Migdal et al, Equation (64)

Percentage Accurate: 99.6% → 99.6%

Time: 13.1s

Alternatives: 15

Speedup: 2.0×

• 43.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[Power[N[Sqrt[a2 ^ 2 + a1 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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. Step-by-step derivation

   1. distribute-lft-in 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. frac-add 99.4%

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

   5. fma-def 99.4%

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

   6. pow2 99.4%

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

   7. pow2 99.4%

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

   8. rem-square-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. fma-udef 99.7%

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

7. Simplified 99.7%

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

8. Taylor expanded in a1 around 0 99.6%

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

   1. distribute-rgt-out 99.6%

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

   2. associate-*r* 99.6%

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

   3. +-commutative 99.6%

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

   4. unpow2 99.6%

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

   5. fma-def 99.6%

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

10. Simplified 99.6%

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

    1. expm1-log1p-u 97.2%

       \[\leadsto \frac{\cos th \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, {a1}^{2}\right)\right)\right)}}{2} \]

    2. expm1-udef 83.1%

       \[\leadsto \frac{\cos th \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \mathsf{fma}\left(a2, a2, {a1}^{2}\right)\right)} - 1\right)}}{2} \]

    3. add-sqr-sqrt 83.1%

       \[\leadsto \frac{\cos th \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \color{blue}{\left(\sqrt{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)}\right)}\right)} - 1\right)}{2} \]

    4. pow2 83.1%

       \[\leadsto \frac{\cos th \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot \color{blue}{{\left(\sqrt{\mathsf{fma}\left(a2, a2, {a1}^{2}\right)}\right)}^{2}}\right)} - 1\right)}{2} \]

    5. fma-udef 83.1%

       \[\leadsto \frac{\cos th \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot {\left(\sqrt{\color{blue}{a2 \cdot a2 + {a1}^{2}}}\right)}^{2}\right)} - 1\right)}{2} \]

    6. pow2 83.1%

       \[\leadsto \frac{\cos th \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot {\left(\sqrt{a2 \cdot a2 + \color{blue}{a1 \cdot a1}}\right)}^{2}\right)} - 1\right)}{2} \]

    7. hypot-def 83.1%

       \[\leadsto \frac{\cos th \cdot \left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot {\color{blue}{\left(\mathsf{hypot}\left(a2, a1\right)\right)}}^{2}\right)} - 1\right)}{2} \]

12. Applied egg-rr 83.1%

    \[\leadsto \frac{\cos th \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{2} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right)} - 1\right)}}{2} \]
13. Step-by-step derivation

    1. expm1-def 97.2%

       \[\leadsto \frac{\cos th \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{2} \cdot {\left(\mathsf{hypot}\left(a2, a1\right)\right)}^{2}\right)\right)}}{2} \]

    2. expm1-log1p 99.7%

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

14. Simplified 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (/ (/ (cos th) (pow 2.0 0.25)) (pow 2.0 0.25)) (+ (* a1 a1) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return ((cos(th) / pow(2.0, 0.25)) / pow(2.0, 0.25)) * ((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) / (2.0d0 ** 0.25d0)) / (2.0d0 ** 0.25d0)) * ((a1 * a1) + (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return ((Math.cos(th) / Math.pow(2.0, 0.25)) / Math.pow(2.0, 0.25)) * ((a1 * a1) + (a2 * a2));
}

Python

def code(a1, a2, th):
	return ((math.cos(th) / math.pow(2.0, 0.25)) / math.pow(2.0, 0.25)) * ((a1 * a1) + (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(Float64(Float64(cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = ((cos(th) / (2.0 ^ 0.25)) / (2.0 ^ 0.25)) * ((a1 * a1) + (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $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. Step-by-step derivation

   1. *-un-lft-identity 99.6%

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

   2. add-sqr-sqrt 99.6%

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

   3. times-frac 99.3%

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

   4. pow1/2 99.3%

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

   5. sqrt-pow1 99.3%

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

   6. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{\color{blue}{0.25}}} \cdot \frac{\cos th}{\sqrt{\sqrt{2}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   7. pow1/2 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\sqrt{\color{blue}{{2}^{0.5}}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   8. sqrt-pow1 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{\color{blue}{{2}^{\left(\frac{0.5}{2}\right)}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   9. metadata-eval 99.3%

      \[\leadsto \left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{\color{blue}{0.25}}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

5. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\left(\frac{1}{{2}^{0.25}} \cdot \frac{\cos th}{{2}^{0.25}}\right)} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
6. Step-by-step derivation

   1. associate-*l/ 99.6%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   2. *-lft-identity 99.6%

      \[\leadsto \frac{\color{blue}{\frac{\cos th}{{2}^{0.25}}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

7. Simplified 99.6%

   \[\leadsto \color{blue}{\frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

8. Final simplification 99.6%

   \[\leadsto \frac{\frac{\cos th}{{2}^{0.25}}}{{2}^{0.25}} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

Alternative 3: 71.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.68)
   (* (cos th) (pow a2 2.0))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = cos(th) * pow(a2, 2.0);
	} 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) <= 0.68d0) then
        tmp = cos(th) * (a2 ** 2.0d0)
    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) <= 0.68) {
		tmp = Math.cos(th) * Math.pow(a2, 2.0);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * Math.sqrt(0.5);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.68:
		tmp = math.cos(th) * math.pow(a2, 2.0)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * math.sqrt(0.5)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.68)
		tmp = Float64(cos(th) * (a2 ^ 2.0));
	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) <= 0.68)
		tmp = cos(th) * (a2 ^ 2.0);
	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], 0.68], N[(N[Cos[th], $MachinePrecision] * N[Power[a2, 2.0], $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) < 0.680000000000000049

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   7. Applied egg-rr 58.5%

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

   8. Taylor expanded in a1 around 0 42.4%

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

   if 0.680000000000000049 < (cos.f64 th)

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 94.7%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot {a2}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 4: 71.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 + 2 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.68)
   (* (cos th) (* a2 (+ a2 (* 2.0 a1))))
   (* (+ (* a1 a1) (* a2 a2)) (sqrt 0.5))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.68) {
		tmp = cos(th) * (a2 * (a2 + (2.0 * 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) <= 0.68d0) then
        tmp = cos(th) * (a2 * (a2 + (2.0d0 * 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) <= 0.68) {
		tmp = Math.cos(th) * (a2 * (a2 + (2.0 * 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) <= 0.68:
		tmp = math.cos(th) * (a2 * (a2 + (2.0 * 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) <= 0.68)
		tmp = Float64(cos(th) * Float64(a2 * Float64(a2 + Float64(2.0 * 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) <= 0.68)
		tmp = cos(th) * (a2 * (a2 + (2.0 * 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], 0.68], N[(N[Cos[th], $MachinePrecision] * N[(a2 * N[(a2 + N[(2.0 * a1), $MachinePrecision]), $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) < 0.680000000000000049

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. distribute-lft-in 99.6%

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

      2. associate-*l/ 99.7%

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

      3. associate-*l/ 99.7%

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

      4. frac-add 99.3%

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

      5. fma-def 99.3%

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

      6. pow2 99.3%

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

      7. pow2 99.3%

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

      8. rem-square-sqrt 99.5%

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

   5. Applied egg-rr 99.5%

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

      1. fma-udef 99.5%

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

      2. *-commutative 99.5%

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

      3. distribute-rgt-out 99.5%

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

      4. distribute-lft-in 99.5%

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

      5. *-commutative 99.5%

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

      6. unpow2 99.5%

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

      7. fma-udef 99.6%

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

   7. Simplified 99.6%

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

   9. Applied egg-rr 58.5%

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

       1. *-commutative 58.5%

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

       2. associate-*l* 58.5%

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

       3. +-commutative 58.5%

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

       4. +-commutative 58.5%

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

   11. Simplified 58.5%

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

   12. Taylor expanded in a2 around inf 40.7%

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

       1. +-commutative 40.7%

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

       2. unpow2 40.7%

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

       3. associate-*r* 40.7%

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

       4. distribute-rgt-out 42.8%

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

   14. Simplified 42.8%

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

   if 0.680000000000000049 < (cos.f64 th)

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 94.7%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot \left(a2 + 2 \cdot a1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 5: 79.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.680000000000000049

   1. Initial program 99.6%

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

      1. distribute-lft-out 99.6%

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

   3. Simplified 99.6%

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

      1. frac-2neg 99.6%

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

      2. div-inv 99.6%

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

   5. Applied egg-rr 99.6%

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

   7. Applied egg-rr 58.5%

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

   if 0.680000000000000049 < (cos.f64 th)

   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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.7%

