Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.4s

Alternatives: 12

Speedup: 1.0×

Specification

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[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. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((sin(t) * eh) * sin(atan(((-eh * t) / ew)))) - ((cos(t) * ew) * cos(atan(((tan(t) * 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
    code = abs((((sin(t) * eh) * sin(atan(((-eh * t) / ew)))) - ((cos(t) * ew) * cos(atan(((tan(t) * eh) / -ew))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $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 t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 4: 83.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot t\_1\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+229}:\\ \;\;\;\;\left|\frac{t\_1 \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (fabs (* (sin (atan (* (/ (sin t) ew) (/ eh (cos t))))) t_1))))
   (if (<= eh -9e+51)
     t_2
     (if (<= eh 2.5e+229)
       (fabs
        (/
         (+ (* t_1 (* (/ (tan t) ew) eh)) (* (cos t) ew))
         (/ 1.0 (pow (+ 1.0 (pow (/ ew (* (tan t) eh)) -2.0)) -0.5))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * t_1));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 2.5e+229) {
		tmp = fabs((((t_1 * ((tan(t) / ew) * eh)) + (cos(t) * ew)) / (1.0 / pow((1.0 + pow((ew / (tan(t) * eh)), -2.0)), -0.5))));
	} else {
		tmp = 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) :: tmp
    t_1 = sin(t) * eh
    t_2 = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * t_1))
    if (eh <= (-9d+51)) then
        tmp = t_2
    else if (eh <= 2.5d+229) then
        tmp = abs((((t_1 * ((tan(t) / ew) * eh)) + (cos(t) * ew)) / (1.0d0 / ((1.0d0 + ((ew / (tan(t) * eh)) ** (-2.0d0))) ** (-0.5d0)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double t_2 = Math.abs((Math.sin(Math.atan(((Math.sin(t) / ew) * (eh / Math.cos(t))))) * t_1));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 2.5e+229) {
		tmp = Math.abs((((t_1 * ((Math.tan(t) / ew) * eh)) + (Math.cos(t) * ew)) / (1.0 / Math.pow((1.0 + Math.pow((ew / (Math.tan(t) * eh)), -2.0)), -0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.fabs((math.sin(math.atan(((math.sin(t) / ew) * (eh / math.cos(t))))) * t_1))
	tmp = 0
	if eh <= -9e+51:
		tmp = t_2
	elif eh <= 2.5e+229:
		tmp = math.fabs((((t_1 * ((math.tan(t) / ew) * eh)) + (math.cos(t) * ew)) / (1.0 / math.pow((1.0 + math.pow((ew / (math.tan(t) * eh)), -2.0)), -0.5))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = abs(Float64(sin(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * t_1))
	tmp = 0.0
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 2.5e+229)
		tmp = abs(Float64(Float64(Float64(t_1 * Float64(Float64(tan(t) / ew) * eh)) + Float64(cos(t) * ew)) / Float64(1.0 / (Float64(1.0 + (Float64(ew / Float64(tan(t) * eh)) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * t_1));
	tmp = 0.0;
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 2.5e+229)
		tmp = abs((((t_1 * ((tan(t) / ew) * eh)) + (cos(t) * ew)) / (1.0 / ((1.0 + ((ew / (tan(t) * eh)) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+51], t$95$2, If[LessEqual[eh, 2.5e+229], N[Abs[N[(N[(N[(t$95$1 * N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 + N[Power[N[(ew / N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999999e51 or 2.50000000000000025e229 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 78.7

