Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.6%

Time: 11.1s

Alternatives: 18

Speedup: 2.0×

• 43.2% 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: 99.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision] * N[Cos[th], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[2.0], $MachinePrecision] / N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

   1. fma-def 99.3%

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

   2. add-sqr-sqrt 99.3%

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

   3. *-un-lft-identity 99.3%

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

   4. times-frac 99.3%

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

   5. hypot-def 99.2%

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

   6. hypot-def 99.2%

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

5. Applied egg-rr 99.2%

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

   1. /-rgt-identity 99.2%

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

   2. associate-*r* 99.5%

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

   3. clear-num 99.5%

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

   4. un-div-inv 99.6%

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

   5. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Final simplification 99.3%

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

Alternative 3: 99.6% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

   1. fma-def 99.3%

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

   2. add-sqr-sqrt 99.3%

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

   3. *-un-lft-identity 99.3%

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

   4. times-frac 99.3%

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

   5. hypot-def 99.2%

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

   6. hypot-def 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in th around inf 99.3%

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

   1. associate-/l* 99.3%

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

   2. unpow2 99.3%

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

   3. fma-def 99.3%

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

   4. unpow2 99.3%

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

8. Simplified 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(\cos th \cdot \frac{-1}{-\sqrt{2}}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (/ -1.0 (- (sqrt 2.0)))) (+ (* a1 a1) (* a2 a2))))

C

double code(double a1, double a2, double th) {
	return (cos(th) * (-1.0 / -sqrt(2.0))) * ((a1 * a1) + (a2 * a2));
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    code = (cos(th) * ((-1.0d0) / -sqrt(2.0d0))) * ((a1 * a1) + (a2 * a2))
end function

Java

public static double code(double a1, double a2, double th) {
	return (Math.cos(th) * (-1.0 / -Math.sqrt(2.0))) * ((a1 * a1) + (a2 * a2));
}

Python

def code(a1, a2, th):
	return (math.cos(th) * (-1.0 / -math.sqrt(2.0))) * ((a1 * a1) + (a2 * a2))

Julia

function code(a1, a2, th)
	return Float64(Float64(cos(th) * Float64(-1.0 / Float64(-sqrt(2.0)))) * Float64(Float64(a1 * a1) + Float64(a2 * a2)))
end

MATLAB

function tmp = code(a1, a2, th)
	tmp = (cos(th) * (-1.0 / -sqrt(2.0))) * ((a1 * a1) + (a2 * a2));
end

Wolfram

code[a1_, a2_, th_] := N[(N[(N[Cos[th], $MachinePrecision] * N[(-1.0 / (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. frac-2neg 99.2%

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

   2. div-inv 99.2%

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

5. Applied egg-rr 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 72.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.65)
   (* (cos th) (* a2 a2))
   (/ (+ (* a1 a1) (* a2 a2)) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.65) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.65d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.65) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.65:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = ((a1 * a1) + (a2 * a2)) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.65)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.65)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = ((a1 * a1) + (a2 * a2)) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.65], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.650000000000000022

   1. Initial program 98.8%

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

      1. distribute-lft-out 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-*r/ 98.9%

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

      4. fma-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in a1 around 0 60.9%

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

      1. unpow2 60.9%

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

   6. Simplified 60.9%

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

   8. Applied egg-rr 35.3%

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

   if 0.650000000000000022 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

      1. frac-2neg 99.5%

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

      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%

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

   6. Taylor expanded in th around 0 91.2%

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

      1. unpow2 91.2%

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

      2. unpow2 91.2%

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

   8. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{a2 \cdot a2 + a1 \cdot a1}{\sqrt{2}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a1 \cdot a1 + a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 6: 99.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Final simplification 99.2%

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

Alternative 7: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq -0.001:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) -0.001)
   (* (+ (* a1 a1) (* a2 a2)) -0.5)
   (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= -0.001) {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= (-0.001d0)) then
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.5d0)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= -0.001) {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= -0.001:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= -0.001)
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.5);
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= -0.001)
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], -0.001], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1e-3

   1. Initial program 98.6%

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

      1. distribute-lft-out 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in th around 0 4.9%

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

   6. Applied egg-rr 54.5%

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

   if -1e-3 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.6%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 60.6%

