Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 8.7s

Alternatives: 12

Speedup: 2.0×

• 44.1% 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 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. associate-/l* 99.4%

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

   5. cos-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Final simplification 99.4%

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

Alternative 2: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.43:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.43)
   (* (+ (* a1 a1) (* a2 a2)) (* (cos th) (- -0.5)))
   (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.43) {
		tmp = ((a1 * a1) + (a2 * a2)) * (cos(th) * -(-0.5));
	} 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) <= 0.43d0) then
        tmp = ((a1 * a1) + (a2 * a2)) * (cos(th) * -(-0.5d0))
    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) <= 0.43) {
		tmp = ((a1 * a1) + (a2 * a2)) * (Math.cos(th) * -(-0.5));
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.43:
		tmp = ((a1 * a1) + (a2 * a2)) * (math.cos(th) * -(-0.5))
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.43)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * Float64(cos(th) * Float64(-(-0.5))));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.43)
		tmp = ((a1 * a1) + (a2 * a2)) * (cos(th) * -(-0.5));
	else
		tmp = (a2 * a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.43], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[th], $MachinePrecision] * (--0.5)), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.429999999999999993

   1. Initial program 98.5%

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

      1. distribute-lft-out 98.5%

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

   3. Simplified 98.5%

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

      1. frac-2neg 98.5%

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

      2. div-inv 98.4%

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

   6. Applied egg-rr 98.4%

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

   8. Applied egg-rr 56.1%

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

   if 0.429999999999999993 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 93.7%

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

   6. Taylor expanded in a1 around 0 54.8%

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

      1. pow2 54.8%

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

   8. Applied egg-rr 54.8%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.43:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \left(\cos th \cdot \left(--0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.71)
   (* (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) <= 0.71) {
		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) <= 0.71d0) 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) <= 0.71) {
		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) <= 0.71:
		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) <= 0.71)
		tmp = Float64(cos(th) * Float64(Float64(a1 * a1) + Float64(a2 * a2)));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.71)
		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], 0.71], N[(N[Cos[th], $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.70999999999999996

   1. Initial program 98.7%

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

      1. distribute-lft-out 98.7%

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

   3. Simplified 98.7%

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

      1. frac-2neg 98.7%

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

      2. div-inv 98.6%

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

   6. Applied egg-rr 98.6%

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

   8. Applied egg-rr 60.1%

      \[\leadsto \color{blue}{\left(0 + \cos th\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 0.70999999999999996 < (cos.f64 th)

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in th around 0 93.8%

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

   6. Taylor expanded in a1 around 0 54.9%

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

      1. pow2 54.9%

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

   8. Applied egg-rr 54.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]
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 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.3%

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

   2. associate-/r/ 99.3%

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

   3. pow1/2 99.3%

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

   4. pow-flip 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

7. Taylor expanded in th around inf 99.2%

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

   1. *-commutative 99.2%

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

9. Simplified 99.2%

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

10. Final simplification 99.2%

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

Alternative 5: 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 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 6: 39.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 69.0%

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

6. Taylor expanded in a1 around 0 41.8%

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

   1. pow2 41.8%

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

   2. div-inv 41.7%

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

   3. pow1/2 41.7%

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

   4. pow-flip 41.7%

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

   5. metadata-eval 41.7%

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

   6. associate-*l* 41.7%

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

   7. add-sqr-sqrt 41.6%

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

   8. sqrt-unprod 41.7%

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

   9. pow-prod-up 41.7%

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

   10. metadata-eval 41.7%

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

   11. metadata-eval 41.7%

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

8. Applied egg-rr 41.7%

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

9. Final simplification 41.7%

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

Alternative 7: 40.0% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 69.0%

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

6. Taylor expanded in a1 around 0 41.8%

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

   1. pow2 41.8%

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

   2. associate-/l* 41.7%

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

8. Applied egg-rr 41.7%

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

9. Final simplification 41.7%

   \[\leadsto a2 \cdot \frac{a2}{\sqrt{2}} \]
10. Add Preprocessing

Alternative 8: 39.9% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 69.0%

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

6. Taylor expanded in a1 around 0 41.8%

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

   1. pow2 41.8%

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

8. Applied egg-rr 41.8%

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

9. Final simplification 41.8%

   \[\leadsto \frac{a2 \cdot a2}{\sqrt{2}} \]
10. Add Preprocessing

Alternative 9: 30.2% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ {a2}^{2} \end{array} \]

