Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 23.1s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

Julia

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

Wolfram

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

Derivation

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. fabs-sub 99.7%

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

   2. sub-neg 99.7%

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

   3. +-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. distribute-rgt-neg-in 99.7%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

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(eh * Float64(tan(t) / Float64(-ew))))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(eh * Float64(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[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

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

5. Final simplification 99.8%

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

Alternative 3: 97.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{\left|ew \cdot \cos t - \left(eh \cdot \sin t\right) \cdot \sin \left(-\tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right|}\right)}^{2} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (pow
  (sqrt
   (fabs
    (-
     (* ew (cos t))
     (* (* eh (sin t)) (sin (- (atan (* (tan t) (/ eh ew)))))))))
  2.0))

C

double code(double eh, double ew, double t) {
	return pow(sqrt(fabs(((ew * cos(t)) - ((eh * sin(t)) * sin(-atan((tan(t) * (eh / ew)))))))), 2.0);
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = sqrt(abs(((ew * cos(t)) - ((eh * sin(t)) * sin(-atan((tan(t) * (eh / ew)))))))) ** 2.0d0
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.pow(Math.sqrt(Math.abs(((ew * Math.cos(t)) - ((eh * Math.sin(t)) * Math.sin(-Math.atan((Math.tan(t) * (eh / ew)))))))), 2.0);
}

Python

def code(eh, ew, t):
	return math.pow(math.sqrt(math.fabs(((ew * math.cos(t)) - ((eh * math.sin(t)) * math.sin(-math.atan((math.tan(t) * (eh / ew)))))))), 2.0)

Julia

function code(eh, ew, t)
	return sqrt(abs(Float64(Float64(ew * cos(t)) - Float64(Float64(eh * sin(t)) * sin(Float64(-atan(Float64(tan(t) * Float64(eh / ew))))))))) ^ 2.0
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = sqrt(abs(((ew * cos(t)) - ((eh * sin(t)) * sin(-atan((tan(t) * (eh / ew)))))))) ^ 2.0;
end

Wolfram

code[eh_, ew_, t_] := N[Power[N[Sqrt[N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - 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]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]

Derivation

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
5. Step-by-step derivation

   1. add-cube-cbrt 98.5%

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

   2. pow3 98.5%

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

6. Applied egg-rr 98.5%

   \[\leadsto \left|\color{blue}{{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}}\right)}^{3}} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right| \]
7. Step-by-step derivation

   1. add-sqr-sqrt 98.2%

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

   2. pow2 98.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\left|{\left(\sqrt[3]{ew \cdot \frac{\cos t}{\mathsf{hypot}\left(1, \frac{eh}{\frac{ew}{\tan t}}\right)}}\right)}^{3} - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right|}\right)}^{2}} \]

8. Applied egg-rr 99.2%

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

9. Taylor expanded in ew around inf 98.5%

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

Alternative 4: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \mathbf{if}\;eh \leq -5.2 \cdot 10^{+36} \lor \neg \left(eh \leq 1.3 \cdot 10^{+149}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t + \frac{\left(ew \cdot \cos t\right) \cdot \cos t\_1}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin t\_1}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ (tan t) (- ew))))))
   (if (or (<= eh -5.2e+36) (not (<= eh 1.3e+149)))
     (fabs (* eh (+ (sin t) (/ (* (* ew (cos t)) (cos t_1)) eh))))
     (fabs (* ew (+ (cos t) (* eh (/ (* (sin t) (sin t_1)) ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh * (tan(t) / -ew)));
	double tmp;
	if ((eh <= -5.2e+36) || !(eh <= 1.3e+149)) {
		tmp = fabs((eh * (sin(t) + (((ew * cos(t)) * cos(t_1)) / eh))));
	} else {
		tmp = fabs((ew * (cos(t) + (eh * ((sin(t) * sin(t_1)) / ew)))));
	}
	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 ((eh <= (-5.2d+36)) .or. (.not. (eh <= 1.3d+149))) then
        tmp = abs((eh * (sin(t) + (((ew * cos(t)) * cos(t_1)) / eh))))
    else
        tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(t_1)) / ew)))))
    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 ((eh <= -5.2e+36) || !(eh <= 1.3e+149)) {
		tmp = Math.abs((eh * (Math.sin(t) + (((ew * Math.cos(t)) * Math.cos(t_1)) / eh))));
	} else {
		tmp = Math.abs((ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(t_1)) / ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((eh * (math.tan(t) / -ew)))
	tmp = 0
	if (eh <= -5.2e+36) or not (eh <= 1.3e+149):
		tmp = math.fabs((eh * (math.sin(t) + (((ew * math.cos(t)) * math.cos(t_1)) / eh))))
	else:
		tmp = math.fabs((ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(t_1)) / ew)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	tmp = 0.0
	if ((eh <= -5.2e+36) || !(eh <= 1.3e+149))
		tmp = abs(Float64(eh * Float64(sin(t) + Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) / eh))));
	else
		tmp = abs(Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(t_1)) / ew)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((eh * (tan(t) / -ew)));
	tmp = 0.0;
	if ((eh <= -5.2e+36) || ~((eh <= 1.3e+149)))
		tmp = abs((eh * (sin(t) + (((ew * cos(t)) * cos(t_1)) / eh))));
	else
		tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(t_1)) / ew)))));
	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[Or[LessEqual[eh, -5.2e+36], N[Not[LessEqual[eh, 1.3e+149]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] + N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.2000000000000003e36 or 1.29999999999999989e149 < eh

   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
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. sin-atan 50.3%

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

      3. associate-*r/ 45.3%

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

      4. add-sqr-sqrt 30.2%

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

      5. sqrt-unprod 18.5%

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

      6. sqr-neg 18.5%

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

      7. sqrt-unprod 15.1%

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

      8. add-sqr-sqrt 45.3%

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

      9. clear-num 45.2%

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

      10. un-div-inv 45.2%

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

      11. hypot-1-def 54.1%

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

      12. add-sqr-sqrt 35.1%

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

   6. Applied egg-rr 54.2%

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

   7. Taylor expanded in eh around -inf 99.3%

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

      1. associate-*r* 99.3%

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

      2. neg-mul-1 99.3%

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

      3. mul-1-neg 99.3%

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

      4. associate-*r* 99.3%

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

      5. mul-1-neg 99.3%

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

      6. distribute-frac-neg2 99.3%

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

      7. associate-*r/ 99.3%

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

   9. Simplified 99.3%

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

   if -5.2000000000000003e36 < eh < 1.29999999999999989e149

   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. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 19.9%