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

      1. unpow2 60.6%

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

      2. associate-*r/ 60.5%

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

      3. associate-*l* 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in th around 0 52.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.001:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 8: 47.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.65) (* (cos th) (* a2 a2)) (* a2 (/ a2 (sqrt 2.0)))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.65) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.65d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.65) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.65:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.65)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.65)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.65], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.650000000000000022

   1. Initial program 98.8%

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

      1. distribute-lft-out 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-*r/ 98.9%

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

      4. fma-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in a1 around 0 60.9%

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

      1. unpow2 60.9%

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

   6. Simplified 60.9%

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

   8. Applied egg-rr 35.3%

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

   if 0.650000000000000022 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 60.4%

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

      1. unpow2 60.4%

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

      2. associate-*r/ 60.3%

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

      3. associate-*l* 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in th around 0 56.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \]

Alternative 9: 47.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.65) (* (cos th) (* a2 a2)) (/ (* a2 a2) (sqrt 2.0))))

C

double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.65) {
		tmp = cos(th) * (a2 * a2);
	} else {
		tmp = (a2 * a2) / sqrt(2.0);
	}
	return tmp;
}

Fortran

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.65d0) then
        tmp = cos(th) * (a2 * a2)
    else
        tmp = (a2 * a2) / sqrt(2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.65) {
		tmp = Math.cos(th) * (a2 * a2);
	} else {
		tmp = (a2 * a2) / Math.sqrt(2.0);
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.65:
		tmp = math.cos(th) * (a2 * a2)
	else:
		tmp = (a2 * a2) / math.sqrt(2.0)
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.65)
		tmp = Float64(cos(th) * Float64(a2 * a2));
	else
		tmp = Float64(Float64(a2 * a2) / sqrt(2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.65)
		tmp = cos(th) * (a2 * a2);
	else
		tmp = (a2 * a2) / sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.65], N[(N[Cos[th], $MachinePrecision] * N[(a2 * a2), $MachinePrecision]), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < 0.650000000000000022

   1. Initial program 98.8%

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

      1. distribute-lft-out 98.8%

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

      2. associate-*l/ 98.8%

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

      3. associate-*r/ 98.9%

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

      4. fma-def 98.9%

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

   3. Simplified 98.9%

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

   4. Taylor expanded in a1 around 0 60.9%

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

      1. unpow2 60.9%

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

   6. Simplified 60.9%

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

   8. Applied egg-rr 35.3%

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

   if 0.650000000000000022 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

      2. associate-*l/ 99.7%

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

      3. associate-*r/ 99.6%

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

      4. fma-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in a1 around 0 60.4%

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

      1. unpow2 60.4%

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

      2. associate-*r/ 60.3%

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

      3. associate-*l* 60.3%

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

   6. Simplified 60.3%

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

   7. Taylor expanded in th around 0 56.2%

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

      1. unpow2 56.2%

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

   9. Simplified 56.2%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq 0.65:\\ \;\;\;\;\cos th \cdot \left(a2 \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{a2 \cdot a2}{\sqrt{2}}\\ \end{array} \]

Alternative 10: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

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. unpow2 60.5%

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

   2. associate-*r/ 60.5%

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

   3. associate-*l* 60.8%

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

6. Simplified 60.8%

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

7. Final simplification 60.8%

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

Alternative 11: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

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. unpow2 60.5%

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

   2. associate-*r/ 60.5%

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

   3. associate-*l* 60.8%

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

6. Simplified 60.8%

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

   1. clear-num 60.8%

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

   2. un-div-inv 60.8%

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

8. Applied egg-rr 60.8%

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

9. Final simplification 60.8%

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

Alternative 12: 57.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

   1. fma-def 99.3%

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

   2. add-sqr-sqrt 99.3%

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

   3. *-un-lft-identity 99.3%

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

   4. times-frac 99.3%

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

   5. hypot-def 99.2%

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

   6. hypot-def 99.2%

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

5. Applied egg-rr 99.2%

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

6. Taylor expanded in a1 around 0 60.5%

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

   1. associate-/l* 60.6%

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

   2. unpow2 60.6%

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

   3. associate-/l* 60.8%

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

8. Simplified 60.8%

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

9. Final simplification 60.8%

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

Alternative 13: 59.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;\cos th \leq -0.001:\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= (cos th) -0.001) (* 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 (cos(th) <= -0.001) {
		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 (cos(th) <= (-0.001d0)) 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 (Math.cos(th) <= -0.001) {
		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 math.cos(th) <= -0.001:
		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 (cos(th) <= -0.001)
		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 (cos(th) <= -0.001)
		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[N[Cos[th], $MachinePrecision], -0.001], N[(t$95$1 * -0.5), $MachinePrecision], N[(t$95$1 * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 th) < -1e-3