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

   7. Applied rewrites 78.7%

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

   if -8.9999999999999999e51 < eh < 2.50000000000000025e229

   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

   4. Applied rewrites 78.0%

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

   6. Applied rewrites 93.1%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{+229}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ t_2 := \frac{\tan t}{ew} \cdot eh\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\left(t\_2 \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot \cos \tan^{-1} t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (sin (atan (* (/ (sin t) ew) (/ eh (cos t))))) (* (sin t) eh))))
        (t_2 (* (/ (tan t) ew) eh)))
   (if (<= eh -9e+51)
     t_1
     (if (<= eh 0.22)
       (* (fabs (+ (* (* t_2 eh) (sin t)) (* (cos t) ew))) (cos (atan t_2)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)));
	double t_2 = (tan(t) / ew) * eh;
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_1;
	} else if (eh <= 0.22) {
		tmp = fabs((((t_2 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_2));
	} 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) :: t_2
    real(8) :: tmp
    t_1 = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)))
    t_2 = (tan(t) / ew) * eh
    if (eh <= (-9d+51)) then
        tmp = t_1
    else if (eh <= 0.22d0) then
        tmp = abs((((t_2 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_2))
    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((Math.sin(Math.atan(((Math.sin(t) / ew) * (eh / Math.cos(t))))) * (Math.sin(t) * eh)));
	double t_2 = (Math.tan(t) / ew) * eh;
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_1;
	} else if (eh <= 0.22) {
		tmp = Math.abs((((t_2 * eh) * Math.sin(t)) + (Math.cos(t) * ew))) * Math.cos(Math.atan(t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((math.sin(t) / ew) * (eh / math.cos(t))))) * (math.sin(t) * eh)))
	t_2 = (math.tan(t) / ew) * eh
	tmp = 0
	if eh <= -9e+51:
		tmp = t_1
	elif eh <= 0.22:
		tmp = math.fabs((((t_2 * eh) * math.sin(t)) + (math.cos(t) * ew))) * math.cos(math.atan(t_2))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * Float64(sin(t) * eh)))
	t_2 = Float64(Float64(tan(t) / ew) * eh)
	tmp = 0.0
	if (eh <= -9e+51)
		tmp = t_1;
	elseif (eh <= 0.22)
		tmp = Float64(abs(Float64(Float64(Float64(t_2 * eh) * sin(t)) + Float64(cos(t) * ew))) * cos(atan(t_2)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)));
	t_2 = (tan(t) / ew) * eh;
	tmp = 0.0;
	if (eh <= -9e+51)
		tmp = t_1;
	elseif (eh <= 0.22)
		tmp = abs((((t_2 * eh) * sin(t)) + (cos(t) * ew))) * cos(atan(t_2));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -9e+51], t$95$1, If[LessEqual[eh, 0.22], N[(N[Abs[N[(N[(N[(t$95$2 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999999e51 or 0.220000000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if -8.9999999999999999e51 < eh < 0.220000000000000001