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

      3. sqrt-unprod 99.5%

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

      4. sqr-neg 99.5%

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

      5. sqrt-unprod 79.4%

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

      6. add-sqr-sqrt 99.4%

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

      7. log1p-expm1-u 99.4%

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

      8. cos-atan 99.4%

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

      9. un-div-inv 99.4%

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

      10. hypot-1-def 99.4%

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

      11. add-sqr-sqrt 50.1%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in ew around inf 99.2%

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

      1. associate-/l* 99.1%

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

      2. mul-1-neg 99.1%

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

      3. distribute-frac-neg2 99.1%

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

      4. associate-*r/ 99.1%

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

   9. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.2 \cdot 10^{+36} \lor \neg \left(eh \leq 1.3 \cdot 10^{+149}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t + \frac{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{eh}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+192} \lor \neg \left(eh \leq 4.8 \cdot 10^{+182}\right):\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -8.8e+192) (not (<= eh 4.8e+182)))
   (fabs (* eh (sin t)))
   (fabs
    (*
     ew
     (+
      (cos t)
      (* eh (/ (* (sin t) (sin (atan (* eh (/ (tan t) (- ew)))))) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.8e+192) || !(eh <= 4.8e+182)) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / -ew))))) / ew)))));
	}
	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 ((eh <= (-8.8d+192)) .or. (.not. (eh <= 4.8d+182))) then
        tmp = abs((eh * sin(t)))
    else
        tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / -ew))))) / ew)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8.8e+192) || !(eh <= 4.8e+182)) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = Math.abs((ew * (Math.cos(t) + (eh * ((Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / -ew))))) / ew)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -8.8e+192) or not (eh <= 4.8e+182):
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = math.fabs((ew * (math.cos(t) + (eh * ((math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / -ew))))) / ew)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -8.8e+192) || !(eh <= 4.8e+182))
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(Float64(ew * Float64(cos(t) + Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))) / ew)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -8.8e+192) || ~((eh <= 4.8e+182)))
		tmp = abs((eh * sin(t)));
	else
		tmp = abs((ew * (cos(t) + (eh * ((sin(t) * sin(atan((eh * (tan(t) / -ew))))) / ew)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -8.8e+192], N[Not[LessEqual[eh, 4.8e+182]], $MachinePrecision]], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[Cos[t], $MachinePrecision] + N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.8000000000000003e192 or 4.80000000000000019e182 < eh

   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
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. sin-atan 37.5%

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

      3. associate-*r/ 29.6%

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

      4. add-sqr-sqrt 13.9%

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

      5. sqrt-unprod 1.3%

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

      6. sqr-neg 1.3%

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

      7. sqrt-unprod 15.7%

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

      8. add-sqr-sqrt 29.6%

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

      9. clear-num 29.5%

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

      10. un-div-inv 29.5%

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

      11. hypot-1-def 38.3%

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

      12. add-sqr-sqrt 16.3%

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

   6. Applied egg-rr 38.4%

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

   7. Taylor expanded in t around 0 37.1%

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

   8. Taylor expanded in eh around -inf 85.3%

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

   if -8.8000000000000003e192 < eh < 4.80000000000000019e182

   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. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. add-sqr-sqrt 21.9%

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

      3. sqrt-unprod 99.6%

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

      4. sqr-neg 99.6%

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

      5. sqrt-unprod 77.5%

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

      6. add-sqr-sqrt 99.5%

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

      7. log1p-expm1-u 99.5%

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

      8. cos-atan 99.5%

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

      9. un-div-inv 99.5%

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

      10. hypot-1-def 99.5%

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

      11. add-sqr-sqrt 54.2%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in ew around inf 98.2%

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

      1. associate-/l* 98.1%

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

      2. mul-1-neg 98.1%

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

      3. distribute-frac-neg2 98.1%

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

      4. associate-*r/ 98.1%

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

   9. Simplified 98.1%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+192} \lor \neg \left(eh \leq 4.8 \cdot 10^{+182}\right):\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\cos t + eh \cdot \frac{\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left|t\_1 \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{if}\;ew \leq -2 \cdot 10^{-79}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 7.4 \cdot 10^{-283}:\\ \;\;\;\;\left|eh \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)\right|\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{+138}:\\ \;\;\;\;t\_1 + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (fabs (* t_1 (cos (atan (* eh (/ (tan t) (- ew)))))))))
   (if (<= ew -2e-79)
     t_2
     (if (<= ew 7.4e-283)
       (fabs (* eh (expm1 (log1p (sin t)))))
       (if (<= ew 4.2e+138)
         (+ t_1 (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = fabs((t_1 * cos(atan((eh * (tan(t) / -ew))))));
	double tmp;
	if (ew <= -2e-79) {
		tmp = t_2;
	} else if (ew <= 7.4e-283) {
		tmp = fabs((eh * expm1(log1p(sin(t)))));
	} else if (ew <= 4.2e+138) {
		tmp = t_1 + ((eh * sin(t)) * sin(atan((tan(t) * (eh / ew)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.abs((t_1 * Math.cos(Math.atan((eh * (Math.tan(t) / -ew))))));
	double tmp;
	if (ew <= -2e-79) {
		tmp = t_2;
	} else if (ew <= 7.4e-283) {
		tmp = Math.abs((eh * Math.expm1(Math.log1p(Math.sin(t)))));
	} else if (ew <= 4.2e+138) {
		tmp = t_1 + ((eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.fabs((t_1 * math.cos(math.atan((eh * (math.tan(t) / -ew))))))
	tmp = 0
	if ew <= -2e-79:
		tmp = t_2
	elif ew <= 7.4e-283:
		tmp = math.fabs((eh * math.expm1(math.log1p(math.sin(t)))))
	elif ew <= 4.2e+138:
		tmp = t_1 + ((eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = abs(Float64(t_1 * cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))))
	tmp = 0.0
	if (ew <= -2e-79)
		tmp = t_2;
	elseif (ew <= 7.4e-283)
		tmp = abs(Float64(eh * expm1(log1p(sin(t)))));
	elseif (ew <= 4.2e+138)
		tmp = Float64(t_1 + Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2e-79], t$95$2, If[LessEqual[ew, 7.4e-283], N[Abs[N[(eh * N[(Exp[N[Log[1 + N[Sin[t], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4.2e+138], N[(t$95$1 + 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]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2e-79 or 4.20000000000000014e138 < 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