   1. Initial program 98.6%

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

      1. distribute-lft-out 98.6%

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

   3. Simplified 98.6%

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

   4. Taylor expanded in th around 0 4.9%

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

   6. Applied egg-rr 54.5%

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

   if -1e-3 < (cos.f64 th)

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 85.0%

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

   6. Applied egg-rr 57.1%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos th \leq -0.001:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.5\\ \end{array} \]

Alternative 14: 32.1% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43} \lor \neg \left(th \leq 1.72 \cdot 10^{+75}\right) \land th \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 6e+43) (and (not (<= th 1.72e+75)) (<= th 1.5e+129)))
   (* a2 a2)
   (* (+ (* a1 a1) (* a2 a2)) -0.5)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 1.5e+129))) {
		tmp = a2 * a2;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -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 <= 6d+43) .or. (.not. (th <= 1.72d+75)) .and. (th <= 1.5d+129)) then
        tmp = a2 * a2
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 1.5e+129))) {
		tmp = a2 * a2;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 6e+43) or (not (th <= 1.72e+75) and (th <= 1.5e+129)):
		tmp = a2 * a2
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 1.5e+129)))
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 6e+43) || (~((th <= 1.72e+75)) && (th <= 1.5e+129)))
		tmp = a2 * a2;
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 6e+43], And[N[Not[LessEqual[th, 1.72e+75]], $MachinePrecision], LessEqual[th, 1.5e+129]]], N[(a2 * a2), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 6.00000000000000033e43 or 1.72e75 < th < 1.50000000000000015e129

   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%

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

      2. associate-*l/ 99.3%

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

      3. associate-*r/ 99.3%

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

      4. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 62.3%

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

      1. unpow2 62.3%

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

   6. Simplified 62.3%

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

   8. Applied egg-rr 39.0%

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

   9. Taylor expanded in th around 0 30.2%

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

       1. unpow2 30.2%

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

   11. Simplified 30.2%

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

   if 6.00000000000000033e43 < th < 1.72e75 or 1.50000000000000015e129 < 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. Taylor expanded in th around 0 20.8%

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

   6. Applied egg-rr 30.3%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43} \lor \neg \left(th \leq 1.72 \cdot 10^{+75}\right) \land th \leq 1.5 \cdot 10^{+129}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \]

Alternative 15: 32.5% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 6 \cdot 10^{+43}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.72 \cdot 10^{+75} \lor \neg \left(th \leq 4.3 \cdot 10^{+130}\right):\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.125\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 6e+43)
     (* a2 a2)
     (if (or (<= th 1.72e+75) (not (<= th 4.3e+130)))
       (* t_1 -0.5)
       (* t_1 0.125)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 6e+43) {
		tmp = a2 * a2;
	} else if ((th <= 1.72e+75) || !(th <= 4.3e+130)) {
		tmp = t_1 * -0.5;
	} else {
		tmp = t_1 * 0.125;
	}
	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 <= 6d+43) then
        tmp = a2 * a2
    else if ((th <= 1.72d+75) .or. (.not. (th <= 4.3d+130))) then
        tmp = t_1 * (-0.5d0)
    else
        tmp = t_1 * 0.125d0
    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 <= 6e+43) {
		tmp = a2 * a2;
	} else if ((th <= 1.72e+75) || !(th <= 4.3e+130)) {
		tmp = t_1 * -0.5;
	} else {
		tmp = t_1 * 0.125;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 6e+43:
		tmp = a2 * a2
	elif (th <= 1.72e+75) or not (th <= 4.3e+130):
		tmp = t_1 * -0.5
	else:
		tmp = t_1 * 0.125
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 6e+43)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.72e+75) || !(th <= 4.3e+130))
		tmp = Float64(t_1 * -0.5);
	else
		tmp = Float64(t_1 * 0.125);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 6e+43)
		tmp = a2 * a2;
	elseif ((th <= 1.72e+75) || ~((th <= 4.3e+130)))
		tmp = t_1 * -0.5;
	else
		tmp = t_1 * 0.125;
	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, 6e+43], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.72e+75], N[Not[LessEqual[th, 4.3e+130]], $MachinePrecision]], N[(t$95$1 * -0.5), $MachinePrecision], N[(t$95$1 * 0.125), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 6.00000000000000033e43

