Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 7.8s

Alternatives: 11

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. Add Preprocessing

Alternative 2: 79.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* (cos th) (* (+ a2 a1) (+ a2 a1)))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	}
	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) <= 0.7d0) then
        tmp = cos(th) * ((a2 + a1) * (a2 + a1))
    else
        tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = Math.cos(th) * ((a2 + a1) * (a2 + a1));
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = math.cos(th) * ((a2 + a1) * (a2 + a1))
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(cos(th) * Float64(Float64(a2 + a1) * Float64(a2 + a1)));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * sqrt(0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = cos(th) * ((a2 + a1) * (a2 + a1));
	else
		tmp = ((a1 * a1) + (a2 * a2)) * sqrt(0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(N[Cos[th], $MachinePrecision] * N[(N[(a2 + a1), $MachinePrecision] * N[(a2 + a1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.69999999999999996

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

      2. cos-neg 99.7%

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

      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* 99.7%

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

      5. cos-neg 99.7%

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

      6. +-commutative 99.7%

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

      7. fma-define 99.7%

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

   3. Simplified 99.7%

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

      1. div-inv 99.6%

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

      2. add-sqr-sqrt 99.6%

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

      3. associate-*l* 99.5%

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

      4. fma-undefine 99.5%

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

      5. hypot-define 99.5%

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

      6. fma-undefine 99.5%

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

      7. hypot-define 99.5%

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

      8. pow1/2 99.5%

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

      9. pow-flip 99.5%

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

      10. metadata-eval 99.5%

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

   6. Applied egg-rr 99.5%

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

   8. Applied egg-rr 59.1%

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

      1. +-commutative 59.1%

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

      2. distribute-lft-out 64.4%

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

      3. +-commutative 64.4%

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

   10. Simplified 64.4%

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

   if 0.69999999999999996 < (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 99.6%

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

      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 0 92.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;\cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

Alternative 4: 99.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

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

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. Final simplification 99.6%

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

Alternative 5: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. Taylor expanded in a2 around inf 54.2%

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

   1. pow2 54.2%

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

7. Applied egg-rr 54.2%

   \[\leadsto \cos th \cdot \frac{\color{blue}{a2 \cdot a2}}{\sqrt{2}} \]
8. Add Preprocessing

Alternative 6: 58.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. div-inv 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. associate-*l* 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. fma-undefine 99.5%

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

   7. hypot-define 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.6%

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in a2 around inf 32.4%

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

   1. *-commutative 32.4%

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

9. Simplified 32.4%

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

10. Taylor expanded in a2 around inf 54.3%

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

11. Final simplification 54.3%

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

Alternative 7: 59.8% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   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. div-inv 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. associate-*l* 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. fma-undefine 99.5%

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

   7. hypot-define 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.6%

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 55.7%

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

   1. +-commutative 55.7%

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

   2. distribute-lft-out 62.0%

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

   3. +-commutative 62.0%

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

10. Simplified 62.0%

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

11. Final simplification 62.0%

    \[\leadsto \cos th \cdot \left(\left(a2 + a1\right) \cdot \left(a2 + a1\right)\right) \]
12. Add Preprocessing

Alternative 8: 46.6% accurate, 46.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 67.1%

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

7. Applied egg-rr 47.3%

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

8. Final simplification 47.3%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5 \]
9. Add Preprocessing

Alternative 9: 45.8% accurate, 46.1× speedup

Math

\[\begin{array}{l} \\ \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25 \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (* (+ (* a1 a1) (* a2 a2)) 0.25))

C

double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * 0.25;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = ((a1 * a1) + (a2 * a2)) * 0.25d0
end function

Java

public static double code(double a1, double a2, double th) {
	return ((a1 * a1) + (a2 * a2)) * 0.25;
}

Python

def code(a1, a2, th):
	return ((a1 * a1) + (a2 * a2)) * 0.25

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * 0.25)
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((a1 * a1) + (a2 * a2)) * 0.25;
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * 0.25), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 67.1%

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

7. Applied egg-rr 46.5%

   \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Final simplification 46.5%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25 \]
9. Add Preprocessing

Alternative 10: 20.4% accurate, 46.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in th around 0 67.1%

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

7. Applied egg-rr 19.8%

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

8. Final simplification 19.8%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5 \]
9. Add Preprocessing

Alternative 11: 3.5% accurate, 415.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. distribute-lft-out 99.6%

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

   2. 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. div-inv 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. associate-*l* 99.5%

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

   4. fma-undefine 99.5%

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

   5. hypot-define 99.5%

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

   6. fma-undefine 99.5%

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

   7. hypot-define 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-flip 99.6%

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

   10. metadata-eval 99.6%

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

6. Applied egg-rr 99.6%

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

8. Applied egg-rr 3.4%

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

   1. *-inverses 3.4%

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

10. Simplified 3.4%

    \[\leadsto \color{blue}{1} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024125 
(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))))