Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%

Time: 8.0s

Alternatives: 8

Speedup: 0.7×

• 44.2% 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: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(a1, a2, th)
	t_1 = Float64(cos(th) / sqrt(2.0))
	return fma(Float64(t_1 * a2), a2, Float64(t_1 * (a1 ^ 2.0)))
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 * a2), $MachinePrecision] * a2 + N[(t$95$1 * N[Power[a1, 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. Step-by-step derivation

   1. +-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. associate-*r* 99.6%

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

   4. fma-def 99.6%

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

   5. pow2 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.71) (* (cos th) (pow a2 2.0)) (* a2 (* a2 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.71) {
		tmp = cos(th) * pow(a2, 2.0);
	} else {
		tmp = 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) <= 0.71d0) then
        tmp = cos(th) * (a2 ** 2.0d0)
    else
        tmp = 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) <= 0.71) {
		tmp = Math.cos(th) * Math.pow(a2, 2.0);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.71:
		tmp = math.cos(th) * math.pow(a2, 2.0)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.71)
		tmp = Float64(cos(th) * (a2 ^ 2.0));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.71)
		tmp = cos(th) * (a2 ^ 2.0);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.71], N[(N[Cos[th], $MachinePrecision] * N[Power[a2, 2.0], $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.70999999999999996

   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. Taylor expanded in a1 around 0 59.2%

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

      1. pow2 59.2%

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

      2. *-commutative 59.2%

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

      3. associate-*r* 59.2%

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

      4. associate-*l/ 59.3%

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

      5. associate-*l/ 59.2%

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

      6. div-inv 59.2%

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

      7. associate-*l* 59.1%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-unprod 59.1%

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

      10. frac-times 59.1%

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

      11. metadata-eval 59.1%

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

      12. rem-square-sqrt 59.2%

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

      13. metadata-eval 59.2%

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

   6. Applied egg-rr 59.2%

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

   8. Applied egg-rr 42.0%

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

      1. +-lft-identity 42.0%

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

   10. Simplified 42.0%

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

   11. Taylor expanded in th around inf 42.0%

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

   if 0.70999999999999996 < (cos.f64 th)

   1. Initial program 99.2%

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

      1. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in a1 around 0 60.5%

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

      1. pow2 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*r* 60.5%

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

      4. associate-*l/ 60.9%

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

      5. associate-*l/ 60.9%

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

      6. div-inv 60.9%

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

      7. associate-*l* 60.9%

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

      8. add-sqr-sqrt 60.9%

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

      9. sqrt-unprod 60.9%

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

      10. frac-times 60.9%

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

      11. metadata-eval 60.9%

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

      12. rem-square-sqrt 61.0%

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

      13. metadata-eval 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in th around inf 61.0%

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

      1. *-commutative 61.0%

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

   9. Simplified 61.0%

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

   10. Taylor expanded in th around 0 56.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 3: 48.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a2}^{2}}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.71) (* (cos th) (pow a2 2.0)) (/ (pow a2 2.0) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.71) {
		tmp = cos(th) * pow(a2, 2.0);
	} else {
		tmp = pow(a2, 2.0) / 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) * (a2 ** 2.0d0)
    else
        tmp = (a2 ** 2.0d0) / 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) * Math.pow(a2, 2.0);
	} else {
		tmp = Math.pow(a2, 2.0) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.71:
		tmp = math.cos(th) * math.pow(a2, 2.0)
	else:
		tmp = math.pow(a2, 2.0) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.71)
		tmp = Float64(cos(th) * (a2 ^ 2.0));
	else
		tmp = Float64((a2 ^ 2.0) / 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) * (a2 ^ 2.0);
	else
		tmp = (a2 ^ 2.0) / 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[Power[a2, 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Power[a2, 2.0], $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.70999999999999996

   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. Taylor expanded in a1 around 0 59.2%

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

      1. pow2 59.2%

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

      2. *-commutative 59.2%

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

      3. associate-*r* 59.2%

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

      4. associate-*l/ 59.3%

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

      5. associate-*l/ 59.2%

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

      6. div-inv 59.2%

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

      7. associate-*l* 59.1%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-unprod 59.1%

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

      10. frac-times 59.1%

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

      11. metadata-eval 59.1%

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

      12. rem-square-sqrt 59.2%

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

      13. metadata-eval 59.2%

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

   6. Applied egg-rr 59.2%

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

   8. Applied egg-rr 42.0%

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

      1. +-lft-identity 42.0%

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

   10. Simplified 42.0%

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

   11. Taylor expanded in th around inf 42.0%

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

   if 0.70999999999999996 < (cos.f64 th)

   1. Initial program 99.2%

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

      1. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in th around 0 93.7%

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

   5. Taylor expanded in a1 around 0 56.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{{a2}^{2}}{\sqrt{2}}\\ \end{array} \]

Alternative 4: 48.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.71) (* a2 (* (cos th) a2)) (* a2 (* a2 (sqrt 0.5)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.71) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = 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) <= 0.71d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = 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) <= 0.71) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = a2 * (a2 * Math.sqrt(0.5));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.71:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 * math.sqrt(0.5))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.71)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(a2 * Float64(a2 * sqrt(0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.71)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = a2 * (a2 * sqrt(0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.71], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.70999999999999996

   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. Taylor expanded in a1 around 0 59.2%