   5. Taylor expanded in ew around inf 86.6%

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

      1. associate-*r* 86.6%

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

      2. *-commutative 86.6%

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

      3. mul-1-neg 86.6%

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

      4. distribute-frac-neg2 86.6%

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

      5. associate-/l* 86.6%

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

   7. Simplified 86.6%

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

   if -2e-79 < ew < 7.4000000000000001e-283

   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
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. sin-atan 60.9%

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

      3. associate-*r/ 60.8%

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

      4. add-sqr-sqrt 20.6%

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

      5. sqrt-unprod 58.4%

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

      6. sqr-neg 58.4%

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

      7. sqrt-unprod 39.6%

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

      8. add-sqr-sqrt 60.5%

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

      9. clear-num 60.4%

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

      10. un-div-inv 60.2%

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

      11. hypot-1-def 83.1%

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

      12. add-sqr-sqrt 37.5%

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

   6. Applied egg-rr 83.5%

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

   7. Taylor expanded in t around 0 75.8%

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

   8. Taylor expanded in eh around -inf 77.0%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   9. Step-by-step derivation

      1. expm1-log1p-u 77.0%

         \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

      2. expm1-undefine 45.3%

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

   10. Applied egg-rr 45.3%

       \[\leadsto \left|eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]
   11. Step-by-step derivation

       1. expm1-define 77.0%

          \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   12. Simplified 77.0%

       \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   if 7.4000000000000001e-283 < ew < 4.20000000000000014e138

   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 79.1%

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

      1. associate-*r/ 79.1%

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

      2. *-commutative 79.1%

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

      3. associate-/l* 79.1%

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

      4. associate-/r/ 79.1%

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

      5. *-commutative 79.1%

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

      6. fma-undefine 79.1%

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

   8. Simplified 79.1%

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

   9. Taylor expanded in ew around inf 78.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 7.4 \cdot 10^{-283}:\\ \;\;\;\;\left|eh \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)\right|\\ \mathbf{elif}\;ew \leq 4.2 \cdot 10^{+138}:\\ \;\;\;\;ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{+235}:\\ \;\;\;\;\left|eh \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)\right|\\ \mathbf{elif}\;t \leq -0.0015:\\ \;\;\;\;\left|t\_1 \cdot \cos t\_2\right|\\ \mathbf{elif}\;t \leq 0.0136:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1 + eh \cdot \sin t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (* eh (/ (tan t) (- ew))))))
   (if (<= t -4.9e+235)
     (fabs (* eh (expm1 (log1p (sin t)))))
     (if (<= t -0.0015)
       (fabs (* t_1 (cos t_2)))
       (if (<= t 0.0136)
         (fabs (+ ew (* eh (* t (sin t_2)))))
         (+ t_1 (* eh (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan((eh * (tan(t) / -ew)));
	double tmp;
	if (t <= -4.9e+235) {
		tmp = fabs((eh * expm1(log1p(sin(t)))));
	} else if (t <= -0.0015) {
		tmp = fabs((t_1 * cos(t_2)));
	} else if (t <= 0.0136) {
		tmp = fabs((ew + (eh * (t * sin(t_2)))));
	} else {
		tmp = t_1 + (eh * sin(t));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan((eh * (Math.tan(t) / -ew)));
	double tmp;
	if (t <= -4.9e+235) {
		tmp = Math.abs((eh * Math.expm1(Math.log1p(Math.sin(t)))));
	} else if (t <= -0.0015) {
		tmp = Math.abs((t_1 * Math.cos(t_2)));
	} else if (t <= 0.0136) {
		tmp = Math.abs((ew + (eh * (t * Math.sin(t_2)))));
	} else {
		tmp = t_1 + (eh * Math.sin(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan((eh * (math.tan(t) / -ew)))
	tmp = 0
	if t <= -4.9e+235:
		tmp = math.fabs((eh * math.expm1(math.log1p(math.sin(t)))))
	elif t <= -0.0015:
		tmp = math.fabs((t_1 * math.cos(t_2)))
	elif t <= 0.0136:
		tmp = math.fabs((ew + (eh * (t * math.sin(t_2)))))
	else:
		tmp = t_1 + (eh * math.sin(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(eh * Float64(tan(t) / Float64(-ew))))
	tmp = 0.0
	if (t <= -4.9e+235)
		tmp = abs(Float64(eh * expm1(log1p(sin(t)))));
	elseif (t <= -0.0015)
		tmp = abs(Float64(t_1 * cos(t_2)));
	elseif (t <= 0.0136)
		tmp = abs(Float64(ew + Float64(eh * Float64(t * sin(t_2)))));
	else
		tmp = Float64(t_1 + Float64(eh * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.9e+235], N[Abs[N[(eh * N[(Exp[N[Log[1 + N[Sin[t], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, -0.0015], N[Abs[N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.0136], N[Abs[N[(ew + N[(eh * N[(t * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 + N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.8999999999999998e235

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-*l* 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

      4. cancel-sign-sub 99.5%

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

      5. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

      1. associate-*r* 99.5%

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

      2. sin-atan 50.3%

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

      3. associate-*r/ 49.7%

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

      4. add-sqr-sqrt 25.0%

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

      5. sqrt-unprod 49.7%

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

      6. sqr-neg 49.7%

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

      7. sqrt-unprod 24.7%

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

      8. add-sqr-sqrt 49.7%

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

      9. clear-num 49.6%

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

      10. un-div-inv 49.4%

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

      11. hypot-1-def 50.4%

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

      12. add-sqr-sqrt 25.9%

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

   6. Applied egg-rr 50.8%

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

   7. Taylor expanded in t around 0 37.3%

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

   8. Taylor expanded in eh around -inf 83.5%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   9. Step-by-step derivation