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

      3. pow1/2 99.7%

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

      4. pow-flip 99.7%

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

      5. metadata-eval 99.7%

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

   5. Applied egg-rr 99.7%

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

   6. Taylor expanded in th around 0 94.7%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.68:\\ \;\;\;\;\cos th \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot \sqrt{0.5}\\ \end{array} \]

Alternative 6: 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. 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%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in th around inf 99.6%

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

7. Final simplification 99.6%

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

Alternative 7: 37.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

3. Simplified 99.6%

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

   1. distribute-lft-in 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. frac-add 99.4%

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

   5. fma-def 99.4%

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

   6. pow2 99.4%

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

   7. pow2 99.4%

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

   8. rem-square-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. fma-udef 99.7%

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

7. Simplified 99.7%

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

9. Applied egg-rr 63.7%

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

    1. *-commutative 63.7%

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

    2. associate-*l* 63.7%

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

    3. +-commutative 63.7%

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

    4. +-commutative 63.7%

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

11. Simplified 63.7%

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

12. Taylor expanded in a2 around inf 38.1%

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

    1. +-commutative 38.1%

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

    2. unpow2 38.1%

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

    3. associate-*r* 38.1%

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

    4. distribute-rgt-out 40.5%

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

14. Simplified 40.5%

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

15. Final simplification 40.5%

    \[\leadsto \cos th \cdot \left(a2 \cdot \left(a2 + 2 \cdot a1\right)\right) \]

Alternative 8: 44.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{a2}^{2} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 9.4e+24) (* (+ (* a1 a1) (* a2 a2)) 0.5) (* (pow a2 2.0) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9.4e+24) {
		tmp = ((a1 * a1) + (a2 * a2)) * 0.5;
	} else {
		tmp = pow(a2, 2.0) * -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 (th <= 9.4d+24) then
        tmp = ((a1 * a1) + (a2 * a2)) * 0.5d0
    else
        tmp = (a2 ** 2.0d0) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 9.4e+24) {
		tmp = ((a1 * a1) + (a2 * a2)) * 0.5;
	} else {
		tmp = Math.pow(a2, 2.0) * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if th <= 9.4e+24:
		tmp = ((a1 * a1) + (a2 * a2)) * 0.5
	else:
		tmp = math.pow(a2, 2.0) * -0.5
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 9.4e+24)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * 0.5);
	else
		tmp = Float64((a2 ^ 2.0) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 9.4e+24)
		tmp = ((a1 * a1) + (a2 * a2)) * 0.5;
	else
		tmp = (a2 ^ 2.0) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[th, 9.4e+24], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(N[Power[a2, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 9.3999999999999999e24

   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. Taylor expanded in th around 0 79.1%

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

   6. Applied egg-rr 56.8%

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

   if 9.3999999999999999e24 < th

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. Taylor expanded in th around 0 22.4%

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

   6. Applied egg-rr 43.5%

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

   7. Taylor expanded in a1 around 0 33.4%

      \[\leadsto -0.5 \cdot \color{blue}{{a2}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;{a2}^{2} \cdot -0.5\\ \end{array} \]

Alternative 9: 45.4% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;t_1 \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.4e+24) (* t_1 0.125) (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = t_1 * 0.125;
	} else {
		tmp = t_1 * -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) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (th <= 9.4d+24) then
        tmp = t_1 * 0.125d0
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = t_1 * 0.125;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.4e+24:
		tmp = t_1 * 0.125
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.4e+24)
		tmp = Float64(t_1 * 0.125);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.4e+24)
		tmp = t_1 * 0.125;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.4e+24], N[(t$95$1 * 0.125), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.3999999999999999e24

   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. Taylor expanded in th around 0 79.1%

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

   6. Applied egg-rr 55.5%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]

   if 9.3999999999999999e24 < th

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. Taylor expanded in th around 0 22.4%

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

   6. Applied egg-rr 43.5%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 10: 45.8% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;0.25 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.4e+24) (* 0.25 t_1) (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = 0.25 * t_1;
	} else {
		tmp = t_1 * -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) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (th <= 9.4d+24) then
        tmp = 0.25d0 * t_1
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = 0.25 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.4e+24:
		tmp = 0.25 * t_1
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.4e+24)
		tmp = Float64(0.25 * t_1);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.4e+24)
		tmp = 0.25 * t_1;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.4e+24], N[(0.25 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.3999999999999999e24

