Migdal et al, Equation (64)

Percentage Accurate: 99.5% → 99.5%
Time: 8.6s
Alternatives: 10
Speedup: 2.0×

Specification

?
\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))
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));
}

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

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));
}

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

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

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

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]]

\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}

Sampling outcomes in binary64 precision:

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 10 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Initial Program: 99.5% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\cos th}{\sqrt{2}}\\ t_1 \cdot \left(a1 \cdot a1\right) + t_1 \cdot \left(a2 \cdot a2\right) \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (/ (cos th) (sqrt 2.0))))
   (+ (* t_1 (* a1 a1)) (* t_1 (* a2 a2)))))
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));
}

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

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));
}

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

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

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

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]]

\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}

Alternative 1: 99.5% accurate, 0.8× speedup?

\[\begin{array}{l} \\ \frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (/ (/ (* (cos th) (pow (hypot a1 a2) 2.0)) (pow 2.0 0.25)) (pow 2.0 0.25)))
double code(double a1, double a2, double th) {
	return ((cos(th) * pow(hypot(a1, a2), 2.0)) / pow(2.0, 0.25)) / pow(2.0, 0.25);
}

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

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

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

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

code[a1_, a2_, th_] := N[(N[(N[(N[Cos[th], $MachinePrecision] * N[Power[N[Sqrt[a1 ^ 2 + a2 ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision] / N[Power[2.0, 0.25], $MachinePrecision]), $MachinePrecision]

\begin{array}{l}

\\
\frac{\frac{\cos th \cdot {\left(\mathsf{hypot}\left(a1, a2\right)\right)}^{2}}{{2}^{0.25}}}{{2}^{0.25}}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 2: 70.8% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7)
   (* a2 (* (cos th) a2))
   (* (sqrt 0.5) (+ (* a1 a1) (* a2 a2)))))
double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = sqrt(0.5) * ((a1 * a1) + (a2 * a2));
	}
	return tmp;
}

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.7d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = sqrt(0.5d0) * ((a1 * a1) + (a2 * a2))
    end if
    code = tmp
end function

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

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

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

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

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

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\cos th \leq 0.7:\\
\;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5} \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 3: 99.6% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \left(\cos th \cdot \sqrt{0.5}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right) \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (* (* (cos th) (sqrt 0.5)) (+ (* a1 a1) (* a2 a2))))
double code(double a1, double a2, double th) {
	return (cos(th) * sqrt(0.5)) * ((a1 * a1) + (a2 * a2));
}

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

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

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

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

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

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

\begin{array}{l}

\\
\left(\cos th \cdot \sqrt{0.5}\right) \cdot \left(a1 \cdot a1 + a2 \cdot a2\right)
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 4: 47.7% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* a2 (* (cos th) a2)) (* a2 (* a2 (sqrt 0.5)))))
double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = a2 * (a2 * sqrt(0.5));
	}
	return tmp;
}

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.7d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = a2 * (a2 * sqrt(0.5d0))
    end if
    code = tmp
end function

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

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

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

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

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

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\cos th \leq 0.7:\\
\;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\

\mathbf{else}:\\
\;\;\;\;a2 \cdot \left(a2 \cdot \sqrt{0.5}\right)\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 5: 47.7% accurate, 2.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos th \leq 0.7:\\ \;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\ \mathbf{else}:\\ \;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= (cos th) 0.7) (* a2 (* (cos th) a2)) (* a2 (/ a2 (sqrt 2.0)))))
double code(double a1, double a2, double th) {
	double tmp;
	if (cos(th) <= 0.7) {
		tmp = a2 * (cos(th) * a2);
	} else {
		tmp = a2 * (a2 / sqrt(2.0));
	}
	return tmp;
}

real(8) function code(a1, a2, th)
    real(8), intent (in) :: a1
    real(8), intent (in) :: a2
    real(8), intent (in) :: th
    real(8) :: tmp
    if (cos(th) <= 0.7d0) then
        tmp = a2 * (cos(th) * a2)
    else
        tmp = a2 * (a2 / sqrt(2.0d0))
    end if
    code = tmp
end function

