Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 7.9s

Alternatives: 11

Speedup: 2.0×

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

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t\_1 \cdot \left(a1 \cdot a1\right) + t\_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 2.0× speedup

Math

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

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(Float64(sqrt(0.5) * 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[(N[Sqrt[0.5], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)} \]
4. Add Preprocessing
5. 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/ 99.5%

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

   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%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in th around inf 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

Alternative 2: 42.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-122)
     (/ (- 1.0 (/ a2 a1)) a1)
     (* a2 (/ a2 (sqrt 2.0))))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} 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) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= (-2d-122)) then
        tmp = (1.0d0 - (a2 / a1)) / a1
    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 t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122:
		tmp = (1.0 - (a2 / a1)) / a1
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-122)
		tmp = Float64(Float64(1.0 - Float64(a2 / a1)) / a1);
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122)
		tmp = (1.0 - (a2 / a1)) / a1;
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-122], N[(N[(1.0 - N[(a2 / a1), $MachinePrecision]), $MachinePrecision] / a1), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.00000000000000012e-122

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 2.2%

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

      1. associate-/r* 2.2%

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

      2. *-inverses 2.2%

         \[\leadsto \frac{\color{blue}{1}}{a2 + a1} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\frac{1}{a2 + a1}} \]

   11. Taylor expanded in a1 around inf 12.0%

       \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{a2}{a1}}{a1}} \]
   12. Step-by-step derivation

       1. mul-1-neg 12.0%

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

       2. unsub-neg 12.0%

          \[\leadsto \frac{\color{blue}{1 - \frac{a2}{a1}}}{a1} \]

   13. Simplified 12.0%

       \[\leadsto \color{blue}{\frac{1 - \frac{a2}{a1}}{a1}} \]

   if -2.00000000000000012e-122 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing

   5. Taylor expanded in th around 0 83.0%

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

   6. Taylor expanded in a1 around 0 49.4%

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

      1. pow2 49.4%

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

      2. associate-/l* 49.5%

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

   8. Applied egg-rr 49.5%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.75\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-122)
     (/ (- 1.0 (/ a2 a1)) a1)
     (* (+ (* a1 a1) (* a2 a2)) 0.75))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * 0.75;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= (-2d-122)) then
        tmp = (1.0d0 - (a2 / a1)) / a1
    else
        tmp = ((a1 * a1) + (a2 * a2)) * 0.75d0
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * 0.75;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122:
		tmp = (1.0 - (a2 / a1)) / a1
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * 0.75
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-122)
		tmp = Float64(Float64(1.0 - Float64(a2 / a1)) / a1);
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * 0.75);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122)
		tmp = (1.0 - (a2 / a1)) / a1;
	else
		tmp = ((a1 * a1) + (a2 * a2)) * 0.75;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-122], N[(N[(1.0 - N[(a2 / a1), $MachinePrecision]), $MachinePrecision] / a1), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * 0.75), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.00000000000000012e-122

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 2.2%

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

      1. associate-/r* 2.2%

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

      2. *-inverses 2.2%

         \[\leadsto \frac{\color{blue}{1}}{a2 + a1} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\frac{1}{a2 + a1}} \]

   11. Taylor expanded in a1 around inf 12.0%

       \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{a2}{a1}}{a1}} \]
   12. Step-by-step derivation

       1. mul-1-neg 12.0%

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

       2. unsub-neg 12.0%

          \[\leadsto \frac{\color{blue}{1 - \frac{a2}{a1}}}{a1} \]

   13. Simplified 12.0%

       \[\leadsto \color{blue}{\frac{1 - \frac{a2}{a1}}{a1}} \]

   if -2.00000000000000012e-122 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 60.8%

      \[\leadsto \left(\color{blue}{0.75} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. Taylor expanded in th around 0 59.4%

      \[\leadsto \color{blue}{0.75} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.75\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 25.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot a1 + a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) -2e-122)
     (/ (- 1.0 (/ a2 a1)) a1)
     (+ (* a1 a1) a2))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} else {
		tmp = (a1 * a1) + 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) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= (-2d-122)) then
        tmp = (1.0d0 - (a2 / a1)) / a1
    else
        tmp = (a1 * a1) + a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122) {
		tmp = (1.0 - (a2 / a1)) / a1;
	} else {
		tmp = (a1 * a1) + a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122:
		tmp = (1.0 - (a2 / a1)) / a1
	else:
		tmp = (a1 * a1) + a2
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= -2e-122)
		tmp = Float64(Float64(1.0 - Float64(a2 / a1)) / a1);
	else
		tmp = Float64(Float64(a1 * a1) + a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= -2e-122)
		tmp = (1.0 - (a2 / a1)) / a1;
	else
		tmp = (a1 * a1) + a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], -2e-122], N[(N[(1.0 - N[(a2 / a1), $MachinePrecision]), $MachinePrecision] / a1), $MachinePrecision], N[(N[(a1 * a1), $MachinePrecision] + a2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < -2.00000000000000012e-122