      1. expm1-log1p-u 83.6%

         \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

      2. expm1-undefine 83.7%

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

   10. Applied egg-rr 83.7%

       \[\leadsto \left|eh \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin t\right)} - 1\right)}\right| \]
   11. Step-by-step derivation

       1. expm1-define 83.6%

          \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   12. Simplified 83.6%

       \[\leadsto \left|eh \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)}\right| \]

   if -4.8999999999999998e235 < t < -0.0015

   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 69.4%

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

      1. associate-*r* 69.4%

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

      2. *-commutative 69.4%

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

      3. mul-1-neg 69.4%

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

      4. distribute-frac-neg2 69.4%

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

      5. associate-/l* 69.4%

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

   7. Simplified 69.4%

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

   if -0.0015 < t < 0.0135999999999999992

   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. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. log1p-expm1-u 100.0%

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

      8. cos-atan 100.0%

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

      9. un-div-inv 100.0%

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

      10. hypot-1-def 100.0%

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

      11. add-sqr-sqrt 52.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. distribute-frac-neg2 98.4%

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

      3. associate-*r/ 98.4%

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

   9. Simplified 98.4%

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

   if 0.0135999999999999992 < t

   1. Initial program 99.4%

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

      1. sub-neg 99.4%

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

      2. associate-*l* 99.4%

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

      3. distribute-rgt-neg-in 99.4%

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

      4. cancel-sign-sub 99.4%

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

      5. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   6. Applied egg-rr 55.6%

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

      1. associate-*r/ 55.7%

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

      2. *-commutative 55.7%

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

      3. associate-/l* 55.7%

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

      4. associate-/r/ 55.7%

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

      5. *-commutative 55.7%

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

      6. fma-undefine 55.7%

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

   8. Simplified 55.7%

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

   9. Taylor expanded in ew around inf 55.3%

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

       1. sin-atan 38.3%

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

       2. metadata-eval 38.3%

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

       3. hypot-undefine 48.6%

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

   11. Applied egg-rr 48.6%

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

       1. associate-/l* 48.5%

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

       2. associate-*r/ 48.4%

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

       3. *-commutative 48.4%

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

       4. associate-/l* 48.3%

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

   13. Simplified 48.3%

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

   14. Taylor expanded in eh around inf 59.9%

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

       1. mul-1-neg 59.9%

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

       2. *-commutative 59.9%

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

       3. distribute-rgt-neg-in 59.9%

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

   16. Simplified 59.9%

       \[\leadsto ew \cdot \cos t - \color{blue}{\sin t \cdot \left(-eh\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{+235}:\\ \;\;\;\;\left|eh \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin t\right)\right)\right|\\ \mathbf{elif}\;t \leq -0.0015:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;t \leq 0.0136:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 72.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.00145 \lor \neg \left(t \leq 0.0135\right):\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.00145) (not (<= t 0.0135)))
   (+ (* ew (cos t)) (* eh (sin t)))
   (fabs (+ ew (* eh (* t (sin (atan (* eh (/ (tan t) (- ew)))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00145) || !(t <= 0.0135)) {
		tmp = (ew * cos(t)) + (eh * sin(t));
	} else {
		tmp = fabs((ew + (eh * (t * sin(atan((eh * (tan(t) / -ew))))))));
	}
	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 ((t <= (-0.00145d0)) .or. (.not. (t <= 0.0135d0))) then
        tmp = (ew * cos(t)) + (eh * sin(t))
    else
        tmp = abs((ew + (eh * (t * sin(atan((eh * (tan(t) / -ew))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.00145) || !(t <= 0.0135)) {
		tmp = (ew * Math.cos(t)) + (eh * Math.sin(t));
	} else {
		tmp = Math.abs((ew + (eh * (t * Math.sin(Math.atan((eh * (Math.tan(t) / -ew))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -0.00145) or not (t <= 0.0135):
		tmp = (ew * math.cos(t)) + (eh * math.sin(t))
	else:
		tmp = math.fabs((ew + (eh * (t * math.sin(math.atan((eh * (math.tan(t) / -ew))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.00145) || !(t <= 0.0135))
		tmp = Float64(Float64(ew * cos(t)) + Float64(eh * sin(t)));
	else
		tmp = abs(Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -0.00145) || ~((t <= 0.0135)))
		tmp = (ew * cos(t)) + (eh * sin(t));
	else
		tmp = abs((ew + (eh * (t * sin(atan((eh * (tan(t) / -ew))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.00145], N[Not[LessEqual[t, 0.0135]], $MachinePrecision]], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00145 or 0.0134999999999999998 < t

   1. Initial program 99.5%

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

      1. sub-neg 99.5%

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

      2. associate-*l* 99.5%

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

      3. distribute-rgt-neg-in 99.5%

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

      4. cancel-sign-sub 99.5%

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

      5. associate-/l* 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 55.9%

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

      1. associate-*r/ 55.9%

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

      2. *-commutative 55.9%

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

      3. associate-/l* 55.9%

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

      4. associate-/r/ 55.9%

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

      5. *-commutative 55.9%

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

      6. fma-undefine 55.9%

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

   8. Simplified 55.9%

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

   9. Taylor expanded in ew around inf 55.3%

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

       1. sin-atan 41.5%

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

       2. metadata-eval 41.5%

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

       3. hypot-undefine 50.1%

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

   11. Applied egg-rr 50.1%

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

       1. associate-/l* 50.0%

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

       2. associate-*r/ 50.0%

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

       3. *-commutative 50.0%

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

       4. associate-/l* 49.9%

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

   13. Simplified 49.9%

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

   14. Taylor expanded in eh around inf 58.1%

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

       1. mul-1-neg 58.1%

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

       2. *-commutative 58.1%

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

       3. distribute-rgt-neg-in 58.1%

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

   16. Simplified 58.1%

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

   if -0.00145 < t < 0.0134999999999999998

   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. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. *-commutative 100.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. sqrt-unprod 100.0%

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

      4. sqr-neg 100.0%

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

      5. sqrt-unprod 100.0%

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

      6. add-sqr-sqrt 100.0%

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

      7. log1p-expm1-u 100.0%

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

      8. cos-atan 100.0%

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

      9. un-div-inv 100.0%

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

      10. hypot-1-def 100.0%

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

      11. add-sqr-sqrt 52.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in t around 0 98.4%

