Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 5.2s

Alternatives: 8

Speedup: 1.0×

Specification

\[\mathsf{TRUE}\left(\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{t\_1}{ew}\right) - t\_1 \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (fabs
    (-
     (* (* ew (cos t)) (cos (atan (/ t_1 ew))))
     (* t_1 (sin (atan (/ (* (- eh) (tan t)) ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	return fabs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan(((-eh * tan(t)) / ew))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = eh * sin(t)
    code = abs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan(((-eh * tan(t)) / ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	return Math.abs((((ew * Math.cos(t)) * Math.cos(Math.atan((t_1 / ew)))) - (t_1 * Math.sin(Math.atan(((-eh * Math.tan(t)) / ew))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	return math.fabs((((ew * math.cos(t)) * math.cos(math.atan((t_1 / ew)))) - (t_1 * math.sin(math.atan(((-eh * math.tan(t)) / ew))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(atan(Float64(t_1 / ew)))) - Float64(t_1 * sin(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = abs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan(((-eh * tan(t)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(t$95$1 / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in eh around 0

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Applied rewrites 99.7%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
6. Add Preprocessing

Alternative 3: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{t\_1}{ew}\right) - t\_1 \cdot \sin \tan^{-1} \left(\frac{-eh}{ew}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (fabs
    (-
     (* (* ew (cos t)) (cos (atan (/ t_1 ew))))
     (* t_1 (sin (atan (/ (- eh) ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	return fabs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan((-eh / ew))))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = eh * sin(t)
    code = abs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan((-eh / ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	return Math.abs((((ew * Math.cos(t)) * Math.cos(Math.atan((t_1 / ew)))) - (t_1 * Math.sin(Math.atan((-eh / ew))))));
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	return math.fabs((((ew * math.cos(t)) * math.cos(math.atan((t_1 / ew)))) - (t_1 * math.sin(math.atan((-eh / ew))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(atan(Float64(t_1 / ew)))) - Float64(t_1 * sin(atan(Float64(Float64(-eh) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = abs((((ew * cos(t)) * cos(atan((t_1 / ew)))) - (t_1 * sin(atan((-eh / ew))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(t$95$1 / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[N[ArcTan[N[((-eh) / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in eh around 0

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Applied rewrites 99.7%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

6. Taylor expanded in eh around 0

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right)\right| \]

8. Applied rewrites 98.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh}}{ew}\right)\right| \]
9. Add Preprocessing

