Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.2s

Alternatives: 18

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (/ (* ew (cos t)) (hypot 1.0 (* (tan t) (/ eh ew))))
   (* eh (* (sin t) (sin (atan (* eh (/ (tan t) (- ew))))))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))) - (eh * (sin(t) * sin(atan((eh * (tan(t) / -ew))))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))) - (eh * (Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / -ew))))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) / math.hypot(1.0, (math.tan(t) * (eh / ew)))) - (eh * (math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / -ew))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))) - Float64(eh * Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))) - (eh * (sin(t) * sin(atan((eh * (tan(t) / -ew))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $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. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

   1. cos-atan 99.8%

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

   2. un-div-inv 99.8%

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

   3. hypot-1-def 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-/l* 99.8%

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

   7. add-sqr-sqrt 51.0%

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

   8. sqrt-unprod 93.6%

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

   9. sqr-neg 93.6%

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

   10. sqrt-unprod 48.8%

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

   11. add-sqr-sqrt 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 87.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \mathbf{if}\;ew \leq -1.1 \cdot 10^{+23}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ \mathbf{elif}\;ew \leq 2.25 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \left(\sin t \cdot \sin t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ (tan t) (- ew))))))
   (if (<= ew -1.1e+23)
     (fabs (* (* ew (cos t)) (cos t_1)))
     (if (<= ew 2.25e+30)
       (fabs
        (-
         (/ ew (hypot 1.0 (* (tan t) (/ eh ew))))
         (* eh (* (sin t) (sin t_1)))))
       (* ew (fabs (cos t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh * (tan(t) / -ew)));
	double tmp;
	if (ew <= -1.1e+23) {
		tmp = fabs(((ew * cos(t)) * cos(t_1)));
	} else if (ew <= 2.25e+30) {
		tmp = fabs(((ew / hypot(1.0, (tan(t) * (eh / ew)))) - (eh * (sin(t) * sin(t_1)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh * (Math.tan(t) / -ew)));
	double tmp;
	if (ew <= -1.1e+23) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(t_1)));
	} else if (ew <= 2.25e+30) {
		tmp = Math.abs(((ew / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))) - (eh * (Math.sin(t) * Math.sin(t_1)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh * (math.tan(t) / -ew)))
	tmp = 0
	if ew <= -1.1e+23:
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(t_1)))
	elif ew <= 2.25e+30:
		tmp = math.fabs(((ew / math.hypot(1.0, (math.tan(t) * (eh / ew)))) - (eh * (math.sin(t) * math.sin(t_1)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	tmp = 0.0
	if (ew <= -1.1e+23)
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(t_1)));
	elseif (ew <= 2.25e+30)
		tmp = abs(Float64(Float64(ew / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))) - Float64(eh * Float64(sin(t) * sin(t_1)))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh * (tan(t) / -ew)));
	tmp = 0.0;
	if (ew <= -1.1e+23)
		tmp = abs(((ew * cos(t)) * cos(t_1)));
	elseif (ew <= 2.25e+30)
		tmp = abs(((ew / hypot(1.0, (tan(t) * (eh / ew)))) - (eh * (sin(t) * sin(t_1)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.1e+23], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.25e+30], N[Abs[N[(N[(ew / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.10000000000000004e23

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around inf 91.4%

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

      1. associate-*r* 91.4%

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

      2. mul-1-neg 91.4%

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

      3. distribute-frac-neg2 91.4%

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

      4. associate-/l* 91.4%

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

   7. Simplified 91.4%

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

   if -1.10000000000000004e23 < ew < 2.24999999999999997e30

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-/l* 99.8%

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

      7. add-sqr-sqrt 52.7%

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

      8. sqrt-unprod 98.7%

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

      9. sqr-neg 98.7%

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

      10. sqrt-unprod 47.0%

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

      11. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 86.7%

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

   if 2.24999999999999997e30 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 71.2%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 71.0%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 91.4%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 91.4%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 91.4%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 91.4%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 91.4%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 91.4%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.1 \cdot 10^{+23}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 2.25 \cdot 10^{+30}:\\ \;\;\;\;\left|\frac{ew}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \mathbf{if}\;t \leq -3.9 \cdot 10^{+186}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{elif}\;t \leq -0.00028 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin t\_1 \cdot \left(t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ (tan t) (- ew))))))
   (if (<= t -3.9e+186)
     (* ew (+ (cos t) (* eh (/ (* (sin t) (sin (atan (/ (* t eh) ew)))) ew))))
     (if (or (<= t -0.00028) (not (<= t 1.15e-24)))
       (fabs (* (* ew (cos t)) (cos t_1)))
       (fabs (- ew (* (sin t_1) (* t eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh * (tan(t) / -ew)));
	double tmp;
	if (t <= -3.9e+186) {
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	} else if ((t <= -0.00028) || !(t <= 1.15e-24)) {
		tmp = fabs(((ew * cos(t)) * cos(t_1)));
	} else {
		tmp = fabs((ew - (sin(t_1) * (t * eh))));
	}
	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 = atan((eh * (tan(t) / -ew)))
    if (t <= (-3.9d+186)) then
        tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)))
    else if ((t <= (-0.00028d0)) .or. (.not. (t <= 1.15d-24))) then
        tmp = abs(((ew * cos(t)) * cos(t_1)))
    else
        tmp = abs((ew - (sin(t_1) * (t * eh))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh * (Math.tan(t) / -ew)));
	double tmp;
	if (t <= -3.9e+186) {
		tmp = ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew)))) / ew)));
	} else if ((t <= -0.00028) || !(t <= 1.15e-24)) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(t_1)));
	} else {
		tmp = Math.abs((ew - (Math.sin(t_1) * (t * eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh * (math.tan(t) / -ew)))
	tmp = 0
	if t <= -3.9e+186:
		tmp = ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan(((t * eh) / ew)))) / ew)))
	elif (t <= -0.00028) or not (t <= 1.15e-24):
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(t_1)))
	else:
		tmp = math.fabs((ew - (math.sin(t_1) * (t * eh))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	tmp = 0.0
	if (t <= -3.9e+186)
		tmp = Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew)))) / ew))));
	elseif ((t <= -0.00028) || !(t <= 1.15e-24))
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(t_1)));
	else
		tmp = abs(Float64(ew - Float64(sin(t_1) * Float64(t * eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh * (tan(t) / -ew)));
	tmp = 0.0;
	if (t <= -3.9e+186)
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	elseif ((t <= -0.00028) || ~((t <= 1.15e-24)))
		tmp = abs(((ew * cos(t)) * cos(t_1)));
	else
		tmp = abs((ew - (sin(t_1) * (t * eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.9e+186], N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -0.00028], N[Not[LessEqual[t, 1.15e-24]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[Sin[t$95$1], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.9000000000000001e186