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

      1. mul-1-neg 98.4%

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

      2. distribute-frac-neg2 98.4%

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

      3. associate-*r/ 98.4%

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

   9. Simplified 98.4%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.00145 \lor \neg \left(t \leq 0.0135\right):\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -6.5 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= ew -6.5e-73)
     (fabs (* ew (cos (atan (* eh (/ (tan t) (- ew)))))))
     (if (<= ew 6e-113) (fabs t_1) (+ (* ew (cos t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (ew <= -6.5e-73) {
		tmp = fabs((ew * cos(atan((eh * (tan(t) / -ew))))));
	} else if (ew <= 6e-113) {
		tmp = fabs(t_1);
	} else {
		tmp = (ew * cos(t)) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = eh * sin(t)
    if (ew <= (-6.5d-73)) then
        tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))))
    else if (ew <= 6d-113) then
        tmp = abs(t_1)
    else
        tmp = (ew * cos(t)) + t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (ew <= -6.5e-73) {
		tmp = Math.abs((ew * Math.cos(Math.atan((eh * (Math.tan(t) / -ew))))));
	} else if (ew <= 6e-113) {
		tmp = Math.abs(t_1);
	} else {
		tmp = (ew * Math.cos(t)) + t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if ew <= -6.5e-73:
		tmp = math.fabs((ew * math.cos(math.atan((eh * (math.tan(t) / -ew))))))
	elif ew <= 6e-113:
		tmp = math.fabs(t_1)
	else:
		tmp = (ew * math.cos(t)) + t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -6.5e-73)
		tmp = abs(Float64(ew * cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	elseif (ew <= 6e-113)
		tmp = abs(t_1);
	else
		tmp = Float64(Float64(ew * cos(t)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -6.5e-73)
		tmp = abs((ew * cos(atan((eh * (tan(t) / -ew))))));
	elseif (ew <= 6e-113)
		tmp = abs(t_1);
	else
		tmp = (ew * cos(t)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -6.5e-73], 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, 6e-113], N[Abs[t$95$1], $MachinePrecision], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.4999999999999999e-73

   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

   5. Taylor expanded in t around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. distribute-frac-neg2 58.1%

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

      3. associate-/l* 58.1%

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

   7. Simplified 58.1%

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

   if -6.4999999999999999e-73 < ew < 6.0000000000000002e-113

   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. associate-*r* 99.8%

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

      2. sin-atan 56.6%

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

      3. associate-*r/ 56.4%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 54.1%

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

      6. sqr-neg 54.1%

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

      7. sqrt-unprod 32.0%

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

      8. add-sqr-sqrt 56.2%

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

      9. clear-num 56.1%

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

      10. un-div-inv 56.0%

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

      11. hypot-1-def 76.8%

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

      12. add-sqr-sqrt 34.7%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in t around 0 69.4%

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

   8. Taylor expanded in eh around -inf 74.6%

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

   if 6.0000000000000002e-113 < ew

   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 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-/l* 77.0%

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

      4. associate-/r/ 77.0%

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

      5. *-commutative 77.0%

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

      6. fma-undefine 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in ew around inf 76.1%

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

       1. sin-atan 67.5%

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

       2. metadata-eval 67.5%

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

       3. hypot-undefine 72.8%

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

   11. Applied egg-rr 72.8%

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

       1. associate-/l* 72.7%

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

       2. associate-*r/ 72.7%

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

       3. *-commutative 72.7%

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

       4. associate-/l* 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in eh around inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

   16. Simplified 75.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -6.5 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -4.6 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{-ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 6.8 \cdot 10^{-113}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= ew -4.6e-73)
     (fabs (* ew (/ 1.0 (hypot 1.0 (* eh (/ (tan t) (- ew)))))))
     (if (<= ew 6.8e-113) (fabs t_1) (+ (* ew (cos t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (ew <= -4.6e-73) {
		tmp = fabs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / -ew))))));
	} else if (ew <= 6.8e-113) {
		tmp = fabs(t_1);
	} else {
		tmp = (ew * cos(t)) + t_1;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (ew <= -4.6e-73) {
		tmp = Math.abs((ew * (1.0 / Math.hypot(1.0, (eh * (Math.tan(t) / -ew))))));
	} else if (ew <= 6.8e-113) {
		tmp = Math.abs(t_1);
	} else {
		tmp = (ew * Math.cos(t)) + t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if ew <= -4.6e-73:
		tmp = math.fabs((ew * (1.0 / math.hypot(1.0, (eh * (math.tan(t) / -ew))))))
	elif ew <= 6.8e-113:
		tmp = math.fabs(t_1)
	else:
		tmp = (ew * math.cos(t)) + t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -4.6e-73)
		tmp = abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(eh * Float64(tan(t) / Float64(-ew)))))));
	elseif (ew <= 6.8e-113)
		tmp = abs(t_1);
	else
		tmp = Float64(Float64(ew * cos(t)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -4.6e-73)
		tmp = abs((ew * (1.0 / hypot(1.0, (eh * (tan(t) / -ew))))));
	elseif (ew <= 6.8e-113)
		tmp = abs(t_1);
	else
		tmp = (ew * cos(t)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -4.6e-73], 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, 6.8e-113], N[Abs[t$95$1], $MachinePrecision], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.59999999999999977e-73

   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

   5. Taylor expanded in t around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. distribute-frac-neg2 58.1%

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

      3. associate-/l* 58.1%

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

   7. Simplified 58.1%

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

      1. cos-atan 57.9%

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

      2. hypot-1-def 58.0%

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

   9. Applied egg-rr 58.0%

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

   if -4.59999999999999977e-73 < ew < 6.8000000000000005e-113

   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. associate-*r* 99.8%

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

      2. sin-atan 56.6%

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

      3. associate-*r/ 56.4%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 54.1%

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

      6. sqr-neg 54.1%

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

      7. sqrt-unprod 32.0%

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

      8. add-sqr-sqrt 56.2%

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

      9. clear-num 56.1%

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

      10. un-div-inv 56.0%

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

      11. hypot-1-def 76.8%

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

      12. add-sqr-sqrt 34.7%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in t around 0 69.4%

