5 * Float64(th * th)) + 1.0)));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (a2 <= 8e+171)
		tmp = ((a2 * a2) + (a1 * a1)) * sqrt(0.5);
	elseif ((a2 <= 7.6e+261) || (~((a2 <= 2e+279)) && (a2 <= 1e+297)))
		tmp = sqrt(0.5) * ((a2 * a2) * ((-0.5 * (th * th)) + 1.0));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[a2, 8e+171], N[(N[(N[(a2 * a2), $MachinePrecision] + N[(a1 * a1), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[a2, 7.6e+261], And[N[Not[LessEqual[a2, 2e+279]], $MachinePrecision], LessEqual[a2, 1e+297]]], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[(N[(-0.5 * N[(th * th), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a2 < 7.99999999999999963e171

   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 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
   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 99.6%

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

      5. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   6. Taylor expanded in th around 0 64.0%

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

      1. *-commutative 64.0%

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

      2. unpow2 64.0%

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

      3. unpow2 64.0%

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

      4. +-commutative 64.0%

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

   8. Simplified 64.0%

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

   if 7.99999999999999963e171 < a2 < 7.6000000000000003e261 or 2.00000000000000012e279 < a2 < 1e297

   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%

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

   3. Simplified 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

      3. pow1/2 100.0%

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

      4. pow-flip 100.0%

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

      5. metadata-eval 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in th around inf 100.0%

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

   7. Taylor expanded in a1 around 0 100.0%

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

      1. unpow2 100.0%

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

      2. *-commutative 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in th around 0 0.0%

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

       1. unpow2 0.0%

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

       2. unpow2 0.0%

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

       3. associate-*r* 0.0%

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

       4. distribute-rgt1-in 90.9%

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

       5. unpow2 90.9%

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

   12. Simplified 90.9%

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

   if 7.6000000000000003e261 < a2 < 2.00000000000000012e279 or 1e297 < 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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in th around 0 83.3%

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

   5. Taylor expanded in a1 around 0 83.3%

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

      1. unpow2 83.3%

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

      2. associate-*r/ 83.3%

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

   7. Simplified 83.3%

      \[\leadsto \color{blue}{a2 \cdot \frac{a2}{\sqrt{2}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a2 \leq 8 \cdot 10^{+171}:\\ \;\;\;\;\left(a2 \cdot a2 + a1 \cdot a1\right) \cdot \sqrt{0.5}\\ \mathbf{elif}\;a2 \leq 7.6 \cdot 10^{+261} \lor \neg \left(a2 \leq 2 \cdot 10^{+279}\right) \land a2 \leq 10^{+297}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(\left(a2 \cdot a2\right) \cdot \left(-0.5 \cdot \left(th \cdot th\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 7: 65.7% 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. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.6%

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

   2. associate-/r/ 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-flip 99.6%

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

   5. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around 0 65.2%

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

   1. *-commutative 65.2%

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

   2. unpow2 65.2%

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

   3. unpow2 65.2%

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

   4. +-commutative 65.2%

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

8. Simplified 65.2%

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

9. Final simplification 65.2%

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

Alternative 8: 39.6% 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. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.2%

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

5. Taylor expanded in a1 around 0 40.4%

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

   1. unpow2 40.4%

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

   2. associate-*r/ 40.3%

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

7. Simplified 40.3%

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

8. Final simplification 40.3%

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

Alternative 9: 39.6% accurate, 4.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $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. distribute-lft-out 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in th around 0 40.4%

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

   1. unpow2 40.4%

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

9. Simplified 40.4%

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

    1. div-inv 40.3%

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

    2. add-sqr-sqrt 40.3%

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

    3. sqrt-unprod 40.3%

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

    4. frac-times 40.3%

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

    5. metadata-eval 40.3%

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

    6. add-sqr-sqrt 40.4%

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

    7. metadata-eval 40.4%

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

11. Applied egg-rr 40.4%

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

12. Final simplification 40.4%

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

Alternative 10: 39.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 / N[(N[Sqrt[2.0], $MachinePrecision] / a2), $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 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.2%