public static double code(double a1, double a2, double th) {
	double tmp;
	if (Math.cos(th) <= 0.7) {
		tmp = a2 * (Math.cos(th) * a2);
	} else {
		tmp = a2 * (a2 / Math.sqrt(2.0));
	}
	return tmp;
}

def code(a1, a2, th):
	tmp = 0
	if math.cos(th) <= 0.7:
		tmp = a2 * (math.cos(th) * a2)
	else:
		tmp = a2 * (a2 / math.sqrt(2.0))
	return tmp

function code(a1, a2, th)
	tmp = 0.0
	if (cos(th) <= 0.7)
		tmp = Float64(a2 * Float64(cos(th) * a2));
	else
		tmp = Float64(a2 * Float64(a2 / sqrt(2.0)));
	end
	return tmp
end

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (cos(th) <= 0.7)
		tmp = a2 * (cos(th) * a2);
	else
		tmp = a2 * (a2 / sqrt(2.0));
	end
	tmp_2 = tmp;
end

code[a1_, a2_, th_] := If[LessEqual[N[Cos[th], $MachinePrecision], 0.7], N[(a2 * N[(N[Cos[th], $MachinePrecision] * a2), $MachinePrecision]), $MachinePrecision], N[(a2 * N[(a2 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;\cos th \leq 0.7:\\
\;\;\;\;a2 \cdot \left(\cos th \cdot a2\right)\\

\mathbf{else}:\\
\;\;\;\;a2 \cdot \frac{a2}{\sqrt{2}}\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 6: 38.3% accurate, 4.0× speedup?

\[\begin{array}{l} \\ a2 \cdot \left(\cos th \cdot a2\right) \end{array} \]
(FPCore (a1 a2 th) :precision binary64 (* a2 (* (cos th) a2)))
double code(double a1, double a2, double th) {
	return a2 * (cos(th) * a2);
}

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

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

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

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

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

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

\begin{array}{l}

\\
a2 \cdot \left(\cos th \cdot a2\right)
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 7: 33.5% accurate, 37.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 220000:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 220000.0) (* a2 a2) (* (+ (* a1 a1) (* a2 a2)) -0.5)))
double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 220000.0) {
		tmp = a2 * a2;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	return tmp;
}

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 <= 220000.0d0) then
        tmp = a2 * a2
    else
        tmp = ((a1 * a1) + (a2 * a2)) * (-0.5d0)
    end if
    code = tmp
end function

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 220000.0) {
		tmp = a2 * a2;
	} else {
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	}
	return tmp;
}

def code(a1, a2, th):
	tmp = 0
	if th <= 220000.0:
		tmp = a2 * a2
	else:
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5
	return tmp

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 220000.0)
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(Float64(Float64(a1 * a1) + Float64(a2 * a2)) * -0.5);
	end
	return tmp
end

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 220000.0)
		tmp = a2 * a2;
	else
		tmp = ((a1 * a1) + (a2 * a2)) * -0.5;
	end
	tmp_2 = tmp;
end

code[a1_, a2_, th_] := If[LessEqual[th, 220000.0], N[(a2 * a2), $MachinePrecision], N[(N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;th \leq 220000:\\
\;\;\;\;a2 \cdot a2\\

\mathbf{else}:\\
\;\;\;\;\left(a1 \cdot a1 + a2 \cdot a2\right) \cdot -0.5\\


\end{array}
\end{array}
Derivation
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  1. Add Preprocessing

Alternative 8: 47.3% accurate, 37.5× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a1 \cdot a1 + a2 \cdot a2\\ \mathbf{if}\;th \leq 220000:\\ \;\;\;\;0.5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot -0.5\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (let* ((t_1 (+ (* a1 a1) (* a2 a2))))
   (if (<= th 220000.0) (* 0.5 t_1) (* t_1 -0.5))))
double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 220000.0) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