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

   8. Taylor expanded in eh around -inf 74.6%

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

   if 6.8000000000000005e-113 < ew

   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 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-/l* 77.0%

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

      4. associate-/r/ 77.0%

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

      5. *-commutative 77.0%

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

      6. fma-undefine 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in ew around inf 76.1%

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

       1. sin-atan 67.5%

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

       2. metadata-eval 67.5%

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

       3. hypot-undefine 72.8%

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

   11. Applied egg-rr 72.8%

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

       1. associate-/l* 72.7%

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

       2. associate-*r/ 72.7%

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

       3. *-commutative 72.7%

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

       4. associate-/l* 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in eh around inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

   16. Simplified 75.0%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.6 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{-ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 6.8 \cdot 10^{-113}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -4.6 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-t\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= ew -4.6e-73)
     (fabs (* ew (cos (atan (/ (* eh (- t)) ew)))))
     (if (<= ew 6e-113) (fabs t_1) (+ (* ew (cos t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (ew <= -4.6e-73) {
		tmp = fabs((ew * cos(atan(((eh * -t) / ew)))));
	} else if (ew <= 6e-113) {
		tmp = fabs(t_1);
	} else {
		tmp = (ew * cos(t)) + t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = eh * sin(t)
    if (ew <= (-4.6d-73)) then
        tmp = abs((ew * cos(atan(((eh * -t) / ew)))))
    else if (ew <= 6d-113) then
        tmp = abs(t_1)
    else
        tmp = (ew * cos(t)) + t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (ew <= -4.6e-73) {
		tmp = Math.abs((ew * Math.cos(Math.atan(((eh * -t) / ew)))));
	} else if (ew <= 6e-113) {
		tmp = Math.abs(t_1);
	} else {
		tmp = (ew * Math.cos(t)) + t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if ew <= -4.6e-73:
		tmp = math.fabs((ew * math.cos(math.atan(((eh * -t) / ew)))))
	elif ew <= 6e-113:
		tmp = math.fabs(t_1)
	else:
		tmp = (ew * math.cos(t)) + t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -4.6e-73)
		tmp = abs(Float64(ew * cos(atan(Float64(Float64(eh * Float64(-t)) / ew)))));
	elseif (ew <= 6e-113)
		tmp = abs(t_1);
	else
		tmp = Float64(Float64(ew * cos(t)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -4.6e-73)
		tmp = abs((ew * cos(atan(((eh * -t) / ew)))));
	elseif (ew <= 6e-113)
		tmp = abs(t_1);
	else
		tmp = (ew * cos(t)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -4.6e-73], N[Abs[N[(ew * N[Cos[N[ArcTan[N[(N[(eh * (-t)), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6e-113], N[Abs[t$95$1], $MachinePrecision], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.59999999999999977e-73

   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

   5. Taylor expanded in t around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. distribute-frac-neg2 58.1%

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

      3. associate-/l* 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in t around 0 57.1%

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

      1. associate-*r/ 57.1%

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

      2. associate-*r* 57.1%

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

      3. neg-mul-1 57.1%

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

   10. Simplified 57.1%

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

   if -4.59999999999999977e-73 < ew < 6.0000000000000002e-113

   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. associate-*r* 99.8%

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

      2. sin-atan 56.6%

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

      3. associate-*r/ 56.4%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 54.1%

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

      6. sqr-neg 54.1%

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

      7. sqrt-unprod 32.0%

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

      8. add-sqr-sqrt 56.2%

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

      9. clear-num 56.1%

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

      10. un-div-inv 56.0%

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

      11. hypot-1-def 76.8%

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

      12. add-sqr-sqrt 34.7%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in t around 0 69.4%

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

   8. Taylor expanded in eh around -inf 74.6%

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

   if 6.0000000000000002e-113 < ew

   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 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-/l* 77.0%

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

      4. associate-/r/ 77.0%

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

      5. *-commutative 77.0%

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

      6. fma-undefine 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in ew around inf 76.1%

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

       1. sin-atan 67.5%

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

       2. metadata-eval 67.5%

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

       3. hypot-undefine 72.8%

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

   11. Applied egg-rr 72.8%

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

       1. associate-/l* 72.7%

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

       2. associate-*r/ 72.7%

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

       3. *-commutative 72.7%

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

       4. associate-/l* 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in eh around inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

   16. Simplified 75.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.6 \cdot 10^{-73}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{eh \cdot \left(-t\right)}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -2.2 \cdot 10^{+148} \lor \neg \left(ew \leq 4.4 \cdot 10^{-90}\right):\\ \;\;\;\;ew \cdot \cos t - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (or (<= ew -2.2e+148) (not (<= ew 4.4e-90)))
     (- (* ew (cos t)) t_1)
     (fabs t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if ((ew <= -2.2e+148) || !(ew <= 4.4e-90)) {
		tmp = (ew * cos(t)) - t_1;
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = eh * sin(t)
    if ((ew <= (-2.2d+148)) .or. (.not. (ew <= 4.4d-90))) then
        tmp = (ew * cos(t)) - t_1
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if ((ew <= -2.2e+148) || !(ew <= 4.4e-90)) {
		tmp = (ew * Math.cos(t)) - t_1;
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if (ew <= -2.2e+148) or not (ew <= 4.4e-90):
		tmp = (ew * math.cos(t)) - t_1
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if ((ew <= -2.2e+148) || !(ew <= 4.4e-90))
		tmp = Float64(Float64(ew * cos(t)) - t_1);
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if ((ew <= -2.2e+148) || ~((ew <= 4.4e-90)))
		tmp = (ew * cos(t)) - t_1;
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -2.2e+148], N[Not[LessEqual[ew, 4.4e-90]], $MachinePrecision]], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.1999999999999999e148 or 4.39999999999999972e-90 < 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 62.8%