   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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\left|\cos t \cdot ew - \left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \left(-\sin t\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\cos t}\\ t_2 := \left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot t\_1\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot t\_1\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (cos t)))
        (t_2 (fabs (* (sin (atan (* (/ (sin t) ew) t_1))) (* (sin t) eh)))))
   (if (<= eh -9e+51)
     t_2
     (if (<= eh 0.22)
       (fabs (* (cos (atan (* (/ (- (sin t)) ew) t_1))) (* (cos t) ew)))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / cos(t);
	double t_2 = fabs((sin(atan(((sin(t) / ew) * t_1))) * (sin(t) * eh)));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 0.22) {
		tmp = fabs((cos(atan(((-sin(t) / ew) * t_1))) * (cos(t) * ew)));
	} else {
		tmp = 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) :: tmp
    t_1 = eh / cos(t)
    t_2 = abs((sin(atan(((sin(t) / ew) * t_1))) * (sin(t) * eh)))
    if (eh <= (-9d+51)) then
        tmp = t_2
    else if (eh <= 0.22d0) then
        tmp = abs((cos(atan(((-sin(t) / ew) * t_1))) * (cos(t) * ew)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / Math.cos(t);
	double t_2 = Math.abs((Math.sin(Math.atan(((Math.sin(t) / ew) * t_1))) * (Math.sin(t) * eh)));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 0.22) {
		tmp = Math.abs((Math.cos(Math.atan(((-Math.sin(t) / ew) * t_1))) * (Math.cos(t) * ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / math.cos(t)
	t_2 = math.fabs((math.sin(math.atan(((math.sin(t) / ew) * t_1))) * (math.sin(t) * eh)))
	tmp = 0
	if eh <= -9e+51:
		tmp = t_2
	elif eh <= 0.22:
		tmp = math.fabs((math.cos(math.atan(((-math.sin(t) / ew) * t_1))) * (math.cos(t) * ew)))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / cos(t))
	t_2 = abs(Float64(sin(atan(Float64(Float64(sin(t) / ew) * t_1))) * Float64(sin(t) * eh)))
	tmp = 0.0
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 0.22)
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-sin(t)) / ew) * t_1))) * Float64(cos(t) * ew)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / cos(t);
	t_2 = abs((sin(atan(((sin(t) / ew) * t_1))) * (sin(t) * eh)));
	tmp = 0.0;
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 0.22)
		tmp = abs((cos(atan(((-sin(t) / ew) * t_1))) * (cos(t) * ew)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+51], t$95$2, If[LessEqual[eh, 0.22], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999999e51 or 0.220000000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if -8.9999999999999999e51 < eh < 0.220000000000000001

   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 ew around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

      \[\leadsto \left|\color{blue}{\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\frac{\left(-\cos t\right) \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (sin (atan (* (/ (sin t) ew) (/ eh (cos t))))) (* (sin t) eh)))))
   (if (<= eh -9e+51)
     t_1
     (if (<= eh 0.22)
       (fabs
        (/ (* (- (cos t)) ew) (/ -1.0 (cos (atan (* (/ (tan t) ew) eh))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_1;
	} else if (eh <= 0.22) {
		tmp = fabs(((-cos(t) * ew) / (-1.0 / cos(atan(((tan(t) / ew) * eh))))));
	} 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((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)))
    if (eh <= (-9d+51)) then
        tmp = t_1
    else if (eh <= 0.22d0) then
        tmp = abs(((-cos(t) * ew) / ((-1.0d0) / cos(atan(((tan(t) / ew) * eh))))))
    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((Math.sin(Math.atan(((Math.sin(t) / ew) * (eh / Math.cos(t))))) * (Math.sin(t) * eh)));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_1;
	} else if (eh <= 0.22) {
		tmp = Math.abs(((-Math.cos(t) * ew) / (-1.0 / Math.cos(Math.atan(((Math.tan(t) / ew) * eh))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((math.sin(t) / ew) * (eh / math.cos(t))))) * (math.sin(t) * eh)))
	tmp = 0
	if eh <= -9e+51:
		tmp = t_1
	elif eh <= 0.22:
		tmp = math.fabs(((-math.cos(t) * ew) / (-1.0 / math.cos(math.atan(((math.tan(t) / ew) * eh))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * Float64(sin(t) * eh)))
	tmp = 0.0
	if (eh <= -9e+51)
		tmp = t_1;
	elseif (eh <= 0.22)
		tmp = abs(Float64(Float64(Float64(-cos(t)) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(tan(t) / ew) * eh))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (sin(t) * eh)));
	tmp = 0.0;
	if (eh <= -9e+51)
		tmp = t_1;
	elseif (eh <= 0.22)
		tmp = abs(((-cos(t) * ew) / (-1.0 / cos(atan(((tan(t) / ew) * eh))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+51], t$95$1, If[LessEqual[eh, 0.22], N[Abs[N[(N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999999e51 or 0.220000000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. *-commutative N/A