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around inf 46.2%

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

      1. associate-/l* 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-*r/ 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in t around 0 71.3%

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

   if -3.9000000000000001e186 < t < -2.7999999999999998e-4 or 1.1500000000000001e-24 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in ew around inf 62.2%

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

      1. associate-*r* 62.2%

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

      2. mul-1-neg 62.2%

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

      3. distribute-frac-neg2 62.2%

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

      4. associate-/l* 62.2%

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

   7. Simplified 62.2%

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

   if -2.7999999999999998e-4 < t < 1.1500000000000001e-24

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l* 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. cos-atan 100.0%

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

      2. un-div-inv 100.0%

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

      3. hypot-1-def 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

      7. add-sqr-sqrt 55.3%

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

      8. sqrt-unprod 90.4%

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

      9. sqr-neg 90.4%

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

      10. sqrt-unprod 44.7%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-*r* 98.2%

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

      4. mul-1-neg 98.2%

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

      5. distribute-frac-neg 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-rgt-neg-out 98.2%

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

      8. distribute-rgt-neg-out 98.2%

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

      9. *-commutative 98.2%

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

      10. distribute-frac-neg 98.2%

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

      11. distribute-frac-neg2 98.2%

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

      12. associate-/l* 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.9 \cdot 10^{+186}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{elif}\;t \leq -0.00028 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{-79}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1\right|\\ \mathbf{elif}\;ew \leq 8.4 \cdot 10^{-156}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin t\_1\right|\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{+20}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ (tan t) (- ew))))))
   (if (<= ew -1.9e-79)
     (fabs (* (* ew (cos t)) (cos t_1)))
     (if (<= ew 8.4e-156)
       (fabs (* (* eh (sin t)) (sin t_1)))
       (if (<= ew 1.9e+20)
         (*
          ew
          (+ (cos t) (* eh (/ (* (sin t) (sin (atan (/ (* t eh) ew)))) ew))))
         (* ew (fabs (cos t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh * (tan(t) / -ew)));
	double tmp;
	if (ew <= -1.9e-79) {
		tmp = fabs(((ew * cos(t)) * cos(t_1)));
	} else if (ew <= 8.4e-156) {
		tmp = fabs(((eh * sin(t)) * sin(t_1)));
	} else if (ew <= 1.9e+20) {
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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 = atan((eh * (tan(t) / -ew)))
    if (ew <= (-1.9d-79)) then
        tmp = abs(((ew * cos(t)) * cos(t_1)))
    else if (ew <= 8.4d-156) then
        tmp = abs(((eh * sin(t)) * sin(t_1)))
    else if (ew <= 1.9d+20) then
        tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((eh * (Math.tan(t) / -ew)));
	double tmp;
	if (ew <= -1.9e-79) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(t_1)));
	} else if (ew <= 8.4e-156) {
		tmp = Math.abs(((eh * Math.sin(t)) * Math.sin(t_1)));
	} else if (ew <= 1.9e+20) {
		tmp = ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew)))) / ew)));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh * (math.tan(t) / -ew)))
	tmp = 0
	if ew <= -1.9e-79:
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(t_1)))
	elif ew <= 8.4e-156:
		tmp = math.fabs(((eh * math.sin(t)) * math.sin(t_1)))
	elif ew <= 1.9e+20:
		tmp = ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan(((t * eh) / ew)))) / ew)))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	tmp = 0.0
	if (ew <= -1.9e-79)
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(t_1)));
	elseif (ew <= 8.4e-156)
		tmp = abs(Float64(Float64(eh * sin(t)) * sin(t_1)));
	elseif (ew <= 1.9e+20)
		tmp = Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew)))) / ew))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh * (tan(t) / -ew)));
	tmp = 0.0;
	if (ew <= -1.9e-79)
		tmp = abs(((ew * cos(t)) * cos(t_1)));
	elseif (ew <= 8.4e-156)
		tmp = abs(((eh * sin(t)) * sin(t_1)));
	elseif (ew <= 1.9e+20)
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.9e-79], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8.4e-156], N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.9e+20], N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -1.9000000000000001e-79

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around inf 84.9%

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

      1. associate-*r* 84.9%

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

      2. mul-1-neg 84.9%

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

      3. distribute-frac-neg2 84.9%

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

      4. associate-/l* 84.9%

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

   7. Simplified 84.9%

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

   if -1.9000000000000001e-79 < ew < 8.40000000000000049e-156

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in ew around 0 74.0%

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

      1. mul-1-neg 74.0%

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

      2. associate-*r* 74.0%

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

      3. mul-1-neg 74.0%

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

      4. distribute-frac-neg2 74.0%

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

      5. associate-/l* 74.0%

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

   7. Simplified 74.0%

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

   if 8.40000000000000049e-156 < ew < 1.9e20

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 77.9%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around inf 76.3%

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

      1. associate-/l* 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-*r/ 76.2%

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

   9. Simplified 76.2%

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

   10. Taylor expanded in t around 0 85.2%

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

   if 1.9e20 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 69.5%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 69.2%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 90.2%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 90.2%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 90.2%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 90.2%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 90.2%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 90.2%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{-79}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 8.4 \cdot 10^{-156}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.9 \cdot 10^{+20}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+146}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(\left(1 - \frac{\tan t}{ew}\right) + -1\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.32 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -2.9e+146)
   (fabs (* ew (cos (atan (* eh (+ (- 1.0 (/ (tan t) ew)) -1.0))))))
   (if (<= ew -1.32e-39)
     (sqrt (pow (* ew (cos t)) 2.0))
     (if (<= ew 5e-301)
       (fabs (- ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* t eh))))
       (if (<= ew 2.4e-163)
         (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew)))))
         (* ew (fabs (cos t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.9e+146) {
		tmp = fabs((ew * cos(atan((eh * ((1.0 - (tan(t) / ew)) + -1.0))))));
	} else if (ew <= -1.32e-39) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= 5e-301) {
		tmp = fabs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	} else if (ew <= 2.4e-163) {
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-2.9d+146)) then
        tmp = abs((ew * cos(atan((eh * ((1.0d0 - (tan(t) / ew)) + (-1.0d0)))))))
    else if (ew <= (-1.32d-39)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= 5d-301) then
        tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))))
    else if (ew <= 2.4d-163) then
        tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.9e+146) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * ((1.0 - (Math.tan(t) / ew)) + -1.0))))));
	} else if (ew <= -1.32e-39) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= 5e-301) {
		tmp = Math.abs((ew - (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (t * eh))));
	} else if (ew <= 2.4e-163) {
		tmp = (eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -2.9e+146:
		tmp = math.fabs((ew * math.cos(math.atan((eh * ((1.0 - (math.tan(t) / ew)) + -1.0))))))
	elif ew <= -1.32e-39:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= 5e-301:
		tmp = math.fabs((ew - (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (t * eh))))
	elif ew <= 2.4e-163:
		tmp = (eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -2.9e+146)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(Float64(1.0 - Float64(tan(t) / ew)) + -1.0))))));
	elseif (ew <= -1.32e-39)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= 5e-301)
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(t * eh))));
	elseif (ew <= 2.4e-163)
		tmp = Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -2.9e+146)
		tmp = abs((ew * cos(atan((eh * ((1.0 - (tan(t) / ew)) + -1.0))))));
	elseif (ew <= -1.32e-39)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= 5e-301)
		tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	elseif (ew <= 2.4e-163)
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -2.9e+146], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(N[(1.0 - N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1.32e-39], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 5e-301], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.4e-163], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if ew < -2.8999999999999998e146