Alternative 4: 91.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{-eh}{ew}\right)\\ \mathbf{if}\;ew \leq -5 \cdot 10^{+83}:\\ \;\;\;\;\left|t\_2 \cdot \cos t\_3 - t\_1 \cdot \sin t\_3\right|\\ \mathbf{elif}\;ew \leq 6.7 \cdot 10^{+78}:\\ \;\;\;\;\left|t\_2 \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-eh\right)}{ew}\right) - t\_1 \cdot \sin \tan^{-1} \left(\frac{\left|-eh\right|}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))) (t_2 (* ew (cos t))) (t_3 (atan (/ (- eh) ew))))
   (if (<= ew -5e+83)
     (fabs (- (* t_2 (cos t_3)) (* t_1 (sin t_3))))
     (if (<= ew 6.7e+78)
       (fabs
        (-
         (* t_2 (cos (atan (/ (* eh (- eh)) ew))))
         (* t_1 (sin (atan (/ (fabs (- eh)) ew))))))
       (fabs t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = atan((-eh / ew));
	double tmp;
	if (ew <= -5e+83) {
		tmp = fabs(((t_2 * cos(t_3)) - (t_1 * sin(t_3))));
	} else if (ew <= 6.7e+78) {
		tmp = fabs(((t_2 * cos(atan(((eh * -eh) / ew)))) - (t_1 * sin(atan((fabs(-eh) / ew))))));
	} else {
		tmp = fabs(t_2);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = eh * sin(t)
    t_2 = ew * cos(t)
    t_3 = atan((-eh / ew))
    if (ew <= (-5d+83)) then
        tmp = abs(((t_2 * cos(t_3)) - (t_1 * sin(t_3))))
    else if (ew <= 6.7d+78) then
        tmp = abs(((t_2 * cos(atan(((eh * -eh) / ew)))) - (t_1 * sin(atan((abs(-eh) / ew))))))
    else
        tmp = abs(t_2)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.atan((-eh / ew));
	double tmp;
	if (ew <= -5e+83) {
		tmp = Math.abs(((t_2 * Math.cos(t_3)) - (t_1 * Math.sin(t_3))));
	} else if (ew <= 6.7e+78) {
		tmp = Math.abs(((t_2 * Math.cos(Math.atan(((eh * -eh) / ew)))) - (t_1 * Math.sin(Math.atan((Math.abs(-eh) / ew))))));
	} else {
		tmp = Math.abs(t_2);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	t_2 = ew * math.cos(t)
	t_3 = math.atan((-eh / ew))
	tmp = 0
	if ew <= -5e+83:
		tmp = math.fabs(((t_2 * math.cos(t_3)) - (t_1 * math.sin(t_3))))
	elif ew <= 6.7e+78:
		tmp = math.fabs(((t_2 * math.cos(math.atan(((eh * -eh) / ew)))) - (t_1 * math.sin(math.atan((math.fabs(-eh) / ew))))))
	else:
		tmp = math.fabs(t_2)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(-eh) / ew))
	tmp = 0.0
	if (ew <= -5e+83)
		tmp = abs(Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * sin(t_3))));
	elseif (ew <= 6.7e+78)
		tmp = abs(Float64(Float64(t_2 * cos(atan(Float64(Float64(eh * Float64(-eh)) / ew)))) - Float64(t_1 * sin(atan(Float64(abs(Float64(-eh)) / ew))))));
	else
		tmp = abs(t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	t_2 = ew * cos(t);
	t_3 = atan((-eh / ew));
	tmp = 0.0;
	if (ew <= -5e+83)
		tmp = abs(((t_2 * cos(t_3)) - (t_1 * sin(t_3))));
	elseif (ew <= 6.7e+78)
		tmp = abs(((t_2 * cos(atan(((eh * -eh) / ew)))) - (t_1 * sin(atan((abs(-eh) / ew))))));
	else
		tmp = abs(t_2);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[((-eh) / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5e+83], N[Abs[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6.7e+78], N[Abs[N[(N[(t$95$2 * N[Cos[N[ArcTan[N[(N[(eh * (-eh)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[N[ArcTan[N[(N[Abs[(-eh)], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$2], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.00000000000000029e83

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.0%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right)\right| \]

   8. Applied rewrites 98.7%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh}}{ew}\right)\right| \]

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-eh}{ew}\right)\right| \]

   11. Applied rewrites 90.1%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-eh}{ew}\right)\right| \]

   if -5.00000000000000029e83 < ew < 6.69999999999999967e78

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right)\right| \]

   8. Applied rewrites 98.9%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh}}{ew}\right)\right| \]

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-1 \cdot \color{blue}{eh}}{ew}\right)\right| \]

   11. Applied rewrites 98.9%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left|-eh\right|}{ew}\right)\right| \]

   12. Taylor expanded in t around 0

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left|-eh\right|}{ew}\right)\right| \]

   14. Applied rewrites 93.4%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-eh\right)}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left|-eh\right|}{ew}\right)\right| \]

   if 6.69999999999999967e78 < ew

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

   5. Applied rewrites 6.9%

      \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

   6. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right) + t \cdot \left(\frac{-1}{2} \cdot \left(ew \cdot \left(t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}\right| \]

   8. Applied rewrites 92.4%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 90.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{-eh}{ew}\right)\\ t_3 := \left|t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\right|\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{-91}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t \leq 3.6 \cdot 10^{-282}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (- eh) ew)))
        (t_3 (fabs (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))))))
   (if (<= t -3.6e-91) t_3 (if (<= t 3.6e-282) (fabs t_1) t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan((-eh / ew));
	double t_3 = fabs(((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))));
	double tmp;
	if (t <= -3.6e-91) {
		tmp = t_3;
	} else if (t <= 3.6e-282) {
		tmp = fabs(t_1);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan((-eh / ew))
    t_3 = abs(((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))))
    if (t <= (-3.6d-91)) then
        tmp = t_3
    else if (t <= 3.6d-282) then
        tmp = abs(t_1)
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan((-eh / ew));
	double t_3 = Math.abs(((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))));
	double tmp;
	if (t <= -3.6e-91) {
		tmp = t_3;
	} else if (t <= 3.6e-282) {
		tmp = Math.abs(t_1);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan((-eh / ew))
	t_3 = math.fabs(((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))))
	tmp = 0
	if t <= -3.6e-91:
		tmp = t_3
	elif t <= 3.6e-282:
		tmp = math.fabs(t_1)
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(-eh) / ew))
	t_3 = abs(Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))))
	tmp = 0.0
	if (t <= -3.6e-91)
		tmp = t_3;
	elseif (t <= 3.6e-282)
		tmp = abs(t_1);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan((-eh / ew));
	t_3 = abs(((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))));
	tmp = 0.0;
	if (t <= -3.6e-91)
		tmp = t_3;
	elseif (t <= 3.6e-282)
		tmp = abs(t_1);
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[((-eh) / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.6e-91], t$95$3, If[LessEqual[t, 3.6e-282], N[Abs[t$95$1], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.6e-91 or 3.5999999999999998e-282 < t