FPCore

(FPCore (a1 a2 th) :precision binary64 (pow a2 2.0))

C

double code(double a1, double a2, double th) {
	return pow(a2, 2.0);
}

Fortran

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return a2 ^ 2.0
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[Power[a2, 2.0], $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. associate-/l* 99.4%

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

   5. cos-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in th around 0 69.1%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. rem-square-sqrt 69.1%

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

   2. +-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. unpow2 69.1%

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

   5. hypot-undefine 69.1%

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

   6. +-commutative 69.1%

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

   7. unpow2 69.1%

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

   8. unpow2 69.1%

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

   9. hypot-undefine 69.1%

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

   10. unpow2 69.1%

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

   11. hypot-undefine 69.1%

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

   12. unpow2 69.1%

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

   13. unpow2 69.1%

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

   14. +-commutative 69.1%

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

   15. unpow2 69.1%

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

   16. unpow2 69.1%

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

   17. hypot-define 69.1%

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

7. Simplified 69.1%

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

9. Applied egg-rr 41.2%

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

10. Taylor expanded in a1 around 0 31.8%

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

11. Final simplification 31.8%

    \[\leadsto {a2}^{2} \]
12. Add Preprocessing

Alternative 10: 31.1% accurate, 83.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. associate-/l* 99.4%

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

   5. cos-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in th around 0 69.1%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. rem-square-sqrt 69.1%

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

   2. +-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. unpow2 69.1%

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

   5. hypot-undefine 69.1%

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

   6. +-commutative 69.1%

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

   7. unpow2 69.1%

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

   8. unpow2 69.1%

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

   9. hypot-undefine 69.1%

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

   10. unpow2 69.1%

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

   11. hypot-undefine 69.1%

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

   12. unpow2 69.1%

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

   13. unpow2 69.1%

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

   14. +-commutative 69.1%

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

   15. unpow2 69.1%

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

   16. unpow2 69.1%

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

   17. hypot-define 69.1%

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

7. Simplified 69.1%

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

9. Applied egg-rr 41.2%

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

10. Taylor expanded in a2 around 0 32.0%

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

11. Final simplification 32.0%

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

Alternative 11: 30.5% accurate, 83.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

   2. cos-neg 99.3%

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

   3. associate-*l/ 99.3%

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

   4. associate-/l* 99.4%

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

   5. cos-neg 99.4%

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

   6. +-commutative 99.4%

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

   7. fma-define 99.4%

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

3. Simplified 99.4%

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

5. Taylor expanded in th around 0 69.1%

   \[\leadsto \color{blue}{\frac{{a1}^{2} + {a2}^{2}}{\sqrt{2}}} \]
6. Step-by-step derivation

   1. rem-square-sqrt 69.1%

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

   2. +-commutative 69.1%

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

   3. unpow2 69.1%

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

   4. unpow2 69.1%

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

   5. hypot-undefine 69.1%

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

   6. +-commutative 69.1%

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

   7. unpow2 69.1%

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

   8. unpow2 69.1%

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

   9. hypot-undefine 69.1%

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

   10. unpow2 69.1%

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

   11. hypot-undefine 69.1%

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

   12. unpow2 69.1%

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

   13. unpow2 69.1%

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

   14. +-commutative 69.1%

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

   15. unpow2 69.1%

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

   16. unpow2 69.1%

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

   17. hypot-define 69.1%

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

7. Simplified 69.1%

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

9. Applied egg-rr 41.2%

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

10. Taylor expanded in a1 around 0 31.7%

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

11. Final simplification 31.7%

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

Alternative 12: 4.1% accurate, 138.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. distribute-lft-out 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in th around 0 69.0%

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

7. Applied egg-rr 3.9%

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

8. Final simplification 3.9%

   \[\leadsto a2 + a1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024042 
(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))))