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

      9. times-frac N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 71.8

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

   7. Applied rewrites 71.8%

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

   if -8.9999999999999999e51 < eh < 0.220000000000000001

   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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

      \[\leadsto \left|\frac{\color{blue}{\left(-ew\right) \cdot \cos t}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\frac{\left(-\cos t\right) \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ t_2 := \left|\frac{\sin t\_1 \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\frac{\left(-\cos t\right) \cdot ew}{\frac{-1}{\cos t\_1}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (tan t) ew) eh)))
        (t_2 (fabs (* (/ (* (sin t_1) (* (sin t) eh)) ew) ew))))
   (if (<= eh -9e+51)
     t_2
     (if (<= eh 0.22) (fabs (/ (* (- (cos t)) ew) (/ -1.0 (cos t_1)))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((tan(t) / ew) * eh));
	double t_2 = fabs((((sin(t_1) * (sin(t) * eh)) / ew) * ew));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 0.22) {
		tmp = fabs(((-cos(t) * ew) / (-1.0 / cos(t_1))));
	} else {
		tmp = 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) :: tmp
    t_1 = atan(((tan(t) / ew) * eh))
    t_2 = abs((((sin(t_1) * (sin(t) * eh)) / ew) * ew))
    if (eh <= (-9d+51)) then
        tmp = t_2
    else if (eh <= 0.22d0) then
        tmp = abs(((-cos(t) * ew) / ((-1.0d0) / cos(t_1))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((Math.tan(t) / ew) * eh));
	double t_2 = Math.abs((((Math.sin(t_1) * (Math.sin(t) * eh)) / ew) * ew));
	double tmp;
	if (eh <= -9e+51) {
		tmp = t_2;
	} else if (eh <= 0.22) {
		tmp = Math.abs(((-Math.cos(t) * ew) / (-1.0 / Math.cos(t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((math.tan(t) / ew) * eh))
	t_2 = math.fabs((((math.sin(t_1) * (math.sin(t) * eh)) / ew) * ew))
	tmp = 0
	if eh <= -9e+51:
		tmp = t_2
	elif eh <= 0.22:
		tmp = math.fabs(((-math.cos(t) * ew) / (-1.0 / math.cos(t_1))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(tan(t) / ew) * eh))
	t_2 = abs(Float64(Float64(Float64(sin(t_1) * Float64(sin(t) * eh)) / ew) * ew))
	tmp = 0.0
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 0.22)
		tmp = abs(Float64(Float64(Float64(-cos(t)) * ew) / Float64(-1.0 / cos(t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((tan(t) / ew) * eh));
	t_2 = abs((((sin(t_1) * (sin(t) * eh)) / ew) * ew));
	tmp = 0.0;
	if (eh <= -9e+51)
		tmp = t_2;
	elseif (eh <= 0.22)
		tmp = abs(((-cos(t) * ew) / (-1.0 / cos(t_1))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -9e+51], t$95$2, If[LessEqual[eh, 0.22], N[Abs[N[(N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision] / N[(-1.0 / N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.9999999999999999e51 or 0.220000000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}{ew} \cdot ew\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

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

   12. Applied rewrites 58.6%

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

   if -8.9999999999999999e51 < eh < 0.220000000000000001

   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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in ew around inf

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
   6. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