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

      1. associate-*r/ 62.8%

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

      2. *-commutative 62.8%

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

      3. associate-/l* 62.8%

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

      4. associate-/r/ 62.8%

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

      5. *-commutative 62.8%

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

      6. fma-undefine 62.8%

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

   8. Simplified 62.8%

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

   9. Taylor expanded in ew around inf 62.1%

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

       1. sin-atan 57.2%

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

       2. metadata-eval 57.2%

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

       3. hypot-undefine 61.3%

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

   11. Applied egg-rr 61.3%

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

       1. associate-/l* 61.2%

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

       2. associate-*r/ 61.2%

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

       3. *-commutative 61.2%

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

       4. associate-/l* 61.2%

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

   13. Simplified 61.2%

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

   14. Taylor expanded in eh around -inf 55.8%

       \[\leadsto ew \cdot \cos t - \color{blue}{eh \cdot \sin t} \]

   if -2.1999999999999999e148 < ew < 4.39999999999999972e-90

   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
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. sin-atan 63.8%

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

      3. associate-*r/ 62.8%

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

      4. add-sqr-sqrt 32.8%

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

      5. sqrt-unprod 59.1%

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

      6. sqr-neg 59.1%

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

      7. sqrt-unprod 29.7%

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

      8. add-sqr-sqrt 62.7%

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

      9. clear-num 62.6%

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

      10. un-div-inv 62.5%

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

      11. hypot-1-def 77.4%

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

      12. add-sqr-sqrt 40.5%

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

   6. Applied egg-rr 77.6%

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

   7. Taylor expanded in t around 0 66.7%

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

   8. Taylor expanded in eh around -inf 64.8%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+148} \lor \neg \left(ew \leq 4.4 \cdot 10^{-90}\right):\\ \;\;\;\;ew \cdot \cos t - eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 64.6% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{t}{-ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 7 \cdot 10^{-113}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= ew -9.8e-74)
     (fabs (* ew (/ 1.0 (hypot 1.0 (* eh (/ t (- ew)))))))
     (if (<= ew 7e-113) (fabs t_1) (+ (* ew (cos t)) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (ew <= -9.8e-74) {
		tmp = fabs((ew * (1.0 / hypot(1.0, (eh * (t / -ew))))));
	} else if (ew <= 7e-113) {
		tmp = fabs(t_1);
	} else {
		tmp = (ew * cos(t)) + t_1;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (ew <= -9.8e-74) {
		tmp = Math.abs((ew * (1.0 / Math.hypot(1.0, (eh * (t / -ew))))));
	} else if (ew <= 7e-113) {
		tmp = Math.abs(t_1);
	} else {
		tmp = (ew * Math.cos(t)) + t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if ew <= -9.8e-74:
		tmp = math.fabs((ew * (1.0 / math.hypot(1.0, (eh * (t / -ew))))))
	elif ew <= 7e-113:
		tmp = math.fabs(t_1)
	else:
		tmp = (ew * math.cos(t)) + t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -9.8e-74)
		tmp = abs(Float64(ew * Float64(1.0 / hypot(1.0, Float64(eh * Float64(t / Float64(-ew)))))));
	elseif (ew <= 7e-113)
		tmp = abs(t_1);
	else
		tmp = Float64(Float64(ew * cos(t)) + t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -9.8e-74)
		tmp = abs((ew * (1.0 / hypot(1.0, (eh * (t / -ew))))));
	elseif (ew <= 7e-113)
		tmp = abs(t_1);
	else
		tmp = (ew * cos(t)) + t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -9.8e-74], N[Abs[N[(ew * N[(1.0 / N[Sqrt[1.0 ^ 2 + N[(eh * N[(t / (-ew)), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 7e-113], N[Abs[t$95$1], $MachinePrecision], N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -9.8000000000000006e-74

   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

   5. Taylor expanded in t around 0 58.1%

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

      1. mul-1-neg 58.1%

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

      2. distribute-frac-neg2 58.1%

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

      3. associate-/l* 58.1%

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

   7. Simplified 58.1%

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

   8. Taylor expanded in t around 0 57.1%

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

      1. associate-*r/ 57.1%

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

      2. associate-*r* 57.1%

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

      3. neg-mul-1 57.1%

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

   10. Simplified 57.1%

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

       1. cos-atan 56.3%

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

       2. hypot-1-def 56.4%

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

       3. associate-/l* 56.4%

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

   12. Applied egg-rr 56.4%

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

   if -9.8000000000000006e-74 < ew < 7.00000000000000057e-113

   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. associate-*r* 99.8%

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

      2. sin-atan 56.6%

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

      3. associate-*r/ 56.4%

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

      4. add-sqr-sqrt 24.0%

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

      5. sqrt-unprod 54.1%

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

      6. sqr-neg 54.1%

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

      7. sqrt-unprod 32.0%

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

      8. add-sqr-sqrt 56.2%

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

      9. clear-num 56.1%

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

      10. un-div-inv 56.0%

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

      11. hypot-1-def 76.8%

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

      12. add-sqr-sqrt 34.7%

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

   6. Applied egg-rr 77.1%

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

   7. Taylor expanded in t around 0 69.4%

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

   8. Taylor expanded in eh around -inf 74.6%

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

   if 7.00000000000000057e-113 < ew

   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 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-/l* 77.0%

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

      4. associate-/r/ 77.0%

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

      5. *-commutative 77.0%

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

      6. fma-undefine 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in ew around inf 76.1%

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

       1. sin-atan 67.5%

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

       2. metadata-eval 67.5%

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

       3. hypot-undefine 72.8%

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

   11. Applied egg-rr 72.8%

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

       1. associate-/l* 72.7%

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

       2. associate-*r/ 72.7%

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

       3. *-commutative 72.7%

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

       4. associate-/l* 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in eh around inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