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 60.3%

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

      1. mul-1-neg 60.3%

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

      2. distribute-frac-neg2 60.3%

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

      3. associate-/l* 60.3%

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

   7. Simplified 60.3%

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

      1. expm1-log1p-u 60.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)\right)}\right)\right| \]

      2. expm1-undefine 60.4%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]

   9. Applied egg-rr 60.4%

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 60.4%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} + \left(-1\right)\right)}\right)\right| \]

       2. log1p-undefine 60.4%

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

       3. rem-exp-log 60.4%

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

       4. distribute-frac-neg2 60.4%

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

       5. unsub-neg 60.4%

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

       6. metadata-eval 60.4%

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

   11. Simplified 60.4%

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

   if -2.8999999999999998e146 < ew < -1.31999999999999997e-39

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 38.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 27.0%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 26.1%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 78.5%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 78.5%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 78.5%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -1.31999999999999997e-39 < ew < 5.00000000000000013e-301

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

      1. cos-atan 99.8%

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

      2. un-div-inv 99.8%

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

      3. hypot-1-def 99.8%

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

      4. associate-*r/ 99.8%

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

      5. *-commutative 99.8%

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

      6. associate-/l* 99.8%

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

      7. add-sqr-sqrt 56.0%

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

      8. sqrt-unprod 99.4%

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

      9. sqr-neg 99.4%

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

      10. sqrt-unprod 43.8%

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

      11. add-sqr-sqrt 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 56.3%

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

      1. mul-1-neg 56.3%

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

      2. unsub-neg 56.3%

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

      3. associate-*r* 56.3%

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

      4. mul-1-neg 56.3%

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

      5. distribute-frac-neg 56.3%

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

      6. *-commutative 56.3%

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

      7. distribute-rgt-neg-out 56.3%

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

      8. distribute-rgt-neg-out 56.3%

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

      9. *-commutative 56.3%

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

      10. distribute-frac-neg 56.3%

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

      11. distribute-frac-neg2 56.3%

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

      12. associate-/l* 56.3%

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

   9. Simplified 56.3%

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

   if 5.00000000000000013e-301 < ew < 2.4000000000000001e-163

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 72.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around 0 59.0%

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

      1. associate-*r* 59.0%

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

      2. *-commutative 59.0%

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

      3. *-commutative 59.0%

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

      4. associate-*r/ 59.0%

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

   9. Simplified 59.0%

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

   if 2.4000000000000001e-163 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 62.0%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 61.7%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 79.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 79.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 79.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 79.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 79.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 79.3%

       \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+146}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(\left(1 - \frac{\tan t}{ew}\right) + -1\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.32 \cdot 10^{-39}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 5 \cdot 10^{-301}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(\left(1 - \frac{\tan t}{ew}\right) + -1\right)\right)\right|\\ \mathbf{if}\;ew \leq -2.9 \cdot 10^{+146}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -2.25 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -7.2 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs (* ew (cos (atan (* eh (+ (- 1.0 (/ (tan t) ew)) -1.0))))))))
   (if (<= ew -2.9e+146)
     t_1
     (if (<= ew -2.25e-159)
       (sqrt (pow (* ew (cos t)) 2.0))
       (if (<= ew -7.2e-290)
         t_1
         (if (<= ew 3.5e-160)
           (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew)))))
           (* ew (fabs (cos t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(atan((eh * ((1.0 - (tan(t) / ew)) + -1.0))))));
	double tmp;
	if (ew <= -2.9e+146) {
		tmp = t_1;
	} else if (ew <= -2.25e-159) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -7.2e-290) {
		tmp = t_1;
	} else if (ew <= 3.5e-160) {
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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((ew * cos(atan((eh * ((1.0d0 - (tan(t) / ew)) + (-1.0d0)))))))
    if (ew <= (-2.9d+146)) then
        tmp = t_1
    else if (ew <= (-2.25d-159)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= (-7.2d-290)) then
        tmp = t_1
    else if (ew <= 3.5d-160) then
        tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(Math.atan((eh * ((1.0 - (Math.tan(t) / ew)) + -1.0))))));
	double tmp;
	if (ew <= -2.9e+146) {
		tmp = t_1;
	} else if (ew <= -2.25e-159) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -7.2e-290) {
		tmp = t_1;
	} else if (ew <= 3.5e-160) {
		tmp = (eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(math.atan((eh * ((1.0 - (math.tan(t) / ew)) + -1.0))))))
	tmp = 0
	if ew <= -2.9e+146:
		tmp = t_1
	elif ew <= -2.25e-159:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -7.2e-290:
		tmp = t_1
	elif ew <= 3.5e-160:
		tmp = (eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(atan(Float64(eh * Float64(Float64(1.0 - Float64(tan(t) / ew)) + -1.0))))))
	tmp = 0.0
	if (ew <= -2.9e+146)
		tmp = t_1;
	elseif (ew <= -2.25e-159)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -7.2e-290)
		tmp = t_1;
	elseif (ew <= 3.5e-160)
		tmp = Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(atan((eh * ((1.0 - (tan(t) / ew)) + -1.0))))));
	tmp = 0.0;
	if (ew <= -2.9e+146)
		tmp = t_1;
	elseif (ew <= -2.25e-159)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -7.2e-290)
		tmp = t_1;
	elseif (ew <= 3.5e-160)
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(N[(1.0 - N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.9e+146], t$95$1, If[LessEqual[ew, -2.25e-159], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -7.2e-290], t$95$1, If[LessEqual[ew, 3.5e-160], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -2.8999999999999998e146 or -2.24999999999999994e-159 < ew < -7.19999999999999959e-290

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 48.4%

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

      1. mul-1-neg 48.4%

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

      2. distribute-frac-neg2 48.4%

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

      3. associate-/l* 48.4%

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

   7. Simplified 48.4%

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

      1. expm1-log1p-u 39.6%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)\right)}\right)\right| \]