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.6%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\color{blue}{eh \cdot \sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-1 \cdot \frac{eh \cdot \sin t}{\cos t}}}{ew}\right)\right| \]

   8. Applied rewrites 98.6%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh}}{ew}\right)\right| \]

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \color{blue}{\sin t}}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-eh}{ew}\right)\right| \]

   11. Applied rewrites 94.0%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-eh}{ew}\right)\right| \]

   if -3.6e-91 < t < 3.5999999999999998e-282

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

   5. Applied rewrites 17.9%

      \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

   6. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right) + t \cdot \left(\frac{-1}{2} \cdot \left(ew \cdot \left(t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}\right| \]

   8. Applied rewrites 86.5%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 74.9% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -7.5 \cdot 10^{+105}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -7.5e+105) t_1 (if (<= eh 4.4e+111) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -7.5e+105) {
		tmp = t_1;
	} else if (eh <= 4.4e+111) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-7.5d+105)) then
        tmp = t_1
    else if (eh <= 4.4d+111) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -7.5e+105) {
		tmp = t_1;
	} else if (eh <= 4.4e+111) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -7.5e+105:
		tmp = t_1
	elif eh <= 4.4e+111:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -7.5e+105)
		tmp = t_1;
	elseif (eh <= 4.4e+111)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -7.5e+105)
		tmp = t_1;
	elseif (eh <= 4.4e+111)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7.5e+105], t$95$1, If[LessEqual[eh, 4.4e+111], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.5000000000000002e105 or 4.39999999999999997e111 < eh

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

   5. Applied rewrites 38.8%

      \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

   6. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right) + t \cdot \left(t \cdot \left(\frac{-1}{2} \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right) + \frac{1}{6} \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}\right| \]

   8. Applied rewrites 74.1%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]

   if -7.5000000000000002e105 < eh < 4.39999999999999997e111

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

   5. Applied rewrites 11.8%

      \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

   6. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right) + t \cdot \left(\frac{-1}{2} \cdot \left(ew \cdot \left(t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}\right| \]

   8. Applied rewrites 82.5%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 60.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew (cos t))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * cos(t)));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * cos(t)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * Math.cos(t)));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * math.cos(t)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * cos(t)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * cos(t)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in eh around inf

   \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

5. Applied rewrites 22.4%

   \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

6. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right) + t \cdot \left(\frac{-1}{2} \cdot \left(ew \cdot \left(t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right) - eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}\right| \]

8. Applied rewrites 61.4%

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
9. Add Preprocessing

Alternative 8: 9.5% accurate, 172.4× speedup

Math

\[\begin{array}{l} \\ \left|-eh\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (- eh)))

C

double code(double eh, double ew, double t) {
	return fabs(-eh);
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(-eh)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(-eh);
}

Python

def code(eh, ew, t):
	return math.fabs(-eh)

Julia

function code(eh, ew, t)
	return abs(Float64(-eh))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(-eh);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[(-eh)], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing

3. Taylor expanded in eh around inf

   \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]

5. Applied rewrites 22.4%

   \[\leadsto \left|\color{blue}{\left(-eh\right) \cdot \tan t}\right| \]

6. Taylor expanded in eh around 0

   \[\leadsto \left|-1 \cdot \color{blue}{\frac{eh \cdot \sin t}{\cos t}}\right| \]

8. Applied rewrites 9.4%

   \[\leadsto \left|-eh\right| \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024321 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  :pre (TRUE)
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))