Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.7s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    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((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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((((eh * Math.sin(t)) * Math.sin(t_1)) - ((ew * Math.cos(t)) * Math.cos(t_1))));
}

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = abs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 4: 92.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-eh}{ew}\\ \mathbf{if}\;eh \leq -8.5 \cdot 10^{+176} \lor \neg \left(eh \leq 1.6 \cdot 10^{+218}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(t\_1 \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(t\_1 \cdot \tan t\right), \cos t, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (- eh) ew)))
   (if (or (<= eh -8.5e+176) (not (<= eh 1.6e+218)))
     (fabs (* (- eh) (* (tanh (asinh (* t_1 (tan (+ t PI))))) (sin t))))
     (fabs
      (*
       (fma
        (cos (atan (* t_1 (tan t))))
        (cos t)
        (/ (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))) ew))
       ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh / ew;
	double tmp;
	if ((eh <= -8.5e+176) || !(eh <= 1.6e+218)) {
		tmp = fabs((-eh * (tanh(asinh((t_1 * tan((t + ((double) M_PI)))))) * sin(t))));
	} else {
		tmp = fabs((fma(cos(atan((t_1 * tan(t)))), cos(t), ((eh * (tanh(((-eh * t) / ew)) * -sin(t))) / ew)) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) / ew)
	tmp = 0.0
	if ((eh <= -8.5e+176) || !(eh <= 1.6e+218))
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(t_1 * tan(Float64(t + pi))))) * sin(t))));
	else
		tmp = abs(Float64(fma(cos(atan(Float64(t_1 * tan(t)))), cos(t), Float64(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))) / ew)) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -8.5e+176], N[Not[LessEqual[eh, 1.6e+218]], $MachinePrecision]], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[N[(t$95$1 * N[Tan[N[(t + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(t$95$1 * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.4999999999999995e176 or 1.59999999999999994e218 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 77.7%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-PI.f64 78.0

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

   7. Applied rewrites 78.0%

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

   if -8.4999999999999995e176 < eh < 1.59999999999999994e218

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.5 \cdot 10^{+176} \lor \neg \left(eh \leq 1.6 \cdot 10^{+218}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right), \cos t, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-eh}{ew}\\ t_2 := \left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(t\_1 \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ t_3 := \left|\mathsf{fma}\left(\cos \tan^{-1} \left(t\_1 \cdot \tan t\right), 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ t_4 := \frac{eh \cdot t}{ew}\\ \mathbf{if}\;eh \leq -7.2 \cdot 10^{+176}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq -4.8 \cdot 10^{-35}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{-41}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \frac{t\_4}{\sqrt{1 + t\_4 \cdot t\_4}} + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{+88}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (- eh) ew))
        (t_2
         (fabs (* (- eh) (* (tanh (asinh (* t_1 (tan (+ t PI))))) (sin t)))))
        (t_3
         (fabs
          (*
           (fma
            (cos (atan (* t_1 (tan t))))
            1.0
            (/ (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))) ew))
           ew)))
        (t_4 (/ (* eh t) ew)))
   (if (<= eh -7.2e+176)
     t_2
     (if (<= eh -4.8e-35)
       t_3
       (if (<= eh 1.06e-41)
         (fabs
          (+
           (* (* eh (sin t)) (/ t_4 (sqrt (+ 1.0 (* t_4 t_4)))))
           (* (* ew (cos t)) (cos (atan (/ (* eh (tan t)) (- ew)))))))
         (if (<= eh 1.06e+88) t_3 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh / ew;
	double t_2 = fabs((-eh * (tanh(asinh((t_1 * tan((t + ((double) M_PI)))))) * sin(t))));
	double t_3 = fabs((fma(cos(atan((t_1 * tan(t)))), 1.0, ((eh * (tanh(((-eh * t) / ew)) * -sin(t))) / ew)) * ew));
	double t_4 = (eh * t) / ew;
	double tmp;
	if (eh <= -7.2e+176) {
		tmp = t_2;
	} else if (eh <= -4.8e-35) {
		tmp = t_3;
	} else if (eh <= 1.06e-41) {
		tmp = fabs((((eh * sin(t)) * (t_4 / sqrt((1.0 + (t_4 * t_4))))) + ((ew * cos(t)) * cos(atan(((eh * tan(t)) / -ew))))));
	} else if (eh <= 1.06e+88) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) / ew)
	t_2 = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(t_1 * tan(Float64(t + pi))))) * sin(t))))
	t_3 = abs(Float64(fma(cos(atan(Float64(t_1 * tan(t)))), 1.0, Float64(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))) / ew)) * ew))
	t_4 = Float64(Float64(eh * t) / ew)
	tmp = 0.0
	if (eh <= -7.2e+176)
		tmp = t_2;
	elseif (eh <= -4.8e-35)
		tmp = t_3;
	elseif (eh <= 1.06e-41)
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * Float64(t_4 / sqrt(Float64(1.0 + Float64(t_4 * t_4))))) + Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))))));
	elseif (eh <= 1.06e+88)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) / ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[N[(t$95$1 * N[Tan[N[(t + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[Cos[N[ArcTan[N[(t$95$1 * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 1.0 + N[(N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -7.2e+176], t$95$2, If[LessEqual[eh, -4.8e-35], t$95$3, If[LessEqual[eh, 1.06e-41], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$4 / N[Sqrt[N[(1.0 + N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.06e+88], t$95$3, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -7.19999999999999983e176 or 1.06000000000000001e88 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 70.3%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-PI.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if -7.19999999999999983e176 < eh < -4.8000000000000003e-35 or 1.06e-41 < eh < 1.06000000000000001e88