      2. expm1-undefine 39.6%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]

   9. Applied egg-rr 39.6%

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 39.6%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} + \left(-1\right)\right)}\right)\right| \]

       2. log1p-undefine 39.6%

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

       3. rem-exp-log 48.5%

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

       4. distribute-frac-neg2 48.5%

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

       5. unsub-neg 48.5%

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

       6. metadata-eval 48.5%

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

   11. Simplified 48.5%

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

   if -2.8999999999999998e146 < ew < -2.24999999999999994e-159

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

   6. Applied egg-rr 34.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 20.5%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 19.4%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 64.2%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 64.2%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 64.2%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -7.19999999999999959e-290 < ew < 3.5000000000000003e-160

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around 0 56.2%

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

      1. associate-*r* 56.2%

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

      2. *-commutative 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-*r/ 56.2%

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

   9. Simplified 56.2%

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

   if 3.5000000000000003e-160 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 62.0%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 61.7%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 79.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 79.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 79.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 79.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 79.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 79.3%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+146}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(\left(1 - \frac{\tan t}{ew}\right) + -1\right)\right)\right|\\ \mathbf{elif}\;ew \leq -2.25 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -7.2 \cdot 10^{-290}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(\left(1 - \frac{\tan t}{ew}\right) + -1\right)\right)\right|\\ \mathbf{elif}\;ew \leq 3.5 \cdot 10^{-160}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -3.9 \cdot 10^{-290}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6.5 \cdot 10^{-163}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.35e+154)
   (fabs (* ew (cos (atan (* eh (+ -1.0 (- 1.0 (/ t ew))))))))
   (if (<= ew -1.8e-159)
     (sqrt (pow (* ew (cos t)) 2.0))
     (if (<= ew -3.9e-290)
       (fabs (* ew (cos (atan (* eh (/ (tan t) (- ew)))))))
       (if (<= ew 6.5e-163)
         (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew)))))
         (* ew (fabs (cos t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = fabs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -1.8e-159) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -3.9e-290) {
		tmp = fabs((ew * cos(atan((eh * (tan(t) / -ew))))));
	} else if (ew <= 6.5e-163) {
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-1.35d+154)) then
        tmp = abs((ew * cos(atan((eh * ((-1.0d0) + (1.0d0 - (t / ew))))))))
    else if (ew <= (-1.8d-159)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= (-3.9d-290)) then
        tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))))
    else if (ew <= 6.5d-163) then
        tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -1.8e-159) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -3.9e-290) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (Math.tan(t) / -ew))))));
	} else if (ew <= 6.5e-163) {
		tmp = (eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.35e+154:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (-1.0 + (1.0 - (t / ew))))))))
	elif ew <= -1.8e-159:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -3.9e-290:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (math.tan(t) / -ew))))))
	elif ew <= 6.5e-163:
		tmp = (eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.35e+154)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(-1.0 + Float64(1.0 - Float64(t / ew))))))));
	elseif (ew <= -1.8e-159)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -3.9e-290)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	elseif (ew <= 6.5e-163)
		tmp = Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.35e+154)
		tmp = abs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	elseif (ew <= -1.8e-159)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -3.9e-290)
		tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))));
	elseif (ew <= 6.5e-163)
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.35e+154], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(-1.0 + N[(1.0 - N[(t / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1.8e-159], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -3.9e-290], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6.5e-163], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if ew < -1.35000000000000003e154

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. distribute-frac-neg2 62.3%

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

      3. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. expm1-log1p-u 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)\right)}\right)\right| \]

      2. expm1-undefine 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]

   9. Applied egg-rr 62.3%

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 62.3%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} + \left(-1\right)\right)}\right)\right| \]

       2. log1p-undefine 62.3%

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

       3. rem-exp-log 62.3%

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

       4. distribute-frac-neg2 62.3%

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

       5. unsub-neg 62.3%

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

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in t around 0 62.3%

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

   if -1.35000000000000003e154 < ew < -1.80000000000000011e-159

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 62.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 62.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -1.80000000000000011e-159 < ew < -3.89999999999999973e-290

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 31.9%

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

      1. mul-1-neg 31.9%

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

      2. distribute-frac-neg2 31.9%

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

      3. associate-/l* 31.9%

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

   7. Simplified 31.9%

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

   if -3.89999999999999973e-290 < ew < 6.4999999999999999e-163

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 67.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around 0 56.2%

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

      1. associate-*r* 56.2%

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

      2. *-commutative 56.2%

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

      3. *-commutative 56.2%

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

      4. associate-*r/ 56.2%

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

   9. Simplified 56.2%

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

   if 6.4999999999999999e-163 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 62.0%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 61.7%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 79.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 79.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 79.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 79.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 79.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 79.3%

       \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -3.9 \cdot 10^{-290}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6.5 \cdot 10^{-163}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+186}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{elif}\;t \leq -0.00033 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -1.16e+186)
   (* ew (+ (cos t) (* eh (/ (* (sin t) (sin (atan (/ (* t eh) ew)))) ew))))
   (if (or (<= t -0.00033) (not (<= t 1.15e-24)))
     (fabs (/ (* ew (cos t)) (hypot 1.0 (* (tan t) (/ eh ew)))))
     (fabs (- ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* t eh)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.16e+186) {
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	} else if ((t <= -0.00033) || !(t <= 1.15e-24)) {
		tmp = fabs(((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))));
	} else {
		tmp = fabs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.16e+186) {
		tmp = ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew)))) / ew)));
	} else if ((t <= -0.00033) || !(t <= 1.15e-24)) {
		tmp = Math.abs(((ew * Math.cos(t)) / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))));
	} else {
		tmp = Math.abs((ew - (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (t * eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -1.16e+186:
		tmp = ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan(((t * eh) / ew)))) / ew)))
	elif (t <= -0.00033) or not (t <= 1.15e-24):
		tmp = math.fabs(((ew * math.cos(t)) / math.hypot(1.0, (math.tan(t) * (eh / ew)))))
	else:
		tmp = math.fabs((ew - (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (t * eh))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -1.16e+186)
		tmp = Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew)))) / ew))));
	elseif ((t <= -0.00033) || !(t <= 1.15e-24))
		tmp = abs(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(t * eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -1.16e+186)
		tmp = ew * (cos(t) + (eh * ((sin(t) * sin(atan(((t * eh) / ew)))) / ew)));
	elseif ((t <= -0.00033) || ~((t <= 1.15e-24)))
		tmp = abs(((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))));
	else
		tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -1.16e+186], N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t, -0.00033], N[Not[LessEqual[t, 1.15e-24]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.15999999999999995e186

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   6. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{2}} \]