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

5. Taylor expanded in a1 around 0 40.4%

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

   1. unpow2 40.4%

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

   2. associate-*r/ 40.3%

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

7. Simplified 40.3%

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

   1. clear-num 40.3%

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

   2. un-div-inv 40.4%

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

9. Applied egg-rr 40.4%

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

10. Final simplification 40.4%

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

Alternative 11: 39.6% accurate, 4.0× speedup

Math

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

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(Float64(a2 * 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[(N[(a2 * a2), $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. distribute-lft-out 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in th around 0 40.4%

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

   1. unpow2 40.4%

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

9. Simplified 40.4%

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

10. Final simplification 40.4%

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

Alternative 12: 27.3% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+24}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+110} \lor \neg \left(th \leq 3.2 \cdot 10^{+155}\right) \land th \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;th \cdot a2\\ \mathbf{else}:\\ \;\;\;\;th \cdot \left(-a2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 4.8e+24)
   (* a1 a1)
   (if (or (<= th 1.3e+110) (and (not (<= th 3.2e+155)) (<= th 3.6e+212)))
     (* th a2)
     (* th (- a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.8e+24) {
		tmp = a1 * a1;
	} else if ((th <= 1.3e+110) || (!(th <= 3.2e+155) && (th <= 3.6e+212))) {
		tmp = th * a2;
	} else {
		tmp = th * -a2;
	}
	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 (th <= 4.8d+24) then
        tmp = a1 * a1
    else if ((th <= 1.3d+110) .or. (.not. (th <= 3.2d+155)) .and. (th <= 3.6d+212)) then
        tmp = th * a2
    else
        tmp = th * -a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.8e+24) {
		tmp = a1 * a1;
	} else if ((th <= 1.3e+110) || (!(th <= 3.2e+155) && (th <= 3.6e+212))) {
		tmp = th * a2;
	} else {
		tmp = th * -a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 4.8e+24:
		tmp = a1 * a1
	elif (th <= 1.3e+110) or (not (th <= 3.2e+155) and (th <= 3.6e+212)):
		tmp = th * a2
	else:
		tmp = th * -a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 4.8e+24)
		tmp = Float64(a1 * a1);
	elseif ((th <= 1.3e+110) || (!(th <= 3.2e+155) && (th <= 3.6e+212)))
		tmp = Float64(th * a2);
	else
		tmp = Float64(th * Float64(-a2));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 4.8e+24)
		tmp = a1 * a1;
	elseif ((th <= 1.3e+110) || (~((th <= 3.2e+155)) && (th <= 3.6e+212)))
		tmp = th * a2;
	else
		tmp = th * -a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 4.8e+24], N[(a1 * a1), $MachinePrecision], If[Or[LessEqual[th, 1.3e+110], And[N[Not[LessEqual[th, 3.2e+155]], $MachinePrecision], LessEqual[th, 3.6e+212]]], N[(th * a2), $MachinePrecision], N[(th * (-a2)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if th < 4.8000000000000001e24

   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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a1 around inf 42.1%

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

      1. unpow2 42.1%

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

   7. Simplified 42.1%

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

   9. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{a1 \cdot a1} \]

   if 4.8000000000000001e24 < th < 1.3e110 or 3.20000000000000012e155 < th < 3.6e212