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 93.8

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

   8. Applied rewrites 93.8%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.9%

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

   if -4.8000000000000003e-35 < eh < 1.06e-41

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. sin-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.2 \cdot 10^{+176}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq -4.8 \cdot 10^{-35}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right), 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{-41}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \frac{\frac{eh \cdot t}{ew}}{\sqrt{1 + \frac{eh \cdot t}{ew} \cdot \frac{eh \cdot t}{ew}}} + \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{+88}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right), 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-eh}{ew}\\ t_2 := \left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(t\_1 \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ t_3 := \cos \tan^{-1} \left(t\_1 \cdot \tan t\right)\\ t_4 := \left|\mathsf{fma}\left(t\_3, 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \mathbf{if}\;eh \leq -7.2 \cdot 10^{+176}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot t\_3\right|\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{+88}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (- eh) ew))
        (t_2
         (fabs (* (- eh) (* (tanh (asinh (* t_1 (tan (+ t PI))))) (sin t)))))
        (t_3 (cos (atan (* t_1 (tan t)))))
        (t_4
         (fabs
          (*
           (fma
            t_3
            1.0
            (/ (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))) ew))
           ew))))
   (if (<= eh -7.2e+176)
     t_2
     (if (<= eh -8.2e-95)
       t_4
       (if (<= eh 1.72e-43)
         (fabs (* (* (cos t) ew) t_3))
         (if (<= eh 1.06e+88) t_4 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh / ew;
	double t_2 = fabs((-eh * (tanh(asinh((t_1 * tan((t + ((double) M_PI)))))) * sin(t))));
	double t_3 = cos(atan((t_1 * tan(t))));
	double t_4 = fabs((fma(t_3, 1.0, ((eh * (tanh(((-eh * t) / ew)) * -sin(t))) / ew)) * ew));
	double tmp;
	if (eh <= -7.2e+176) {
		tmp = t_2;
	} else if (eh <= -8.2e-95) {
		tmp = t_4;
	} else if (eh <= 1.72e-43) {
		tmp = fabs(((cos(t) * ew) * t_3));
	} else if (eh <= 1.06e+88) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) / ew)
	t_2 = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(t_1 * tan(Float64(t + pi))))) * sin(t))))
	t_3 = cos(atan(Float64(t_1 * tan(t))))
	t_4 = abs(Float64(fma(t_3, 1.0, Float64(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))) / ew)) * ew))
	tmp = 0.0
	if (eh <= -7.2e+176)
		tmp = t_2;
	elseif (eh <= -8.2e-95)
		tmp = t_4;
	elseif (eh <= 1.72e-43)
		tmp = abs(Float64(Float64(cos(t) * ew) * t_3));
	elseif (eh <= 1.06e+88)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) / ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[N[(t$95$1 * N[Tan[N[(t + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[ArcTan[N[(t$95$1 * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[(t$95$3 * 1.0 + N[(N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7.2e+176], t$95$2, If[LessEqual[eh, -8.2e-95], t$95$4, If[LessEqual[eh, 1.72e-43], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.06e+88], t$95$4, t$95$2]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -7.19999999999999983e176 or 1.06000000000000001e88 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 70.3%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-PI.f64 70.6