   7. Taylor expanded in ew around inf 46.2%

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

      1. associate-/l* 46.0%

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

      2. *-commutative 46.0%

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

      3. associate-*r/ 46.0%

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

   9. Simplified 46.0%

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

   10. Taylor expanded in t around 0 71.3%

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

   if -1.15999999999999995e186 < t < -3.3e-4 or 1.1500000000000001e-24 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

      1. cos-atan 99.6%

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

      2. un-div-inv 99.6%

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

      3. hypot-1-def 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.6%

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

      7. add-sqr-sqrt 50.6%

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

      8. sqrt-unprod 95.5%

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

      9. sqr-neg 95.5%

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

      10. sqrt-unprod 49.0%

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

      11. add-sqr-sqrt 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. sin-mult 64.8%

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

      2. associate-*r/ 64.8%

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

   8. Applied egg-rr 62.0%

      \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
   9. Step-by-step derivation

      1. +-inverses 62.0%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. mul0-rgt 62.0%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \frac{\color{blue}{0}}{2}\right| \]

      3. metadata-eval 62.0%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{0}\right| \]

   10. Simplified 62.0%

       \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{0}\right| \]

   if -3.3e-4 < t < 1.1500000000000001e-24

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l* 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. cos-atan 100.0%

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

      2. un-div-inv 100.0%

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

      3. hypot-1-def 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

      7. add-sqr-sqrt 55.3%

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

      8. sqrt-unprod 90.4%

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

      9. sqr-neg 90.4%

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

      10. sqrt-unprod 44.7%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-*r* 98.2%

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

      4. mul-1-neg 98.2%

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

      5. distribute-frac-neg 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-rgt-neg-out 98.2%

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

      8. distribute-rgt-neg-out 98.2%

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

      9. *-commutative 98.2%

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

      10. distribute-frac-neg 98.2%

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

      11. distribute-frac-neg2 98.2%

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

      12. associate-/l* 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.16 \cdot 10^{+186}:\\ \;\;\;\;ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)}{ew}\right)\\ \mathbf{elif}\;t \leq -0.00033 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -2.2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.5e+154)
   (fabs (* ew (cos (atan (* eh (+ -1.0 (- 1.0 (/ t ew))))))))
   (if (<= ew -2.2e-159)
     (sqrt (pow (* ew (cos t)) 2.0))
     (if (<= ew -1e-310)
       (fabs (* ew (cos (atan (* eh (/ (tan t) (- ew)))))))
       (if (<= ew 1.5e-144)
         (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
         (* ew (fabs (cos t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.5e+154) {
		tmp = fabs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -2.2e-159) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = fabs((ew * cos(atan((eh * (tan(t) / -ew))))));
	} else if (ew <= 1.5e-144) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-1.5d+154)) then
        tmp = abs((ew * cos(atan((eh * ((-1.0d0) + (1.0d0 - (t / ew))))))))
    else if (ew <= (-2.2d-159)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= (-1d-310)) then
        tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))))
    else if (ew <= 1.5d-144) then
        tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.5e+154) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -2.2e-159) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (Math.tan(t) / -ew))))));
	} else if (ew <= 1.5e-144) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.5e+154:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (-1.0 + (1.0 - (t / ew))))))))
	elif ew <= -2.2e-159:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -1e-310:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (math.tan(t) / -ew))))))
	elif ew <= 1.5e-144:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.5e+154)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(-1.0 + Float64(1.0 - Float64(t / ew))))))));
	elseif (ew <= -2.2e-159)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	elseif (ew <= 1.5e-144)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.5e+154)
		tmp = abs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	elseif (ew <= -2.2e-159)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))));
	elseif (ew <= 1.5e-144)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.5e+154], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(-1.0 + N[(1.0 - N[(t / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -2.2e-159], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1e-310], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.5e-144], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if ew < -1.50000000000000013e154

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. distribute-frac-neg2 62.3%

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

      3. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. expm1-log1p-u 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)\right)}\right)\right| \]

      2. expm1-undefine 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]

   9. Applied egg-rr 62.3%

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 62.3%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} + \left(-1\right)\right)}\right)\right| \]

       2. log1p-undefine 62.3%

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

       3. rem-exp-log 62.3%

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

       4. distribute-frac-neg2 62.3%

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

       5. unsub-neg 62.3%

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

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in t around 0 62.3%

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

   if -1.50000000000000013e154 < ew < -2.2e-159

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 62.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 62.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -2.2e-159 < ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 29.1%

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

      1. mul-1-neg 29.1%

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

      2. distribute-frac-neg2 29.1%

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

      3. associate-/l* 29.1%

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

   7. Simplified 29.1%

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

   if -9.999999999999969e-311 < ew < 1.4999999999999999e-144

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 1.4999999999999999e-144 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

       \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.5 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -2.2 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.5 \cdot 10^{-144}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00025 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.00025) (not (<= t 1.15e-24)))
   (fabs (/ (* ew (cos t)) (hypot 1.0 (* (tan t) (/ eh ew)))))
   (fabs (- ew (* (sin (atan (* eh (/ (tan t) (- ew))))) (* t eh))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00025) || !(t <= 1.15e-24)) {
		tmp = fabs(((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))));
	} else {
		tmp = fabs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00025) || !(t <= 1.15e-24)) {
		tmp = Math.abs(((ew * Math.cos(t)) / Math.hypot(1.0, (Math.tan(t) * (eh / ew)))));
	} else {
		tmp = Math.abs((ew - (Math.sin(Math.atan((eh * (Math.tan(t) / -ew)))) * (t * eh))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.00025) or not (t <= 1.15e-24):
		tmp = math.fabs(((ew * math.cos(t)) / math.hypot(1.0, (math.tan(t) * (eh / ew)))))
	else:
		tmp = math.fabs((ew - (math.sin(math.atan((eh * (math.tan(t) / -ew)))) * (t * eh))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.00025) || !(t <= 1.15e-24))
		tmp = abs(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = abs(Float64(ew - Float64(sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))) * Float64(t * eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.00025) || ~((t <= 1.15e-24)))
		tmp = abs(((ew * cos(t)) / hypot(1.0, (tan(t) * (eh / ew)))));
	else
		tmp = abs((ew - (sin(atan((eh * (tan(t) / -ew)))) * (t * eh))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.00025], N[Not[LessEqual[t, 1.15e-24]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew - N[(N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.5000000000000001e-4 or 1.1500000000000001e-24 < t