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in a1 around 0 69.2%

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

      1. unpow2 69.2%

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

      2. *-commutative 69.2%

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

   6. Simplified 69.2%

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

   7. Taylor expanded in th around 0 1.7%

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

      1. unpow2 1.7%

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

      2. unpow2 1.7%

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

      3. associate-*r* 1.7%

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

      4. distribute-rgt1-in 10.4%

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

      5. unpow2 10.4%

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

   9. Simplified 10.4%

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

   10. Taylor expanded in th around inf 10.4%

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

       1. unpow2 10.4%

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

       2. unpow2 10.4%

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

   12. Simplified 10.4%

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

   14. Applied egg-rr 3.8%

       \[\leadsto \color{blue}{th \cdot a2} \]

   if 1.3e110 < th < 3.20000000000000012e155 or 3.6e212 < th

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 49.5%

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

      1. unpow2 49.5%

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

      2. *-commutative 49.5%

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

   6. Simplified 49.5%

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

   7. Taylor expanded in th around 0 14.6%

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

      1. unpow2 14.6%

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

      2. unpow2 14.6%

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

      3. associate-*r* 14.6%

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

      4. distribute-rgt1-in 32.0%

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

      5. unpow2 32.0%

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

   9. Simplified 32.0%

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

   10. Taylor expanded in th around inf 32.0%

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

       1. unpow2 32.0%

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

       2. unpow2 32.0%

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

   12. Simplified 32.0%

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

   14. Applied egg-rr 11.8%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.8 \cdot 10^{+24}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{elif}\;th \leq 1.3 \cdot 10^{+110} \lor \neg \left(th \leq 3.2 \cdot 10^{+155}\right) \land th \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;th \cdot a2\\ \mathbf{else}:\\ \;\;\;\;th \cdot \left(-a2\right)\\ \end{array} \]

Alternative 13: 30.1% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8 \cdot 10^{+76} \lor \neg \left(th \leq 3.2 \cdot 10^{+155}\right) \land th \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 7.8e+76) (and (not (<= th 3.2e+155)) (<= th 3.6e+212)))
   (* a2 a2)
   (* a2 (/ a2 -2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 7.8e+76) || (!(th <= 3.2e+155) && (th <= 3.6e+212))) {
		tmp = a2 * a2;
	} else {
		tmp = a2 * (a2 / -2.0);
	}
	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 ((th <= 7.8d+76) .or. (.not. (th <= 3.2d+155)) .and. (th <= 3.6d+212)) then
        tmp = a2 * a2
    else
        tmp = a2 * (a2 / (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 7.8e+76) || (!(th <= 3.2e+155) && (th <= 3.6e+212))) {
		tmp = a2 * a2;
	} else {
		tmp = a2 * (a2 / -2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 7.8e+76) or (not (th <= 3.2e+155) and (th <= 3.6e+212)):
		tmp = a2 * a2
	else:
		tmp = a2 * (a2 / -2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 7.8e+76) || (!(th <= 3.2e+155) && (th <= 3.6e+212)))
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(a2 * Float64(a2 / -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 7.8e+76) || (~((th <= 3.2e+155)) && (th <= 3.6e+212)))
		tmp = a2 * a2;
	else
		tmp = a2 * (a2 / -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 7.8e+76], And[N[Not[LessEqual[th, 3.2e+155]], $MachinePrecision], LessEqual[th, 3.6e+212]]], N[(a2 * a2), $MachinePrecision], N[(a2 * N[(a2 / -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 7.79999999999999979e76 or 3.20000000000000012e155 < th < 3.6e212

   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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 70.0%

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

   5. Taylor expanded in a1 around 0 43.2%

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

      1. unpow2 43.2%

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

      2. associate-*r/ 43.2%

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

   7. Simplified 43.2%

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

      1. frac-2neg 43.2%

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

      2. div-inv 43.1%

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

   9. Applied egg-rr 43.1%

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

   11. Applied egg-rr 29.1%

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

   if 7.79999999999999979e76 < th < 3.20000000000000012e155 or 3.6e212 < th

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.5%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 53.4%