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

      1. pow2 59.2%

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

      2. *-commutative 59.2%

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

      3. associate-*r* 59.2%

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

      4. associate-*l/ 59.3%

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

      5. associate-*l/ 59.2%

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

      6. div-inv 59.2%

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

      7. associate-*l* 59.1%

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

      8. add-sqr-sqrt 59.1%

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

      9. sqrt-unprod 59.1%

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

      10. frac-times 59.1%

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

      11. metadata-eval 59.1%

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

      12. rem-square-sqrt 59.2%

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

      13. metadata-eval 59.2%

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

   6. Applied egg-rr 59.2%

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

   8. Applied egg-rr 42.0%

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

      1. +-lft-identity 42.0%

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

   10. Simplified 42.0%

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

   if 0.70999999999999996 < (cos.f64 th)

   1. Initial program 99.2%

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

      1. distribute-lft-out 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in a1 around 0 60.5%

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

      1. pow2 60.5%

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

      2. *-commutative 60.5%

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

      3. associate-*r* 60.5%

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

      4. associate-*l/ 60.9%

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

      5. associate-*l/ 60.9%

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

      6. div-inv 60.9%

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

      7. associate-*l* 60.9%

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

      8. add-sqr-sqrt 60.9%

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

      9. sqrt-unprod 60.9%

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

      10. frac-times 60.9%

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

      11. metadata-eval 60.9%

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

      12. rem-square-sqrt 61.0%

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

      13. metadata-eval 61.0%

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in th around inf 61.0%

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

      1. *-commutative 61.0%

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

   9. Simplified 61.0%

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

   10. Taylor expanded in th around 0 56.3%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.71:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \]

Alternative 5: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double a1, double a2, double th) {
	return 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 = a2 * (a2 * (cos(th) * sqrt(0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $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. Taylor expanded in a1 around 0 60.1%

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

   1. pow2 60.1%

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

   2. *-commutative 60.1%

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

   3. associate-*r* 60.1%

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

   4. associate-*l/ 60.4%

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

   5. associate-*l/ 60.4%

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

   6. div-inv 60.4%

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

   7. associate-*l* 60.3%

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

   8. add-sqr-sqrt 60.3%

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

   9. sqrt-unprod 60.3%

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

   10. frac-times 60.3%

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

   11. metadata-eval 60.3%

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

   12. rem-square-sqrt 60.5%

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

   13. metadata-eval 60.5%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in th around inf 60.4%

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

   1. *-commutative 60.4%

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

9. Simplified 60.4%

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

10. Final simplification 60.4%

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

Alternative 6: 57.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[Sqrt[0.5], $MachinePrecision]), $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. Taylor expanded in a1 around 0 60.1%

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

   1. pow2 60.1%

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

   2. *-commutative 60.1%

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

   3. associate-*r* 60.1%

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

   4. associate-*l/ 60.4%

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

   5. associate-*l/ 60.4%

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

   6. div-inv 60.4%

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

   7. associate-*l* 60.3%

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

   8. add-sqr-sqrt 60.3%

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

   9. sqrt-unprod 60.3%

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

   10. frac-times 60.3%

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

   11. metadata-eval 60.3%

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

   12. rem-square-sqrt 60.5%

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

   13. metadata-eval 60.5%

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

6. Applied egg-rr 60.5%

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

7. Final simplification 60.5%

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

Alternative 7: 40.5% 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. Taylor expanded in a1 around 0 60.1%

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

   1. pow2 60.1%

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

   2. *-commutative 60.1%

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

   3. associate-*r* 60.1%

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

   4. associate-*l/ 60.4%

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

   5. associate-*l/ 60.4%

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

   6. div-inv 60.4%

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

   7. associate-*l* 60.3%

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

   8. add-sqr-sqrt 60.3%

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

   9. sqrt-unprod 60.3%

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

   10. frac-times 60.3%

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

   11. metadata-eval 60.3%

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

   12. rem-square-sqrt 60.5%

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

   13. metadata-eval 60.5%

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

6. Applied egg-rr 60.5%

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

7. Taylor expanded in th around inf 60.4%

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

   1. *-commutative 60.4%

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

9. Simplified 60.4%

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

10. Taylor expanded in th around 0 45.4%

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

11. Final simplification 45.4%

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

Alternative 8: 30.8% accurate, 138.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(a2 * a2), $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. Taylor expanded in a1 around 0 60.1%

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

   1. pow2 60.1%

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

   2. *-commutative 60.1%

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

   3. associate-*r* 60.1%

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

   4. associate-*l/ 60.4%

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

   5. associate-*l/ 60.4%

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

   6. div-inv 60.4%

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

   7. associate-*l* 60.3%

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

   8. add-sqr-sqrt 60.3%

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

   9. sqrt-unprod 60.3%

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

   10. frac-times 60.3%

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

   11. metadata-eval 60.3%

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

   12. rem-square-sqrt 60.5%

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

   13. metadata-eval 60.5%

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

6. Applied egg-rr 60.5%

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

8. Applied egg-rr 40.4%

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

   1. +-lft-identity 40.4%

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

10. Simplified 40.4%

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

11. Taylor expanded in th around 0 33.7%

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

12. Final simplification 33.7%

    \[\leadsto a2 \cdot a2 \]

Reproduce

herbie shell --seed 2023331 
(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))))