   1. Initial program 99.6%

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

      1. sub-neg 99.6%

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

      2. associate-*l* 99.6%

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

      3. distribute-rgt-neg-in 99.6%

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

      4. cancel-sign-sub 99.6%

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

      5. associate-/l* 99.6%

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

   3. Simplified 99.6%

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

      1. cos-atan 99.6%

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

      2. un-div-inv 99.6%

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

      3. hypot-1-def 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

      6. associate-/l* 99.6%

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

      7. add-sqr-sqrt 47.6%

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

      8. sqrt-unprod 96.2%

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

      9. sqr-neg 96.2%

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

      10. sqrt-unprod 52.0%

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

      11. add-sqr-sqrt 99.6%

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

   6. Applied egg-rr 99.6%

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

      1. sin-mult 59.9%

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

      2. associate-*r/ 59.9%

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

   8. Applied egg-rr 57.5%

      \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{\frac{eh \cdot \left(\cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right) - \cos \left(t + \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}{2}}\right| \]
   9. Step-by-step derivation

      1. +-inverses 57.5%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \frac{eh \cdot \color{blue}{0}}{2}\right| \]

      2. mul0-rgt 57.5%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \frac{\color{blue}{0}}{2}\right| \]

      3. metadata-eval 57.5%

         \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{0}\right| \]

   10. Simplified 57.5%

       \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} - \color{blue}{0}\right| \]

   if -2.5000000000000001e-4 < t < 1.1500000000000001e-24

   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. Step-by-step derivation

      1. sub-neg 100.0%

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

      2. associate-*l* 100.0%

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

      3. distribute-rgt-neg-in 100.0%

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

      4. cancel-sign-sub 100.0%

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

      5. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. cos-atan 100.0%

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

      2. un-div-inv 100.0%

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

      3. hypot-1-def 100.0%

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

      4. associate-*r/ 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

      7. add-sqr-sqrt 55.3%

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

      8. sqrt-unprod 90.4%

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

      9. sqr-neg 90.4%

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

      10. sqrt-unprod 44.7%

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

      11. add-sqr-sqrt 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0 98.2%

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

      1. mul-1-neg 98.2%

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

      2. unsub-neg 98.2%

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

      3. associate-*r* 98.2%

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

      4. mul-1-neg 98.2%

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

      5. distribute-frac-neg 98.2%

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

      6. *-commutative 98.2%

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

      7. distribute-rgt-neg-out 98.2%

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

      8. distribute-rgt-neg-out 98.2%

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

      9. *-commutative 98.2%

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

      10. distribute-frac-neg 98.2%

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

      11. distribute-frac-neg2 98.2%

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

      12. associate-/l* 98.2%

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

   9. Simplified 98.2%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00025 \lor \neg \left(t \leq 1.15 \cdot 10^{-24}\right):\\ \;\;\;\;\left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew - \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right) \cdot \left(t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-145}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.35e+154)
   (fabs (* ew (cos (atan (* eh (+ -1.0 (- 1.0 (/ t ew))))))))
   (if (<= ew -1.8e-159)
     (sqrt (pow (* ew (cos t)) 2.0))
     (if (<= ew -1e-310)
       (fabs (* ew (/ 1.0 (hypot 1.0 (* eh (/ (tan t) ew))))))
       (if (<= ew 2.9e-145)
         (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
         (* ew (fabs (cos t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = fabs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -1.8e-159) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = fabs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / ew))))));
	} else if (ew <= 2.9e-145) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	} else if (ew <= -1.8e-159) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = Math.abs((ew * (1.0 / Math.hypot(1.0, (eh * (Math.tan(t) / ew))))));
	} else if (ew <= 2.9e-145) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.35e+154:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (-1.0 + (1.0 - (t / ew))))))))
	elif ew <= -1.8e-159:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -1e-310:
		tmp = math.fabs((ew * (1.0 / math.hypot(1.0, (eh * (math.tan(t) / ew))))))
	elif ew <= 2.9e-145:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.35e+154)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(-1.0 + Float64(1.0 - Float64(t / ew))))))));
	elseif (ew <= -1.8e-159)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(eh * Float64(tan(t) / ew))))));
	elseif (ew <= 2.9e-145)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.35e+154)
		tmp = abs((ew * cos(atan((eh * (-1.0 + (1.0 - (t / ew))))))));
	elseif (ew <= -1.8e-159)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = abs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / ew))))));
	elseif (ew <= 2.9e-145)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.35e+154], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[(-1.0 + N[(1.0 - N[(t / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1.8e-159], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1e-310], N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.9e-145], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if ew < -1.35000000000000003e154

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 62.3%

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

      1. mul-1-neg 62.3%

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

      2. distribute-frac-neg2 62.3%

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

      3. associate-/l* 62.3%

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

   7. Simplified 62.3%

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

      1. expm1-log1p-u 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)\right)}\right)\right| \]

      2. expm1-undefine 62.3%

         \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]

   9. Applied egg-rr 62.3%

      \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} - 1\right)}\right)\right| \]
   10. Step-by-step derivation

       1. sub-neg 62.3%

          \[\leadsto \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\tan t}{-ew}\right)} + \left(-1\right)\right)}\right)\right| \]

       2. log1p-undefine 62.3%

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

       3. rem-exp-log 62.3%

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

       4. distribute-frac-neg2 62.3%

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

       5. unsub-neg 62.3%

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

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

   12. Taylor expanded in t around 0 62.3%

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

   if -1.35000000000000003e154 < ew < -1.80000000000000011e-159

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 62.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 62.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -1.80000000000000011e-159 < ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 29.1%

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

      1. mul-1-neg 29.1%

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

      2. distribute-frac-neg2 29.1%

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

      3. associate-/l* 29.1%

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

   7. Simplified 29.1%

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

      1. cos-atan 28.8%

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

      2. hypot-1-def 28.8%

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

      3. associate-*r/ 28.8%

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

   9. Applied egg-rr 28.8%

      \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{-ew}\right)}}\right| \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 28.8%

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

       2. distribute-frac-neg 28.8%

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

       3. *-commutative 28.8%

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

       4. distribute-rgt-neg-out 28.8%

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

       5. hypot-undefine 28.8%

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

       6. distribute-rgt-neg-out 28.8%

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

       7. *-commutative 28.8%

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

       8. distribute-frac-neg 28.8%

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

       9. distribute-frac-neg2 28.8%

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

       10. distribute-rgt-neg-out 28.8%

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

       11. *-commutative 28.8%

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

       12. distribute-frac-neg 28.8%

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

       13. distribute-frac-neg2 28.8%

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

   11. Simplified 28.8%

       \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}}\right| \]

   if -9.999999999999969e-311 < ew < 2.89999999999999984e-145

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 2.89999999999999984e-145 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