   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. Taylor expanded in th around 0 79.1%

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

   6. Applied egg-rr 55.9%

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

   if 9.3999999999999999e24 < th

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. Taylor expanded in th around 0 22.4%

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

   6. Applied egg-rr 43.5%

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

4. Final simplification 53.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;0.25 \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 11: 46.5% accurate, 37.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;t_1 \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 9.4e+24) (* t_1 0.5) (* t_1 -0.5))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = t_1 * 0.5;
	} else {
		tmp = t_1 * -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) :: t_1
    real(8) :: tmp
    t_1 = (a1 * a1) + (a2 * a2)
    if (th <= 9.4d+24) then
        tmp = t_1 * 0.5d0
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 9.4e+24) {
		tmp = t_1 * 0.5;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 9.4e+24:
		tmp = t_1 * 0.5
	else:
		tmp = t_1 * -0.5
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 9.4e+24)
		tmp = Float64(t_1 * 0.5);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 9.4e+24)
		tmp = t_1 * 0.5;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 9.4e+24], N[(t$95$1 * 0.5), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if th < 9.3999999999999999e24

   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. Taylor expanded in th around 0 79.1%

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

   6. Applied egg-rr 56.8%

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

   if 9.3999999999999999e24 < th

   1. Initial program 99.8%

      \[\frac{\cos th}{\sqrt{2}} \cdot \left(a1 \cdot a1\right) + \frac{\cos th}{\sqrt{2}} \cdot \left(a2 \cdot a2\right) \]
   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. Taylor expanded in th around 0 22.4%

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

   6. Applied egg-rr 43.5%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 9.4 \cdot 10^{+24}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 12: 19.9% 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. Taylor expanded in th around 0 66.5%

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

6. Applied egg-rr 23.3%

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

7. Final simplification 23.3%

   \[\leadsto \left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5 \]

Alternative 13: 19.6% accurate, 51.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $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. Taylor expanded in th around 0 66.5%

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

6. Applied egg-rr 23.0%

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

7. Final simplification 23.0%

   \[\leadsto a1 \cdot \left(-a1\right) - a2 \cdot a2 \]

Alternative 14: 4.0% accurate, 83.0× speedup

Math

\[\begin{array}{l} \\ \frac{a2 + a1}{2} \end{array} \]

FPCore

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

C

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

Java

public static double code(double a1, double a2, double th) {
	return (a2 + a1) / 2.0;
}

Python

def code(a1, a2, th):
	return (a2 + a1) / 2.0

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(a2 + a1), $MachinePrecision] / 2.0), $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. Step-by-step derivation

   1. distribute-lft-in 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. frac-add 99.4%

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

   5. fma-def 99.4%

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

   6. pow2 99.4%

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

   7. pow2 99.4%

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

   8. rem-square-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. fma-udef 99.7%

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

7. Simplified 99.7%

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

9. Applied egg-rr 4.2%

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

    1. associate-/l* 4.2%

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

    2. associate-/r/ 4.2%

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

    3. *-inverses 4.2%

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

    4. *-lft-identity 4.2%

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

    5. +-commutative 4.2%

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

11. Simplified 4.2%

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

12. Final simplification 4.2%

    \[\leadsto \frac{a2 + a1}{2} \]

Alternative 15: 3.5% accurate, 415.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(a1, a2, th)
	return 0.5
end

MATLAB

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

Wolfram

code[a1_, a2_, th_] := 0.5

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. Step-by-step derivation

   1. distribute-lft-in 99.6%

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

   2. associate-*l/ 99.7%

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

   3. associate-*l/ 99.7%

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

   4. frac-add 99.4%

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

   5. fma-def 99.4%

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

   6. pow2 99.4%

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

   7. pow2 99.4%

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

   8. rem-square-sqrt 99.7%

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

5. Applied egg-rr 99.7%

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

   1. fma-udef 99.7%

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

   2. *-commutative 99.7%

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

   3. distribute-rgt-out 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. *-commutative 99.7%

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

   6. unpow2 99.7%

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

   7. fma-udef 99.7%

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

7. Simplified 99.7%

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

9. Applied egg-rr 3.3%

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

    1. *-inverses 3.3%

       \[\leadsto \frac{\color{blue}{1}}{2} \]

11. Simplified 3.3%

    \[\leadsto \frac{\color{blue}{1}}{2} \]

12. Final simplification 3.3%

    \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023321 
(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))))