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 2.2%

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

      1. associate-/r* 2.2%

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

      2. *-inverses 2.2%

         \[\leadsto \frac{\color{blue}{1}}{a2 + a1} \]

   10. Simplified 2.2%

       \[\leadsto \color{blue}{\frac{1}{a2 + a1}} \]

   11. Taylor expanded in a1 around inf 12.0%

       \[\leadsto \color{blue}{\frac{1 + -1 \cdot \frac{a2}{a1}}{a1}} \]
   12. Step-by-step derivation

       1. mul-1-neg 12.0%

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

       2. unsub-neg 12.0%

          \[\leadsto \frac{\color{blue}{1 - \frac{a2}{a1}}}{a1} \]

   13. Simplified 12.0%

       \[\leadsto \color{blue}{\frac{1 - \frac{a2}{a1}}{a1}} \]

   if -2.00000000000000012e-122 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.7%

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

   8. Applied egg-rr 57.8%

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

      1. *-inverses 57.8%

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

   10. Simplified 57.8%

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

   12. Applied egg-rr 39.2%

       \[\leadsto 1 \cdot \left(a1 \cdot a1 + \color{blue}{\log \left(e^{a2}\right)}\right) \]
   13. Step-by-step derivation

       1. rem-log-exp 31.2%

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

   14. Simplified 31.2%

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

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq -2 \cdot 10^{-122}:\\ \;\;\;\;\frac{1 - \frac{a2}{a1}}{a1}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot a1 + a2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 24.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq 0:\\ \;\;\;\;\frac{a1 + a2}{-2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot a1 + a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) 0.0)
     (/ (+ a1 a2) -2.0)
     (+ (* a1 a1) a2))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0) {
		tmp = (a1 + a2) / -2.0;
	} else {
		tmp = (a1 * a1) + 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) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0d0) then
        tmp = (a1 + a2) / (-2.0d0)
    else
        tmp = (a1 * a1) + a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0) {
		tmp = (a1 + a2) / -2.0;
	} else {
		tmp = (a1 * a1) + a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0:
		tmp = (a1 + a2) / -2.0
	else:
		tmp = (a1 * a1) + a2
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= 0.0)
		tmp = Float64(Float64(a1 + a2) / -2.0);
	else
		tmp = Float64(Float64(a1 * a1) + a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0)
		tmp = (a1 + a2) / -2.0;
	else
		tmp = (a1 * a1) + a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(a1 + a2), $MachinePrecision] / -2.0), $MachinePrecision], N[(N[(a1 * a1), $MachinePrecision] + a2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < 0.0

   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. Add Preprocessing

   5. Taylor expanded in th around 0 17.8%

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

   7. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\frac{a2 + a1}{-2}} \]

   if 0.0 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 56.6%

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

      1. *-inverses 56.6%

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

   10. Simplified 56.6%

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

   12. Applied egg-rr 35.7%

       \[\leadsto 1 \cdot \left(a1 \cdot a1 + \color{blue}{\log \left(e^{a2}\right)}\right) \]
   13. Step-by-step derivation

       1. rem-log-exp 34.3%

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

   14. Simplified 34.3%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq 0:\\ \;\;\;\;\frac{a1 + a2}{-2}\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot a1 + a2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 4.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ \mathbf{if}\;\left(a1 \cdot a1\right) \cdot t\_1 + \left(a2 \cdot a2\right) \cdot t\_1 \leq 0:\\ \;\;\;\;\frac{a1 + a2}{-2}\\ \mathbf{else}:\\ \;\;\;\;a1 + a2\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (if (<= (+ (* (* a1 a1) t_1) (* (* a2 a2) t_1)) 0.0)
     (/ (+ a1 a2) -2.0)
     (+ a1 a2))))