   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%

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

      2. associate-*l/ 99.3%

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

      3. associate-*r/ 99.3%

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

      4. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 62.1%

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

      1. unpow2 62.1%

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

   6. Simplified 62.1%

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

   8. Applied egg-rr 38.9%

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

   9. Taylor expanded in th around 0 30.1%

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

       1. unpow2 30.1%

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

   11. Simplified 30.1%

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

   if 6.00000000000000033e43 < th < 1.72e75 or 4.29999999999999984e130 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 21.4%

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

   6. Applied egg-rr 28.5%

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

   if 1.72e75 < th < 4.29999999999999984e130

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 40.8%

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

   6. Applied egg-rr 40.8%

      \[\leadsto \color{blue}{0.125} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.72 \cdot 10^{+75} \lor \neg \left(th \leq 4.3 \cdot 10^{+130}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.125\\ \end{array} \]

Alternative 16: 32.5% accurate, 27.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 6 \cdot 10^{+43}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.72 \cdot 10^{+75} \lor \neg \left(th \leq 1.5 \cdot 10^{+129}\right):\\ \;\;\;\;t_1 \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 6e+43)
     (* a2 a2)
     (if (or (<= th 1.72e+75) (not (<= th 1.5e+129)))
       (* t_1 -0.5)
       (* t_1 0.25)))))

C

double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 6e+43) {
		tmp = a2 * a2;
	} else if ((th <= 1.72e+75) || !(th <= 1.5e+129)) {
		tmp = t_1 * -0.5;
	} else {
		tmp = t_1 * 0.25;
	}
	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 <= 6d+43) then
        tmp = a2 * a2
    else if ((th <= 1.72d+75) .or. (.not. (th <= 1.5d+129))) then
        tmp = t_1 * (-0.5d0)
    else
        tmp = t_1 * 0.25d0
    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 <= 6e+43) {
		tmp = a2 * a2;
	} else if ((th <= 1.72e+75) || !(th <= 1.5e+129)) {
		tmp = t_1 * -0.5;
	} else {
		tmp = t_1 * 0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 6e+43:
		tmp = a2 * a2
	elif (th <= 1.72e+75) or not (th <= 1.5e+129):
		tmp = t_1 * -0.5
	else:
		tmp = t_1 * 0.25
	return tmp

Julia

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 6e+43)
		tmp = Float64(a2 * a2);
	elseif ((th <= 1.72e+75) || !(th <= 1.5e+129))
		tmp = Float64(t_1 * -0.5);
	else
		tmp = Float64(t_1 * 0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 6e+43)
		tmp = a2 * a2;
	elseif ((th <= 1.72e+75) || ~((th <= 1.5e+129)))
		tmp = t_1 * -0.5;
	else
		tmp = t_1 * 0.25;
	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, 6e+43], N[(a2 * a2), $MachinePrecision], If[Or[LessEqual[th, 1.72e+75], N[Not[LessEqual[th, 1.5e+129]], $MachinePrecision]], N[(t$95$1 * -0.5), $MachinePrecision], N[(t$95$1 * 0.25), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if th < 6.00000000000000033e43

   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%

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

      2. associate-*l/ 99.3%

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

      3. associate-*r/ 99.3%

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

      4. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 62.1%

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

      1. unpow2 62.1%

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

   6. Simplified 62.1%

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

   8. Applied egg-rr 38.9%

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

   9. Taylor expanded in th around 0 30.1%

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

       1. unpow2 30.1%

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

   11. Simplified 30.1%

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

   if 6.00000000000000033e43 < th < 1.72e75 or 1.50000000000000015e129 < 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. Taylor expanded in th around 0 20.8%

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

   6. Applied egg-rr 30.3%

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

   if 1.72e75 < th < 1.50000000000000015e129

   1. Initial program 99.7%

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

      1. distribute-lft-out 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in th around 0 45.3%