       \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]
3. Recombined 5 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \left(-1 + \left(1 - \frac{t}{ew}\right)\right)\right)\right|\\ \mathbf{elif}\;ew \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 2.9 \cdot 10^{-145}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \mathbf{if}\;ew \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -2.25 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{-143}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (/ 1.0 (hypot 1.0 (* eh (/ (tan t) ew))))))))
   (if (<= ew -1.35e+154)
     t_1
     (if (<= ew -2.25e-159)
       (sqrt (pow (* ew (cos t)) 2.0))
       (if (<= ew -1e-310)
         t_1
         (if (<= ew 3.1e-143)
           (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
           (* ew (fabs (cos t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / ew))))));
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = t_1;
	} else if (ew <= -2.25e-159) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = t_1;
	} else if (ew <= 3.1e-143) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * (1.0 / Math.hypot(1.0, (eh * (Math.tan(t) / ew))))));
	double tmp;
	if (ew <= -1.35e+154) {
		tmp = t_1;
	} else if (ew <= -2.25e-159) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = t_1;
	} else if (ew <= 3.1e-143) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * (1.0 / math.hypot(1.0, (eh * (math.tan(t) / ew))))))
	tmp = 0
	if ew <= -1.35e+154:
		tmp = t_1
	elif ew <= -2.25e-159:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -1e-310:
		tmp = t_1
	elif ew <= 3.1e-143:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(eh * Float64(tan(t) / ew))))))
	tmp = 0.0
	if (ew <= -1.35e+154)
		tmp = t_1;
	elseif (ew <= -2.25e-159)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = t_1;
	elseif (ew <= 3.1e-143)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / ew))))));
	tmp = 0.0;
	if (ew <= -1.35e+154)
		tmp = t_1;
	elseif (ew <= -2.25e-159)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = t_1;
	elseif (ew <= 3.1e-143)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.35e+154], t$95$1, If[LessEqual[ew, -2.25e-159], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1e-310], t$95$1, If[LessEqual[ew, 3.1e-143], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -1.35000000000000003e154 or -2.24999999999999994e-159 < ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. distribute-frac-neg2 46.9%

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

      3. associate-/l* 46.9%

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

   7. Simplified 46.9%

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

      1. cos-atan 46.8%

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

      2. hypot-1-def 46.8%

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

      3. associate-*r/ 46.8%

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

   9. Applied egg-rr 46.8%

      \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, \frac{eh \cdot \tan t}{-ew}\right)}}\right| \]
   10. Step-by-step derivation

       1. distribute-frac-neg2 46.8%

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

       2. distribute-frac-neg 46.8%

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

       3. *-commutative 46.8%

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

       4. distribute-rgt-neg-out 46.8%

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

       5. hypot-undefine 46.8%

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

       6. distribute-rgt-neg-out 46.8%

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

       7. *-commutative 46.8%

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

       8. distribute-frac-neg 46.8%

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

       9. distribute-frac-neg2 46.8%

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

       10. distribute-rgt-neg-out 46.8%

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

       11. *-commutative 46.8%

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

       12. distribute-frac-neg 46.8%

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

       13. distribute-frac-neg2 46.8%

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

   11. Simplified 46.8%

       \[\leadsto \left|ew \cdot \color{blue}{\frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}}\right| \]

   if -1.35000000000000003e154 < ew < -2.24999999999999994e-159

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 62.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 62.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -9.999999999999969e-311 < ew < 3.10000000000000007e-143

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 3.10000000000000007e-143 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

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

Alternative 13: 59.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{-t}{ew}\right)\right|\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{+154}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -2.4 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 9.6 \cdot 10^{-146}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos (atan (* eh (/ (- t) ew))))))))
   (if (<= ew -1.65e+154)
     t_1
     (if (<= ew -2.4e-161)
       (sqrt (pow (* ew (cos t)) 2.0))
       (if (<= ew -1e-310)
         t_1
         (if (<= ew 9.6e-146)
           (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
           (* ew (fabs (cos t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(atan((eh * (-t / ew))))));
	double tmp;
	if (ew <= -1.65e+154) {
		tmp = t_1;
	} else if (ew <= -2.4e-161) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = t_1;
	} else if (ew <= 9.6e-146) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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((ew * cos(atan((eh * (-t / ew))))))
    if (ew <= (-1.65d+154)) then
        tmp = t_1
    else if (ew <= (-2.4d-161)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= (-1d-310)) then
        tmp = t_1
    else if (ew <= 9.6d-146) then
        tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(Math.atan((eh * (-t / ew))))));
	double tmp;
	if (ew <= -1.65e+154) {
		tmp = t_1;
	} else if (ew <= -2.4e-161) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= -1e-310) {
		tmp = t_1;
	} else if (ew <= 9.6e-146) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(math.atan((eh * (-t / ew))))))
	tmp = 0
	if ew <= -1.65e+154:
		tmp = t_1
	elif ew <= -2.4e-161:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= -1e-310:
		tmp = t_1
	elif ew <= 9.6e-146:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(atan(Float64(eh * Float64(Float64(-t) / ew))))))
	tmp = 0.0
	if (ew <= -1.65e+154)
		tmp = t_1;
	elseif (ew <= -2.4e-161)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = t_1;
	elseif (ew <= 9.6e-146)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(atan((eh * (-t / ew))))));
	tmp = 0.0;
	if (ew <= -1.65e+154)
		tmp = t_1;
	elseif (ew <= -2.4e-161)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= -1e-310)
		tmp = t_1;
	elseif (ew <= 9.6e-146)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[N[ArcTan[N[(eh * N[((-t) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.65e+154], t$95$1, If[LessEqual[ew, -2.4e-161], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, -1e-310], t$95$1, If[LessEqual[ew, 9.6e-146], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -1.65e154 or -2.39999999999999999e-161 < ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.9%

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

      2. associate-*l* 99.9%

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

      3. distribute-rgt-neg-in 99.9%

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

      4. cancel-sign-sub 99.9%

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

      5. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in t around 0 46.9%

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

      1. mul-1-neg 46.9%

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

      2. distribute-frac-neg2 46.9%

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

      3. associate-/l* 46.9%

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

   7. Simplified 46.9%

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

   8. Taylor expanded in t around 0 45.8%

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

      1. mul-1-neg 45.8%

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

      2. associate-/l* 45.8%

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

      3. distribute-rgt-neg-in 45.8%

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

      4. distribute-neg-frac2 45.8%

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

   10. Simplified 45.8%

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

   if -1.65e154 < ew < -2.39999999999999999e-161

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 19.4%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 18.3%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 62.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 62.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -9.999999999999969e-311 < ew < 9.6000000000000006e-146