      \[\leadsto \left|\frac{\color{blue}{\left(-ew\right) \cdot \cos t}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -9 \cdot 10^{+51}:\\ \;\;\;\;\left|\frac{\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \mathbf{elif}\;eh \leq 0.22:\\ \;\;\;\;\left|\frac{\left(-\cos t\right) \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\frac{\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \mathbf{if}\;eh \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.195:\\ \;\;\;\;\left|\left(\left(-\cos t\right) \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (/ (* (sin (atan (* (/ (tan t) ew) eh))) (* (sin t) eh)) ew)
           ew))))
   (if (<= eh -1.65e+49)
     t_1
     (if (<= eh 0.195)
       (fabs (* (* (- (cos t)) ew) (cos (atan (* (/ t ew) eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((((sin(atan(((tan(t) / ew) * eh))) * (sin(t) * eh)) / ew) * ew));
	double tmp;
	if (eh <= -1.65e+49) {
		tmp = t_1;
	} else if (eh <= 0.195) {
		tmp = fabs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))));
	} 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((((sin(atan(((tan(t) / ew) * eh))) * (sin(t) * eh)) / ew) * ew))
    if (eh <= (-1.65d+49)) then
        tmp = t_1
    else if (eh <= 0.195d0) then
        tmp = abs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))))
    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((((Math.sin(Math.atan(((Math.tan(t) / ew) * eh))) * (Math.sin(t) * eh)) / ew) * ew));
	double tmp;
	if (eh <= -1.65e+49) {
		tmp = t_1;
	} else if (eh <= 0.195) {
		tmp = Math.abs(((-Math.cos(t) * ew) * Math.cos(Math.atan(((t / ew) * eh)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((((math.sin(math.atan(((math.tan(t) / ew) * eh))) * (math.sin(t) * eh)) / ew) * ew))
	tmp = 0
	if eh <= -1.65e+49:
		tmp = t_1
	elif eh <= 0.195:
		tmp = math.fabs(((-math.cos(t) * ew) * math.cos(math.atan(((t / ew) * eh)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(sin(atan(Float64(Float64(tan(t) / ew) * eh))) * Float64(sin(t) * eh)) / ew) * ew))
	tmp = 0.0
	if (eh <= -1.65e+49)
		tmp = t_1;
	elseif (eh <= 0.195)
		tmp = abs(Float64(Float64(Float64(-cos(t)) * ew) * cos(atan(Float64(Float64(t / ew) * eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((((sin(atan(((tan(t) / ew) * eh))) * (sin(t) * eh)) / ew) * ew));
	tmp = 0.0;
	if (eh <= -1.65e+49)
		tmp = t_1;
	elseif (eh <= 0.195)
		tmp = abs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[(N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.65e+49], t$95$1, If[LessEqual[eh, 0.195], N[Abs[N[(N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.6499999999999999e49 or 0.19500000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}{ew} \cdot ew\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

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

   12. Applied rewrites 58.6%

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

   if -1.6499999999999999e49 < eh < 0.19500000000000001

   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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in ew around inf

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 77.8

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

   10. Applied rewrites 77.8%

       \[\leadsto \left|\frac{\color{blue}{\left(-ew\right) \cdot \cos t}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 77.8