C

double code(double a1, double a2, double th) {
	double t_1 = cos(th) / sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0) {
		tmp = (a1 + a2) / -2.0;
	} else {
		tmp = a1 + 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) :: t_1
    real(8) :: tmp
    t_1 = cos(th) / sqrt(2.0d0)
    if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0d0) then
        tmp = (a1 + a2) / (-2.0d0)
    else
        tmp = a1 + a2
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = Math.cos(th) / Math.sqrt(2.0);
	double tmp;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0) {
		tmp = (a1 + a2) / -2.0;
	} else {
		tmp = a1 + a2;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = math.cos(th) / math.sqrt(2.0)
	tmp = 0
	if (((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0:
		tmp = (a1 + a2) / -2.0
	else:
		tmp = a1 + a2
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	tmp = 0.0
	if (Float64(Float64(Float64(a1 * a1) * t_1) + Float64(Float64(a2 * a2) * t_1)) <= 0.0)
		tmp = Float64(Float64(a1 + a2) / -2.0);
	else
		tmp = Float64(a1 + a2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = cos(th) / sqrt(2.0);
	tmp = 0.0;
	if ((((a1 * a1) * t_1) + ((a2 * a2) * t_1)) <= 0.0)
		tmp = (a1 + a2) / -2.0;
	else
		tmp = a1 + a2;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[Cos[th], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(a1 * a1), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[(a2 * a2), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(a1 + a2), $MachinePrecision] / -2.0), $MachinePrecision], N[(a1 + a2), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2))) < 0.0

   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. Add Preprocessing

   5. Taylor expanded in th around 0 17.8%

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

   7. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\frac{a2 + a1}{-2}} \]

   if 0.0 < (+.f64 (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a1 a1)) (*.f64 (/.f64 (cos.f64 th) (sqrt.f64 #s(literal 2 binary64))) (*.f64 a2 a2)))

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. cos-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. associate-/l* 99.5%

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

      5. cos-neg 99.5%

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

      6. +-commutative 99.5%

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

      7. fma-define 99.5%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. add-sqr-sqrt 99.5%

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

      4. pow2 99.5%

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

      5. fma-undefine 99.5%

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

      6. hypot-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 2.8%

      \[\leadsto \color{blue}{e^{\log \left(a2 + a1\right) - \log \cos th}} \]
   9. Step-by-step derivation

      1. exp-diff 2.8%

         \[\leadsto \color{blue}{\frac{e^{\log \left(a2 + a1\right)}}{e^{\log \cos th}}} \]

      2. rem-exp-log 3.4%

         \[\leadsto \frac{\color{blue}{a2 + a1}}{e^{\log \cos th}} \]

      3. rem-exp-log 3.4%

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

   10. Simplified 3.4%

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

   11. Taylor expanded in th around 0 3.4%

       \[\leadsto \color{blue}{a1 + a2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 3.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(a1 \cdot a1\right) \cdot \frac{\cos th}{\sqrt{2}} + \left(a2 \cdot a2\right) \cdot \frac{\cos th}{\sqrt{2}} \leq 0:\\ \;\;\;\;\frac{a1 + a2}{-2}\\ \mathbf{else}:\\ \;\;\;\;a1 + a2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.87:\\ \;\;\;\;t\_1 \cdot \left(\cos th \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.87) (* t_1 (* (cos th) 0.75)) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.87) {
		tmp = t_1 * (cos(th) * 0.75);
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.87d0) then
        tmp = t_1 * (cos(th) * 0.75d0)
    else
        tmp = sqrt(0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (Math.cos(th) <= 0.87) {
		tmp = t_1 * (Math.cos(th) * 0.75);
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.87:
		tmp = t_1 * (math.cos(th) * 0.75)
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.87)
		tmp = Float64(t_1 * Float64(cos(th) * 0.75));
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.87)
		tmp = t_1 * (cos(th) * 0.75);
	else
		tmp = sqrt(0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.87], N[(t$95$1 * N[(N[Cos[th], $MachinePrecision] * 0.75), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.869999999999999996

   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. Add Preprocessing
   5. 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/ 99.5%

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

      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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 62.8%

      \[\leadsto \left(\color{blue}{0.75} \cdot \cos th\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 0.869999999999999996 < (cos.f64 th)

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 94.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.87:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot 0.75\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) 0.68) (* (cos th) t_1) (* (sqrt 0.5) t_1))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = cos(th) * t_1;
	} else {
		tmp = sqrt(0.5) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (cos(th) <= 0.68d0) then
        tmp = cos(th) * t_1
    else
        tmp = sqrt(0.5d0) * t_1
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (Math.cos(th) <= 0.68) {
		tmp = Math.cos(th) * t_1;
	} else {
		tmp = Math.sqrt(0.5) * t_1;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = math.cos(th) * t_1
	else:
		tmp = math.sqrt(0.5) * t_1
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(cos(th) * t_1);
	else
		tmp = Float64(sqrt(0.5) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (cos(th) <= 0.68)
		tmp = cos(th) * t_1;
	else
		tmp = sqrt(0.5) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Cos[th], $MachinePrecision], 0.68], N[(N[Cos[th], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sqrt[0.5], $MachinePrecision] * t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   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. Add Preprocessing
   5. 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/ 99.5%