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

   6. Applied egg-rr 45.3%

      \[\leadsto \color{blue}{0.25} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{elif}\;th \leq 1.72 \cdot 10^{+75} \lor \neg \left(th \leq 1.5 \cdot 10^{+129}\right):\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot 0.25\\ \end{array} \]

Alternative 17: 30.1% accurate, 37.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43} \lor \neg \left(th \leq 1.72 \cdot 10^{+75}\right) \land th \leq 4.3 \cdot 10^{+130}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot -0.25\\ \end{array} \end{array} \]

FPCore

(FPCore (a1 a2 th)
 :precision binary64
 (if (or (<= th 6e+43) (and (not (<= th 1.72e+75)) (<= th 4.3e+130)))
   (* a2 a2)
   (* (* a2 a2) -0.25)))

C

double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 4.3e+130))) {
		tmp = a2 * a2;
	} else {
		tmp = (a2 * a2) * -0.25;
	}
	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 <= 6d+43) .or. (.not. (th <= 1.72d+75)) .and. (th <= 4.3d+130)) then
        tmp = a2 * a2
    else
        tmp = (a2 * a2) * (-0.25d0)
    end if
    code = tmp
end function

Java

public static double code(double a1, double a2, double th) {
	double tmp;
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 4.3e+130))) {
		tmp = a2 * a2;
	} else {
		tmp = (a2 * a2) * -0.25;
	}
	return tmp;
}

Python

def code(a1, a2, th):
	tmp = 0
	if (th <= 6e+43) or (not (th <= 1.72e+75) and (th <= 4.3e+130)):
		tmp = a2 * a2
	else:
		tmp = (a2 * a2) * -0.25
	return tmp

Julia

function code(a1, a2, th)
	tmp = 0.0
	if ((th <= 6e+43) || (!(th <= 1.72e+75) && (th <= 4.3e+130)))
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(Float64(a2 * a2) * -0.25);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if ((th <= 6e+43) || (~((th <= 1.72e+75)) && (th <= 4.3e+130)))
		tmp = a2 * a2;
	else
		tmp = (a2 * a2) * -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[a1_, a2_, th_] := If[Or[LessEqual[th, 6e+43], And[N[Not[LessEqual[th, 1.72e+75]], $MachinePrecision], LessEqual[th, 4.3e+130]]], N[(a2 * a2), $MachinePrecision], N[(N[(a2 * a2), $MachinePrecision] * -0.25), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if th < 6.00000000000000033e43 or 1.72e75 < th < 4.29999999999999984e130

   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%

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

      2. associate-*l/ 99.3%

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

      3. associate-*r/ 99.3%

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

      4. fma-def 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in a1 around 0 62.5%

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

      1. unpow2 62.5%

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

   6. Simplified 62.5%

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

   8. Applied egg-rr 39.3%

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

   9. Taylor expanded in th around 0 30.1%

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

       1. unpow2 30.1%

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

   11. Simplified 30.1%

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

   if 6.00000000000000033e43 < th < 1.72e75 or 4.29999999999999984e130 < th

   1. Initial program 99.5%

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

      1. distribute-lft-out 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in th around 0 21.4%

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

   6. Applied egg-rr 27.6%

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

   7. Taylor expanded in a1 around 0 14.0%

      \[\leadsto \color{blue}{-0.25 \cdot {a2}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 14.0%

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

   9. Simplified 14.0%

      \[\leadsto \color{blue}{-0.25 \cdot \left(a2 \cdot a2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;th \leq 6 \cdot 10^{+43} \lor \neg \left(th \leq 1.72 \cdot 10^{+75}\right) \land th \leq 4.3 \cdot 10^{+130}:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a2 \cdot a2\right) \cdot -0.25\\ \end{array} \]

Alternative 18: 30.1% 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 99.2%

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

   1. distribute-lft-out 99.2%

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

   2. associate-*l/ 99.3%

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

   3. associate-*r/ 99.3%

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

   4. fma-def 99.3%

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

3. Simplified 99.3%

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

4. Taylor expanded in a1 around 0 60.6%

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

   1. unpow2 60.6%

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

6. Simplified 60.6%

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

8. Applied egg-rr 37.4%

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

9. Taylor expanded in th around 0 28.0%

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

    1. unpow2 28.0%

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

11. Simplified 28.0%

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

12. Final simplification 28.0%

    \[\leadsto a2 \cdot a2 \]

Reproduce

herbie shell --seed 2023229 
(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))))