   16. Simplified 75.0%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.8 \cdot 10^{-74}:\\ \;\;\;\;\left|ew \cdot \frac{1}{\mathsf{hypot}\left(1, eh \cdot \frac{t}{-ew}\right)}\right|\\ \mathbf{elif}\;ew \leq 7 \cdot 10^{-113}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;ew \leq -2.5 \cdot 10^{+144}:\\ \;\;\;\;t\_1 - t\_2\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1 + t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (* eh (sin t))))
   (if (<= ew -2.5e+144)
     (- t_1 t_2)
     (if (<= ew 6e-113) (fabs t_2) (+ t_1 t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = eh * sin(t);
	double tmp;
	if (ew <= -2.5e+144) {
		tmp = t_1 - t_2;
	} else if (ew <= 6e-113) {
		tmp = fabs(t_2);
	} else {
		tmp = t_1 + t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = eh * sin(t)
    if (ew <= (-2.5d+144)) then
        tmp = t_1 - t_2
    else if (ew <= 6d-113) then
        tmp = abs(t_2)
    else
        tmp = t_1 + t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = eh * Math.sin(t);
	double tmp;
	if (ew <= -2.5e+144) {
		tmp = t_1 - t_2;
	} else if (ew <= 6e-113) {
		tmp = Math.abs(t_2);
	} else {
		tmp = t_1 + t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = eh * math.sin(t)
	tmp = 0
	if ew <= -2.5e+144:
		tmp = t_1 - t_2
	elif ew <= 6e-113:
		tmp = math.fabs(t_2)
	else:
		tmp = t_1 + t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if (ew <= -2.5e+144)
		tmp = Float64(t_1 - t_2);
	elseif (ew <= 6e-113)
		tmp = abs(t_2);
	else
		tmp = Float64(t_1 + t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = eh * sin(t);
	tmp = 0.0;
	if (ew <= -2.5e+144)
		tmp = t_1 - t_2;
	elseif (ew <= 6e-113)
		tmp = abs(t_2);
	else
		tmp = t_1 + t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.5e+144], N[(t$95$1 - t$95$2), $MachinePrecision], If[LessEqual[ew, 6e-113], N[Abs[t$95$2], $MachinePrecision], N[(t$95$1 + t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.5e144

   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 21.4%

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

      1. associate-*r/ 21.4%

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

      2. *-commutative 21.4%

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

      3. associate-/l* 21.4%

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

      4. associate-/r/ 21.4%

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

      5. *-commutative 21.4%

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

      6. fma-undefine 21.4%

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

   8. Simplified 21.4%

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

   9. Taylor expanded in ew around inf 21.4%

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

       1. sin-atan 21.4%

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

       2. metadata-eval 21.4%

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

       3. hypot-undefine 21.4%

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

   11. Applied egg-rr 21.4%

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

       1. associate-/l* 21.4%

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

       2. associate-*r/ 21.4%

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

       3. *-commutative 21.4%

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

       4. associate-/l* 21.4%

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

   13. Simplified 21.4%

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

   14. Taylor expanded in eh around -inf 21.4%

       \[\leadsto ew \cdot \cos t - \color{blue}{eh \cdot \sin t} \]

   if -2.5e144 < ew < 6.0000000000000002e-113

   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
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. sin-atan 64.4%

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

      3. associate-*r/ 63.4%

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

      4. add-sqr-sqrt 32.1%

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

      5. sqrt-unprod 60.3%

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

      6. sqr-neg 60.3%

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

      7. sqrt-unprod 31.1%

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

      8. add-sqr-sqrt 63.3%

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

      9. clear-num 63.2%

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

      10. un-div-inv 63.1%

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

      11. hypot-1-def 78.6%

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

      12. add-sqr-sqrt 40.1%

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

   6. Applied egg-rr 78.9%

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

   7. Taylor expanded in t around 0 67.4%

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

   8. Taylor expanded in eh around -inf 63.3%

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

   if 6.0000000000000002e-113 < ew

   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 77.0%

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

      1. associate-*r/ 77.0%

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

      2. *-commutative 77.0%

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

      3. associate-/l* 77.0%

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

      4. associate-/r/ 77.0%

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

      5. *-commutative 77.0%

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

      6. fma-undefine 77.0%

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

   8. Simplified 77.0%

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

   9. Taylor expanded in ew around inf 76.1%

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

       1. sin-atan 67.5%

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

       2. metadata-eval 67.5%

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

       3. hypot-undefine 72.8%

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

   11. Applied egg-rr 72.8%

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

       1. associate-/l* 72.7%

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

       2. associate-*r/ 72.7%

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

       3. *-commutative 72.7%

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

       4. associate-/l* 72.7%

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

   13. Simplified 72.7%

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

   14. Taylor expanded in eh around inf 75.0%

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

       1. mul-1-neg 75.0%

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

       2. *-commutative 75.0%

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

       3. distribute-rgt-neg-in 75.0%

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

   16. Simplified 75.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.5 \cdot 10^{+144}:\\ \;\;\;\;ew \cdot \cos t - eh \cdot \sin t\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-113}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \sin t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 42.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|eh \cdot \sin t\right| \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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
5. Step-by-step derivation

   1. associate-*r* 99.7%

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

   2. sin-atan 77.7%

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

   3. associate-*r/ 76.0%

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

   4. add-sqr-sqrt 40.8%

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

   5. sqrt-unprod 66.4%

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

   6. sqr-neg 66.4%

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

   7. sqrt-unprod 35.0%

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

   8. add-sqr-sqrt 75.8%

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

   9. clear-num 75.8%

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

   10. un-div-inv 75.7%

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

   11. hypot-1-def 83.9%

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

   12. add-sqr-sqrt 45.0%

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

6. Applied egg-rr 84.1%

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

7. Taylor expanded in t around 0 63.3%

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

8. Taylor expanded in eh around -inf 45.0%

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

Alternative 16: 18.6% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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
5. Step-by-step derivation

   1. associate-*r* 99.7%

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

   2. sin-atan 77.7%

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

   3. associate-*r/ 76.0%

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

   4. add-sqr-sqrt 40.8%

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

   5. sqrt-unprod 66.4%

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

   6. sqr-neg 66.4%

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

   7. sqrt-unprod 35.0%

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

   8. add-sqr-sqrt 75.8%

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

   9. clear-num 75.8%

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

   10. un-div-inv 75.7%

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

   11. hypot-1-def 83.9%

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

   12. add-sqr-sqrt 45.0%

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

6. Applied egg-rr 84.1%

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

7. Taylor expanded in t around 0 63.3%

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

8. Taylor expanded in eh around -inf 45.0%

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

9. Taylor expanded in t around 0 21.7%

   \[\leadsto \left|\color{blue}{eh \cdot t}\right| \]
10. Step-by-step derivation

    1. *-commutative 21.7%

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

11. Simplified 21.7%

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

12. Final simplification 21.7%

    \[\leadsto \left|eh \cdot t\right| \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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)))))))