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

      1. unpow2 53.4%

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

      2. *-commutative 53.4%

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

   6. Simplified 53.4%

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

   7. Taylor expanded in th around 0 16.5%

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

      1. unpow2 16.5%

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

   9. Simplified 16.5%

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

       1. div-inv 16.5%

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

       2. add-sqr-sqrt 16.5%

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

       3. sqrt-unprod 16.5%

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

       4. frac-times 16.5%

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

       5. metadata-eval 16.5%

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

       6. add-sqr-sqrt 16.5%

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

       7. metadata-eval 16.5%

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

   11. Applied egg-rr 16.5%

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

   13. Applied egg-rr 26.8%

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

       1. associate-/r/ 26.8%

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

   15. Simplified 26.8%

       \[\leadsto \color{blue}{\frac{a2}{-2} \cdot a2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8 \cdot 10^{+76} \lor \neg \left(th \leq 3.2 \cdot 10^{+155}\right) \land th \leq 3.6 \cdot 10^{+212}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{-2}\\ \end{array} \]

Alternative 14: 18.1% accurate, 58.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;a2 \cdot \left(th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{-2}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 7.8e+76) (* a2 (* th a2)) (* a2 (/ a2 -2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8e+76) {
		tmp = a2 * (th * a2);
	} else {
		tmp = a2 * (a2 / -2.0);
	}
	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 (th <= 7.8d+76) then
        tmp = a2 * (th * a2)
    else
        tmp = a2 * (a2 / (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 7.8e+76) {
		tmp = a2 * (th * a2);
	} else {
		tmp = a2 * (a2 / -2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 7.8e+76:
		tmp = a2 * (th * a2)
	else:
		tmp = a2 * (a2 / -2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 7.8e+76)
		tmp = Float64(a2 * Float64(th * a2));
	else
		tmp = Float64(a2 * Float64(a2 / -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 7.8e+76)
		tmp = a2 * (th * a2);
	else
		tmp = a2 * (a2 / -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 7.8e+76], N[(a2 * N[(th * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 7.79999999999999979e76

   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 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.7%

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

      4. fma-def 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in a1 around 0 61.5%

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

      1. unpow2 61.5%

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

      2. *-commutative 61.5%

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

   6. Simplified 61.5%

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

   7. Taylor expanded in th around 0 25.7%

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

      1. unpow2 25.7%

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

      2. unpow2 25.7%

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

      3. associate-*r* 25.7%

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

      4. distribute-rgt1-in 47.6%

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

      5. unpow2 47.6%

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

   9. Simplified 47.6%

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

   10. Taylor expanded in th around inf 14.6%

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

       1. unpow2 14.6%

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

       2. unpow2 14.6%

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

   12. Simplified 14.6%

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

   14. Applied egg-rr 18.3%

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

   if 7.79999999999999979e76 < th

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. associate-*l/ 99.5%

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

      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 54.6%

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

      1. unpow2 54.6%

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

      2. *-commutative 54.6%

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

   6. Simplified 54.6%

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

   7. Taylor expanded in th around 0 17.3%

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

      1. unpow2 17.3%

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

   9. Simplified 17.3%

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

       1. div-inv 17.3%

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

       2. add-sqr-sqrt 17.3%

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

       3. sqrt-unprod 17.3%

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

       4. frac-times 17.3%

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

       5. metadata-eval 17.3%

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

       6. add-sqr-sqrt 17.3%

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

       7. metadata-eval 17.3%

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

   11. Applied egg-rr 17.3%

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

   13. Applied egg-rr 27.0%

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

       1. associate-/r/ 27.0%

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

   15. Simplified 27.0%

       \[\leadsto \color{blue}{\frac{a2}{-2} \cdot a2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 7.8 \cdot 10^{+76}:\\ \;\;\;\;a2 \cdot \left(th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{-2}\\ \end{array} \]

Alternative 15: 29.7% accurate, 69.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

def code(a1, a2, th):
	return a2 * (a2 * -(-0.5))

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * (--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. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.2%

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

5. Taylor expanded in a1 around 0 40.4%

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

   1. unpow2 40.4%

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

   2. associate-*r/ 40.3%

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

7. Simplified 40.3%

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

   1. frac-2neg 40.3%

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

   2. div-inv 40.3%

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

9. Applied egg-rr 40.3%

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

11. Applied egg-rr 28.0%

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

12. Final simplification 28.0%

    \[\leadsto a2 \cdot \left(a2 \cdot \left(--0.5\right)\right) \]