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

   12. Applied rewrites 77.8%

       \[\leadsto \left|\color{blue}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\left(-\cos t\right) \cdot ew\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;\left|\frac{\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \mathbf{elif}\;eh \leq 0.195:\\ \;\;\;\;\left|\left(\left(-\cos t\right) \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(\sin t \cdot eh\right)}{ew} \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\frac{\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)}{ew} \cdot \left(\sin t \cdot eh\right)\right) \cdot ew\right|\\ \mathbf{if}\;eh \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.195:\\ \;\;\;\;\left|\left(\left(-\cos t\right) \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs (* (* (/ (sin (atan (/ (* eh t) ew))) ew) (* (sin t) eh)) ew))))
   (if (<= eh -1.65e+49)
     t_1
     (if (<= eh 0.195)
       (fabs (* (* (- (cos t)) ew) (cos (atan (* (/ t ew) eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((((sin(atan(((eh * t) / ew))) / ew) * (sin(t) * eh)) * ew));
	double tmp;
	if (eh <= -1.65e+49) {
		tmp = t_1;
	} else if (eh <= 0.195) {
		tmp = fabs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))));
	} 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((((sin(atan(((eh * t) / ew))) / ew) * (sin(t) * eh)) * ew))
    if (eh <= (-1.65d+49)) then
        tmp = t_1
    else if (eh <= 0.195d0) then
        tmp = abs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))))
    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((((Math.sin(Math.atan(((eh * t) / ew))) / ew) * (Math.sin(t) * eh)) * ew));
	double tmp;
	if (eh <= -1.65e+49) {
		tmp = t_1;
	} else if (eh <= 0.195) {
		tmp = Math.abs(((-Math.cos(t) * ew) * Math.cos(Math.atan(((t / ew) * eh)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((((math.sin(math.atan(((eh * t) / ew))) / ew) * (math.sin(t) * eh)) * ew))
	tmp = 0
	if eh <= -1.65e+49:
		tmp = t_1
	elif eh <= 0.195:
		tmp = math.fabs(((-math.cos(t) * ew) * math.cos(math.atan(((t / ew) * eh)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(sin(atan(Float64(Float64(eh * t) / ew))) / ew) * Float64(sin(t) * eh)) * ew))
	tmp = 0.0
	if (eh <= -1.65e+49)
		tmp = t_1;
	elseif (eh <= 0.195)
		tmp = abs(Float64(Float64(Float64(-cos(t)) * ew) * cos(atan(Float64(Float64(t / ew) * eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((((sin(atan(((eh * t) / ew))) / ew) * (sin(t) * eh)) * ew));
	tmp = 0.0;
	if (eh <= -1.65e+49)
		tmp = t_1;
	elseif (eh <= 0.195)
		tmp = abs(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[(N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.65e+49], t$95$1, If[LessEqual[eh, 0.195], N[Abs[N[(N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.6499999999999999e49 or 0.19500000000000001 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in ew around 0

      \[\leadsto \left|\frac{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)}{ew} \cdot ew\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 58.4%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|\left(\left(\sin t \cdot eh\right) \cdot \frac{\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)}{ew}\right) \cdot ew\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 58.5%

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

   if -1.6499999999999999e49 < eh < 0.19500000000000001

   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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 79.4

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in ew around inf

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 77.8

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

   10. Applied rewrites 77.8%

       \[\leadsto \left|\frac{\color{blue}{\left(-ew\right) \cdot \cos t}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
   11. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/r/ N/A

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

       4. /-rgt-identity N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 77.8

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

   12. Applied rewrites 77.8%

       \[\leadsto \left|\color{blue}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\left(-\cos t\right) \cdot ew\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.65 \cdot 10^{+49}:\\ \;\;\;\;\left|\left(\frac{\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)}{ew} \cdot \left(\sin t \cdot eh\right)\right) \cdot ew\right|\\ \mathbf{elif}\;eh \leq 0.195:\\ \;\;\;\;\left|\left(\left(-\cos t\right) \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)}{ew} \cdot \left(\sin t \cdot eh\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.4% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((-cos(t) * ew) * cos(atan(((t / ew) * 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(((-cos(t) * ew) * cos(atan(((t / ew) * eh)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 63.0%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 56.0

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

7. Applied rewrites 56.0%

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

8. Taylor expanded in ew around inf

   \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 53.9

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

10. Applied rewrites 53.9%

    \[\leadsto \left|\frac{\color{blue}{\left(-ew\right) \cdot \cos t}}{\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)}}\right| \]
11. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. associate-/r/ N/A

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

    4. /-rgt-identity N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 53.9

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

12. Applied rewrites 53.9%

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

13. Final simplification 53.9%

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

Alternative 12: 42.0% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew}{1}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ ew 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((ew / 1.0));
}

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 / 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew / 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((ew / 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(ew / 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew / 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 43.2%

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

6. Taylor expanded in t around 0

   \[\leadsto \left|\cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot ew\right| \]
7. Step-by-step derivation

8. Applied rewrites 42.5%

   \[\leadsto \left|\cos \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right) \cdot ew\right| \]
9. Step-by-step derivation

10. Applied rewrites 41.7%

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

11. Taylor expanded in ew around inf

    \[\leadsto \left|\frac{ew}{1}\right| \]
12. Step-by-step derivation

13. Applied rewrites 43.4%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))