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 9.6000000000000006e-146 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{+154}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{-t}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq -2.4 \cdot 10^{-161}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{-t}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 9.6 \cdot 10^{-146}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{{\left(ew \cdot \cos t\right)}^{2}}\\ \mathbf{elif}\;ew \leq 3.4 \cdot 10^{-144}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1e-310)
   (sqrt (pow (* ew (cos t)) 2.0))
   (if (<= ew 3.4e-144)
     (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
     (* ew (fabs (cos t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = sqrt(pow((ew * cos(t)), 2.0));
	} else if (ew <= 3.4e-144) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-1d-310)) then
        tmp = sqrt(((ew * cos(t)) ** 2.0d0))
    else if (ew <= 3.4d-144) then
        tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = Math.sqrt(Math.pow((ew * Math.cos(t)), 2.0));
	} else if (ew <= 3.4e-144) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1e-310:
		tmp = math.sqrt(math.pow((ew * math.cos(t)), 2.0))
	elif ew <= 3.4e-144:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1e-310)
		tmp = sqrt((Float64(ew * cos(t)) ^ 2.0));
	elseif (ew <= 3.4e-144)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1e-310)
		tmp = sqrt(((ew * cos(t)) ^ 2.0));
	elseif (ew <= 3.4e-144)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1e-310], N[Sqrt[N[Power[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.4e-144], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 15.5%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 14.0%

         \[\leadsto \color{blue}{\sqrt{ew \cdot \cos t} \cdot \sqrt{ew \cdot \cos t}} \]

      2. sqrt-unprod 31.7%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. pow2 31.7%

         \[\leadsto \sqrt{\color{blue}{{\left(ew \cdot \cos t\right)}^{2}}} \]

   9. Applied egg-rr 31.7%

      \[\leadsto \color{blue}{\sqrt{{\left(ew \cdot \cos t\right)}^{2}}} \]

   if -9.999999999999969e-311 < ew < 3.40000000000000017e-144

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 3.40000000000000017e-144 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

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

Alternative 15: 41.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1 \cdot 10^{-310}:\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{elif}\;ew \leq 4.7 \cdot 10^{-147}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1e-310)
   (* ew (cos t))
   (if (<= ew 4.7e-147)
     (+ ew (* eh (* t (sin (atan (/ (* t eh) ew))))))
     (* ew (fabs (cos t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = ew * cos(t);
	} else if (ew <= 4.7e-147) {
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-1d-310)) then
        tmp = ew * cos(t)
    else if (ew <= 4.7d-147) then
        tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))))
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1e-310) {
		tmp = ew * Math.cos(t);
	} else if (ew <= 4.7e-147) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((t * eh) / ew)))));
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1e-310:
		tmp = ew * math.cos(t)
	elif ew <= 4.7e-147:
		tmp = ew + (eh * (t * math.sin(math.atan(((t * eh) / ew)))))
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1e-310)
		tmp = Float64(ew * cos(t));
	elseif (ew <= 4.7e-147)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1e-310)
		tmp = ew * cos(t);
	elseif (ew <= 4.7e-147)
		tmp = ew + (eh * (t * sin(atan(((t * eh) / ew)))));
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1e-310], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 4.7e-147], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.999999999999969e-311

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 15.5%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]

   if -9.999999999999969e-311 < ew < 4.69999999999999989e-147

   1. Initial program 99.7%

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

      1. sub-neg 99.7%

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

      2. associate-*l* 99.7%

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

      3. distribute-rgt-neg-in 99.7%

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

      4. cancel-sign-sub 99.7%

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

      5. associate-/l* 99.7%

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

   3. Simplified 99.7%

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

   6. Applied egg-rr 68.1%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in t around 0 43.5%

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

   8. Taylor expanded in t around 0 44.7%

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

      1. *-commutative 44.7%

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

   10. Simplified 44.7%

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

   if 4.69999999999999989e-147 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 73.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 64.1%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 63.8%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 81.3%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 81.3%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 81.3%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 81.3%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 81.3%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 81.3%

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

Alternative 16: 38.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.5 \cdot 10^{-302}:\\ \;\;\;\;ew \cdot \cos t\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \left|\cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -2.5e-302) (* ew (cos t)) (* ew (fabs (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.5e-302) {
		tmp = ew * cos(t);
	} else {
		tmp = ew * fabs(cos(t));
	}
	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) :: tmp
    if (ew <= (-2.5d-302)) then
        tmp = ew * cos(t)
    else
        tmp = ew * abs(cos(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -2.5e-302) {
		tmp = ew * Math.cos(t);
	} else {
		tmp = ew * Math.abs(Math.cos(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -2.5e-302:
		tmp = ew * math.cos(t)
	else:
		tmp = ew * math.fabs(math.cos(t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -2.5e-302)
		tmp = Float64(ew * cos(t));
	else
		tmp = Float64(ew * abs(cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -2.5e-302)
		tmp = ew * cos(t);
	else
		tmp = ew * abs(cos(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -2.5e-302], N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision], N[(ew * N[Abs[N[Cos[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.50000000000000017e-302

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 23.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 15.6%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]

   if -2.50000000000000017e-302 < ew

   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. Step-by-step derivation

      1. sub-neg 99.8%

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

      2. associate-*l* 99.8%

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

      3. distribute-rgt-neg-in 99.8%

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

      4. cancel-sign-sub 99.8%

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

      5. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   6. Applied egg-rr 71.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

   7. Taylor expanded in ew around inf 49.9%

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 49.4%

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)} \]

      2. sqrt-unprod 62.7%

         \[\leadsto ew \cdot \color{blue}{\sqrt{\cos t \cdot \cos t}} \]

      3. pow2 62.7%

         \[\leadsto ew \cdot \sqrt{\color{blue}{{\cos t}^{2}}} \]

   9. Applied egg-rr 62.7%

      \[\leadsto ew \cdot \color{blue}{\sqrt{{\cos t}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 62.7%

          \[\leadsto ew \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}} \]

       2. rem-sqrt-square 62.7%

          \[\leadsto ew \cdot \color{blue}{\left|\cos t\right|} \]

   11. Simplified 62.7%

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

Alternative 17: 32.1% accurate, 8.9× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return 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 = ew * cos(t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(ew * N[Cos[t], $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. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

7. Taylor expanded in ew around inf 33.5%

   \[\leadsto \color{blue}{ew \cdot \cos t} \]
8. Add Preprocessing

Alternative 18: 22.5% accurate, 921.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return 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 = ew
end function

Java

public static double code(double eh, double ew, double t) {
	return ew;
}

Python

def code(eh, ew, t):
	return ew

Julia

function code(eh, ew, t)
	return ew
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

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. Step-by-step derivation

   1. sub-neg 99.8%

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

   2. associate-*l* 99.8%

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

   3. distribute-rgt-neg-in 99.8%

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

   4. cancel-sign-sub 99.8%

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

   5. associate-/l* 99.8%

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

3. Simplified 99.8%

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

6. Applied egg-rr 48.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)}\right)}^{3}} \]

7. Taylor expanded in t around 0 22.0%

   \[\leadsto \color{blue}{ew} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024113 
(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)))))))