Alternative 16: 27.4% accurate, 82.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;th \cdot a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 3.2e+24) (* a1 a1) (* th a2)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 3.2e+24) {
		tmp = a1 * a1;
	} else {
		tmp = th * a2;
	}
	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 (th <= 3.2d+24) then
        tmp = a1 * a1
    else
        tmp = th * a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 3.2e+24) {
		tmp = a1 * a1;
	} else {
		tmp = th * a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 3.2e+24:
		tmp = a1 * a1
	else:
		tmp = th * a2
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 3.2e+24)
		tmp = Float64(a1 * a1);
	else
		tmp = Float64(th * a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 3.2e+24)
		tmp = a1 * a1;
	else
		tmp = th * a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 3.2e+24], N[(a1 * a1), $MachinePrecision], N[(th * a2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 3.1999999999999997e24

   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 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in th around 0 73.4%

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

   5. Taylor expanded in a1 around inf 42.1%

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

      1. unpow2 42.1%

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

   7. Simplified 42.1%

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

   9. Applied egg-rr 34.5%

      \[\leadsto \color{blue}{a1 \cdot a1} \]

   if 3.1999999999999997e24 < th

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.6%

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

      2. associate-*l/ 99.5%

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

      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 59.3%

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

      1. unpow2 59.3%

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

      2. *-commutative 59.3%

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

   6. Simplified 59.3%

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

   7. Taylor expanded in th around 0 8.1%

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

      1. unpow2 8.1%

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

      2. unpow2 8.1%

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

      3. associate-*r* 8.1%

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

      4. distribute-rgt1-in 21.2%

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

      5. unpow2 21.2%

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

   9. Simplified 21.2%

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

   10. Taylor expanded in th around inf 21.2%

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

       1. unpow2 21.2%

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

       2. unpow2 21.2%

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

   12. Simplified 21.2%

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

   14. Applied egg-rr 13.5%

       \[\leadsto \color{blue}{th \cdot a2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 3.2 \cdot 10^{+24}:\\ \;\;\;\;a1 \cdot a1\\ \mathbf{else}:\\ \;\;\;\;th \cdot a2\\ \end{array} \]

Alternative 17: 17.0% accurate, 83.0× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double a1, double a2, double th) {
	return a2 * (th * a2);
}

Python

def code(a1, a2, th):
	return a2 * (th * a2)

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(th * a2), $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 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in th around 0 23.9%

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

   1. unpow2 23.9%

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

   2. unpow2 23.9%

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

   3. associate-*r* 23.9%

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

   4. distribute-rgt1-in 45.0%

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

   5. unpow2 45.0%

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

9. Simplified 45.0%

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

10. Taylor expanded in th around inf 16.1%

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

    1. unpow2 16.1%

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

    2. unpow2 16.1%

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

12. Simplified 16.1%

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

14. Applied egg-rr 17.7%

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

15. Final simplification 17.7%

    \[\leadsto a2 \cdot \left(th \cdot a2\right) \]

Alternative 18: 29.4% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 \cdot a1 \end{array} \]

FPCore

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

C

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

Java

public static double code(double a1, double a2, double th) {
	return a1 * a1;
}

Python

def code(a1, a2, th):
	return a1 * a1

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a1 * a1), $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 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.2%

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

5. Taylor expanded in a1 around inf 38.0%

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

   1. unpow2 38.0%

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

7. Simplified 38.0%

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

9. Applied egg-rr 31.7%

   \[\leadsto \color{blue}{a1 \cdot a1} \]

10. Final simplification 31.7%

    \[\leadsto a1 \cdot a1 \]

Alternative 19: 3.5% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 1.0)

C

double code(double a1, double a2, double th) {
	return 1.0;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = 1.0d0
end function

Java

public static double code(double a1, double a2, double th) {
	return 1.0;
}

Python

def code(a1, a2, th):
	return 1.0

Julia

function code(a1, a2, th)
	return 1.0
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = 1.0;
end