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 <= 220000.0d0) then
        tmp = 0.5d0 * t_1
    else
        tmp = t_1 * (-0.5d0)
    end if
    code = tmp
end function

public static double code(double a1, double a2, double th) {
	double t_1 = (a1 * a1) + (a2 * a2);
	double tmp;
	if (th <= 220000.0) {
		tmp = 0.5 * t_1;
	} else {
		tmp = t_1 * -0.5;
	}
	return tmp;
}

def code(a1, a2, th):
	t_1 = (a1 * a1) + (a2 * a2)
	tmp = 0
	if th <= 220000.0:
		tmp = 0.5 * t_1
	else:
		tmp = t_1 * -0.5
	return tmp

function code(a1, a2, th)
	t_1 = Float64(Float64(a1 * a1) + Float64(a2 * a2))
	tmp = 0.0
	if (th <= 220000.0)
		tmp = Float64(0.5 * t_1);
	else
		tmp = Float64(t_1 * -0.5);
	end
	return tmp
end

function tmp_2 = code(a1, a2, th)
	t_1 = (a1 * a1) + (a2 * a2);
	tmp = 0.0;
	if (th <= 220000.0)
		tmp = 0.5 * t_1;
	else
		tmp = t_1 * -0.5;
	end
	tmp_2 = tmp;
end

code[a1_, a2_, th_] := Block[{t$95$1 = N[(N[(a1 * a1), $MachinePrecision] + N[(a2 * a2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[th, 220000.0], N[(0.5 * t$95$1), $MachinePrecision], N[(t$95$1 * -0.5), $MachinePrecision]]]

\begin{array}{l}

\\
\begin{array}{l}
t_1 := a1 \cdot a1 + a2 \cdot a2\\
\mathbf{if}\;th \leq 220000:\\
\;\;\;\;0.5 \cdot t_1\\

\mathbf{else}:\\
\;\;\;\;t_1 \cdot -0.5\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 9: 33.3% accurate, 41.2× speedup?

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;th \leq 220000:\\ \;\;\;\;a2 \cdot a2\\ \mathbf{else}:\\ \;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\ \end{array} \end{array} \]
(FPCore (a1 a2 th)
 :precision binary64
 (if (<= th 220000.0) (* a2 a2) (- (* a1 (- a1)) (* a2 a2))))
double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 220000.0) {
		tmp = a2 * a2;
	} else {
		tmp = (a1 * -a1) - (a2 * a2);
	}
	return tmp;
}

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 <= 220000.0d0) then
        tmp = a2 * a2
    else
        tmp = (a1 * -a1) - (a2 * a2)
    end if
    code = tmp
end function

public static double code(double a1, double a2, double th) {
	double tmp;
	if (th <= 220000.0) {
		tmp = a2 * a2;
	} else {
		tmp = (a1 * -a1) - (a2 * a2);
	}
	return tmp;
}

def code(a1, a2, th):
	tmp = 0
	if th <= 220000.0:
		tmp = a2 * a2
	else:
		tmp = (a1 * -a1) - (a2 * a2)
	return tmp

function code(a1, a2, th)
	tmp = 0.0
	if (th <= 220000.0)
		tmp = Float64(a2 * a2);
	else
		tmp = Float64(Float64(a1 * Float64(-a1)) - Float64(a2 * a2));
	end
	return tmp
end

function tmp_2 = code(a1, a2, th)
	tmp = 0.0;
	if (th <= 220000.0)
		tmp = a2 * a2;
	else
		tmp = (a1 * -a1) - (a2 * a2);
	end
	tmp_2 = tmp;
end

code[a1_, a2_, th_] := If[LessEqual[th, 220000.0], N[(a2 * a2), $MachinePrecision], N[(N[(a1 * (-a1)), $MachinePrecision] - N[(a2 * a2), $MachinePrecision]), $MachinePrecision]]

\begin{array}{l}

\\
\begin{array}{l}
\mathbf{if}\;th \leq 220000:\\
\;\;\;\;a2 \cdot a2\\

\mathbf{else}:\\
\;\;\;\;a1 \cdot \left(-a1\right) - a2 \cdot a2\\


\end{array}
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Alternative 10: 30.3% accurate, 138.3× speedup?

\[\begin{array}{l} \\ a2 \cdot a2 \end{array} \]
(FPCore (a1 a2 th) :precision binary64 (* a2 a2))
double code(double a1, double a2, double th) {
	return a2 * a2;
}

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

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

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

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

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

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

\begin{array}{l}

\\
a2 \cdot a2
\end{array}
Derivation
    &prev;&pcontext;&pcontext2;&ctx;
  1. Add Preprocessing

Reproduce

?
herbie shell --seed 2024008 
(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))))