Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 12.2s

Alternatives: 7

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 99.7

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

7. Simplified 99.7%

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

Alternative 2: 58.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot {2}^{-0.5}\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (cos th) (* (* a2 a2) (pow 2.0 -0.5))))

C

double code(double a1, double a2, double th) {
	return cos(th) * ((a2 * a2) * pow(2.0, -0.5));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = cos(th) * ((a2 * a2) * (2.0d0 ** (-0.5d0)))
end function

Java

public static double code(double a1, double a2, double th) {
	return Math.cos(th) * ((a2 * a2) * Math.pow(2.0, -0.5));
}

Python

def code(a1, a2, th):
	return math.cos(th) * ((a2 * a2) * math.pow(2.0, -0.5))

Julia

function code(a1, a2, th)
	return Float64(cos(th) * Float64(Float64(a2 * a2) * (2.0 ^ -0.5)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = cos(th) * ((a2 * a2) * (2.0 ^ -0.5));
end

Wolfram

code[a1_, a2_, th_] := N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 * a2), $MachinePrecision] * N[Power[2.0, -0.5], $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. Add Preprocessing

3. Taylor expanded in a1 around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 58.9

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

5. Simplified 58.9%

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

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. pow1/2 N/A

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

   4. pow-flip N/A

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

   5. metadata-eval N/A

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

   6. associate-*r* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. pow-lowering-pow.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 59.0

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

7. Applied egg-rr 59.0%

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

8. Final simplification 59.0%

   \[\leadsto \cos th \cdot \left(\left(a2 \cdot a2\right) \cdot {2}^{-0.5}\right) \]
9. Add Preprocessing

Alternative 3: 58.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around inf

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

   1. *-commutative N/A

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

   2. associate-*l* N/A

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

   3. *-commutative N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 N/A

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

   8. +-lowering-+.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. cos-lowering-cos.f64 99.7

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

7. Simplified 99.7%

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

8. Taylor expanded in a1 around 0

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

   1. unpow2 N/A

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

   2. *-lowering-*.f64 59.0

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

10. Simplified 59.0%

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

Alternative 4: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in a1 around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 N/A

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

   5. unpow2 N/A

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

   6. associate-*l* N/A

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

   7. *-lowering-*.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. cos-lowering-cos.f64 59.0

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

7. Simplified 59.0%

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

Alternative 5: 64.6% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 4.3 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{\sqrt{2}}{a2}}{0 - a2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 4.3e+100)
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))
   (/ -1.0 (/ (/ (sqrt 2.0) a2) (- 0.0 a2)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.3e+100) {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = -1.0 / ((sqrt(2.0) / a2) / (0.0 - 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 <= 4.3d+100) then
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    else
        tmp = (-1.0d0) / ((sqrt(2.0d0) / a2) / (0.0d0 - a2))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 4.3e+100) {
		tmp = Math.sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	} else {
		tmp = -1.0 / ((Math.sqrt(2.0) / a2) / (0.0 - a2));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 4.3e+100:
		tmp = math.sqrt(0.5) * ((a1 * a1) + (a2 * a2))
	else:
		tmp = -1.0 / ((math.sqrt(2.0) / a2) / (0.0 - a2))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 4.3e+100)
		tmp = Float64(sqrt(0.5) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	else
		tmp = Float64(-1.0 / Float64(Float64(sqrt(2.0) / a2) / Float64(0.0 - a2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 4.3e+100)
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	else
		tmp = -1.0 / ((sqrt(2.0) / a2) / (0.0 - a2));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 4.3e+100], N[(N[Sqrt[0.5], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(N[Sqrt[2.0], $MachinePrecision] / a2), $MachinePrecision] / N[(0.0 - a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 4.29999999999999993e100

   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. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-lft-out N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. associate-*l* N/A

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

      5. *-lowering-*.f64 N/A

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

      6. pow1/2 N/A

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

      7. pow-flip N/A

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

      8. pow-lowering-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 N/A

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

      11. cos-lowering-cos.f64 N/A

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

      12. +-lowering-+.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. *-lowering-*.f64 99.7

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in th around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. +-lowering-+.f64 N/A

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 71.3

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

   7. Simplified 71.3%

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

   if 4.29999999999999993e100 < 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. Add Preprocessing

   3. Taylor expanded in a1 around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. unpow2 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 63.3

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

   5. Simplified 63.3%

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

   6. Taylor expanded in th around 0

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

      1. /-lowering-/.f64 N/A

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

      2. unpow2 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 20.8

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

   8. Simplified 20.8%

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

      1. clear-num N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. /-lowering-/.f64 N/A

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

      5. neg-sub0 N/A

         \[\leadsto \frac{-1}{\color{blue}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]

      6. --lowering--.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{0 - \frac{\sqrt{2}}{a2 \cdot a2}}} \]

      7. associate-/r* N/A

         \[\leadsto \frac{-1}{0 - \color{blue}{\frac{\frac{\sqrt{2}}{a2}}{a2}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{0 - \color{blue}{\frac{\frac{\sqrt{2}}{a2}}{a2}}} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{-1}{0 - \frac{\color{blue}{\frac{\sqrt{2}}{a2}}}{a2}} \]

      10. sqrt-lowering-sqrt.f64 27.2

          \[\leadsto \frac{-1}{0 - \frac{\frac{\color{blue}{\sqrt{2}}}{a2}}{a2}} \]

   10. Applied egg-rr 27.2%

       \[\leadsto \color{blue}{\frac{-1}{0 - \frac{\frac{\sqrt{2}}{a2}}{a2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 4.3 \cdot 10^{+100}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\frac{\sqrt{2}}{a2}}{0 - a2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. distribute-lft-out N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l* N/A

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

   5. *-lowering-*.f64 N/A

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

   6. pow1/2 N/A

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

   7. pow-flip N/A

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

   8. pow-lowering-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. *-lowering-*.f64 N/A

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

   11. cos-lowering-cos.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. *-lowering-*.f64 99.7

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

4. Applied egg-rr 99.7%

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

5. Taylor expanded in th around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

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

   3. +-lowering-+.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. unpow2 N/A

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

   7. *-lowering-*.f64 63.4

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

7. Simplified 63.4%

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

Alternative 7: 40.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in a1 around 0

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

   1. /-lowering-/.f64 N/A

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

   2. *-lowering-*.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. cos-lowering-cos.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 58.9

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

5. Simplified 58.9%

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

6. Taylor expanded in th around 0

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

   1. /-lowering-/.f64 N/A

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

   2. unpow2 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. sqrt-lowering-sqrt.f64 37.9

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

8. Simplified 37.9%

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

   1. /-rgt-identity N/A

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

   2. clear-num N/A

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

   3. /-lowering-/.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-flip N/A

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

   6. pow-lowering-pow.f64 N/A

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

   7. metadata-eval 37.9

      \[\leadsto \frac{a2 \cdot a2}{\frac{1}{{2}^{\color{blue}{-0.5}}}} \]

10. Applied egg-rr 37.9%

    \[\leadsto \frac{a2 \cdot a2}{\color{blue}{\frac{1}{{2}^{-0.5}}}} \]

11. Taylor expanded in a2 around 0

    \[\leadsto \color{blue}{{a2}^{2} \cdot \sqrt{\frac{1}{2}}} \]
12. Step-by-step derivation

    1. *-lowering-*.f64 N/A

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 N/A

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

    4. sqrt-lowering-sqrt.f64 37.9

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

13. Simplified 37.9%

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

14. Final simplification 37.9%

    \[\leadsto \sqrt{0.5} \cdot \left(a2 \cdot a2\right) \]
15. 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))))