Wolfram

code[a1_, a2_, th_] := 1.0

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 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in th around 0 40.4%

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

   1. unpow2 40.4%

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

9. Simplified 40.4%

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

    1. div-inv 40.3%

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

    2. add-sqr-sqrt 40.3%

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

    3. sqrt-unprod 40.3%

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

    4. frac-times 40.3%

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

    5. metadata-eval 40.3%

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

    6. add-sqr-sqrt 40.4%

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

    7. metadata-eval 40.4%

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

11. Applied egg-rr 40.4%

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

13. Applied egg-rr 3.4%

    \[\leadsto \color{blue}{\frac{a2}{a2}} \]
14. Step-by-step derivation

    1. *-inverses 3.4%

       \[\leadsto \color{blue}{1} \]

15. Simplified 3.4%

    \[\leadsto \color{blue}{1} \]

16. Final simplification 3.4%

    \[\leadsto 1 \]

Alternative 20: 3.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ a1 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 a1)

C

double code(double a1, double a2, double th) {
	return 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 = a1
end function

Java

public static double code(double a1, double a2, double th) {
	return a1;
}

Python

def code(a1, a2, th):
	return a1

Julia

function code(a1, a2, th)
	return a1
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a1

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 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in th around 0 65.2%

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

5. Taylor expanded in a1 around inf 38.0%

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

   1. unpow2 38.0%

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

7. Simplified 38.0%

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

9. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{\left|a1\right|} \]
10. Step-by-step derivation

    1. unpow1 4.7%

       \[\leadsto \left|\color{blue}{{a1}^{1}}\right| \]

    2. *-inverses 4.7%

       \[\leadsto \left|{a1}^{\color{blue}{\left(\frac{a2}{a2}\right)}}\right| \]

    3. sqr-pow 2.3%

       \[\leadsto \left|\color{blue}{{a1}^{\left(\frac{\frac{a2}{a2}}{2}\right)} \cdot {a1}^{\left(\frac{\frac{a2}{a2}}{2}\right)}}\right| \]

    4. fabs-sqr 2.3%

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

    5. sqr-pow 3.5%

       \[\leadsto \color{blue}{{a1}^{\left(\frac{a2}{a2}\right)}} \]

    6. *-inverses 3.5%

       \[\leadsto {a1}^{\color{blue}{1}} \]

    7. unpow1 3.5%

       \[\leadsto \color{blue}{a1} \]

11. Simplified 3.5%

    \[\leadsto \color{blue}{a1} \]

12. Final simplification 3.5%

    \[\leadsto a1 \]

Alternative 21: 3.6% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ a2 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 a2)

C

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

Java

public static double code(double a1, double a2, double th) {
	return a2;
}

Python

def code(a1, a2, th):
	return a2

Julia

function code(a1, a2, th)
	return a2
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := a2

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 99.6%

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

   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/ 99.7%

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

   4. fma-def 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

   2. *-commutative 60.6%

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

6. Simplified 60.6%

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

7. Taylor expanded in th around 0 40.4%

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

   1. unpow2 40.4%

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

9. Simplified 40.4%

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

    1. div-inv 40.3%

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

    2. add-sqr-sqrt 40.3%

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

    3. sqrt-unprod 40.3%

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

    4. frac-times 40.3%

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

    5. metadata-eval 40.3%

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

    6. add-sqr-sqrt 40.4%

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

    7. metadata-eval 40.4%

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

11. Applied egg-rr 40.4%

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

13. Applied egg-rr 3.5%

    \[\leadsto \color{blue}{0 + a2} \]
14. Step-by-step derivation

    1. +-lft-identity 3.5%

       \[\leadsto \color{blue}{a2} \]

15. Simplified 3.5%

    \[\leadsto \color{blue}{a2} \]

16. Final simplification 3.5%

    \[\leadsto a2 \]

Reproduce

herbie shell --seed 2023192 
(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))))