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

      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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 63.2%

      \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   9. Step-by-step derivation

      1. +-lft-identity 63.2%

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

   10. Simplified 63.2%

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

   if 0.680000000000000049 < (cos.f64 th)

   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. Add Preprocessing
   5. 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%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in th around 0 90.2%

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

Alternative 9: 55.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.615:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.615)
   (* (cos th) (+ (* a1 a1) (* a2 a2)))
   (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.615) {
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	} 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 (cos(th) <= 0.615d0) then
        tmp = cos(th) * ((a1 * a1) + (a2 * a2))
    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 (Math.cos(th) <= 0.615) {
		tmp = Math.cos(th) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.615:
		tmp = math.cos(th) * ((a1 * a1) + (a2 * a2))
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.615)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	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 (cos(th) <= 0.615)
		tmp = cos(th) * ((a1 * a1) + (a2 * a2));
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.615], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.61499999999999999

   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. Add Preprocessing
   5. 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/ 99.5%

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

      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%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 63.6%

      \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
   9. Step-by-step derivation

      1. +-lft-identity 63.6%

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

   10. Simplified 63.6%

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

   if 0.61499999999999999 < (cos.f64 th)

   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. Add Preprocessing

   5. Taylor expanded in th around 0 89.2%

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

   6. Taylor expanded in a1 around 0 50.6%

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

      1. pow2 50.6%

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

      2. associate-/l* 50.6%

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

   8. Applied egg-rr 50.6%

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

Alternative 10: 47.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.615:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.615)
   (* (+ a1 a2) (* (cos th) a2))
   (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.615) {
		tmp = (a1 + a2) * (cos(th) * a2);
	} 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 (cos(th) <= 0.615d0) then
        tmp = (a1 + a2) * (cos(th) * a2)
    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 (Math.cos(th) <= 0.615) {
		tmp = (a1 + a2) * (Math.cos(th) * a2);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.615:
		tmp = (a1 + a2) * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.615)
		tmp = Float64(Float64(a1 + a2) * Float64(cos(th) * a2));
	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 (cos(th) <= 0.615)
		tmp = (a1 + a2) * (cos(th) * a2);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.615], N[(N[(a1 + a2), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.61499999999999999

   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. cos-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-/l* 99.6%

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

      5. cos-neg 99.6%

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

      6. +-commutative 99.6%

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

      7. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. add-sqr-sqrt 99.6%

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

      4. pow2 99.6%

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

      5. fma-undefine 99.6%

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

      6. hypot-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   8. Applied egg-rr 63.6%

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

   9. Taylor expanded in a2 around inf 44.0%

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

   if 0.61499999999999999 < (cos.f64 th)

   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. Add Preprocessing

   5. Taylor expanded in th around 0 89.2%

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

   6. Taylor expanded in a1 around 0 50.6%

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

      1. pow2 50.6%

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

      2. associate-/l* 50.6%

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

   8. Applied egg-rr 50.6%

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

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.615:\\ \;\;\;\;\left(a1 + a2\right) \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 4.1% accurate, 138.3× speedup

Math

\[\begin{array}{l} \\ a1 + a2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out 99.5%

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

   2. cos-neg 99.5%

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

   3. associate-*l/ 99.6%

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

   4. associate-/l* 99.5%

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

   5. cos-neg 99.5%

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

   6. +-commutative 99.5%

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

   7. fma-define 99.5%

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

3. Simplified 99.5%

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

   1. clear-num 99.5%

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

   2. un-div-inv 99.6%

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

   3. add-sqr-sqrt 99.6%

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

   4. pow2 99.6%

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

   5. fma-undefine 99.6%

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

   6. hypot-define 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 2.0%

   \[\leadsto \color{blue}{e^{\log \left(a2 + a1\right) - \log \cos th}} \]
9. Step-by-step derivation

   1. exp-diff 2.0%

      \[\leadsto \color{blue}{\frac{e^{\log \left(a2 + a1\right)}}{e^{\log \cos th}}} \]

   2. rem-exp-log 2.5%

      \[\leadsto \frac{\color{blue}{a2 + a1}}{e^{\log \cos th}} \]

   3. rem-exp-log 3.7%

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

10. Simplified 3.7%

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

11. Taylor expanded in th around 0 3.7%

    \[\leadsto \color{blue}{a1 + a2} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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))))