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

   7. Applied rewrites 70.6%

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

   if -7.19999999999999983e176 < eh < -8.1999999999999995e-95 or 1.72000000000000005e-43 < eh < 1.06000000000000001e88

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 95.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 94.6

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

   8. Applied rewrites 94.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.8%

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

   if -8.1999999999999995e-95 < eh < 1.72000000000000005e-43

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.2 \cdot 10^{+176}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq -8.2 \cdot 10^{-95}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right), 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \mathbf{elif}\;eh \leq 1.72 \cdot 10^{-43}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right|\\ \mathbf{elif}\;eh \leq 1.06 \cdot 10^{+88}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right), 1, \frac{eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-eh}{ew}\\ \mathbf{if}\;eh \leq -1.9 \cdot 10^{+116} \lor \neg \left(eh \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(t\_1 \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(t\_1 \cdot \tan t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (- eh) ew)))
   (if (or (<= eh -1.9e+116) (not (<= eh 6.8e+87)))
     (fabs (* (- eh) (* (tanh (asinh (* t_1 (tan (+ t PI))))) (sin t))))
     (fabs (* (* (cos t) ew) (cos (atan (* t_1 (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh / ew;
	double tmp;
	if ((eh <= -1.9e+116) || !(eh <= 6.8e+87)) {
		tmp = fabs((-eh * (tanh(asinh((t_1 * tan((t + ((double) M_PI)))))) * sin(t))));
	} else {
		tmp = fabs(((cos(t) * ew) * cos(atan((t_1 * tan(t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh / ew
	tmp = 0
	if (eh <= -1.9e+116) or not (eh <= 6.8e+87):
		tmp = math.fabs((-eh * (math.tanh(math.asinh((t_1 * math.tan((t + math.pi))))) * math.sin(t))))
	else:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan((t_1 * math.tan(t))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) / ew)
	tmp = 0.0
	if ((eh <= -1.9e+116) || !(eh <= 6.8e+87))
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(t_1 * tan(Float64(t + pi))))) * sin(t))));
	else
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(t_1 * tan(t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh / ew;
	tmp = 0.0;
	if ((eh <= -1.9e+116) || ~((eh <= 6.8e+87)))
		tmp = abs((-eh * (tanh(asinh((t_1 * tan((t + pi))))) * sin(t))));
	else
		tmp = abs(((cos(t) * ew) * cos(atan((t_1 * tan(t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -1.9e+116], N[Not[LessEqual[eh, 6.8e+87]], $MachinePrecision]], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[N[(t$95$1 * N[Tan[N[(t + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(t$95$1 * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8999999999999999e116 or 6.8000000000000004e87 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 69.1%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-PI.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if -1.8999999999999999e116 < eh < 6.8000000000000004e87

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.9 \cdot 10^{+116} \lor \neg \left(eh \leq 6.8 \cdot 10^{+87}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan \left(t + \pi\right)\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.9 \cdot 10^{+116} \lor \neg \left(eh \leq 6.5 \cdot 10^{+53}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.9e+116) (not (<= eh 6.5e+53)))
   (fabs (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))))
   (fabs (* (* (cos t) ew) (cos (atan (* (/ (- eh) ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.9e+116) || !(eh <= 6.5e+53)) {
		tmp = fabs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	} else {
		tmp = fabs(((cos(t) * ew) * cos(atan(((-eh / ew) * tan(t))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-1.9d+116)) .or. (.not. (eh <= 6.5d+53))) then
        tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))))
    else
        tmp = abs(((cos(t) * ew) * cos(atan(((-eh / ew) * tan(t))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.9e+116) || !(eh <= 6.5e+53)) {
		tmp = Math.abs((eh * (Math.tanh(((-eh * t) / ew)) * -Math.sin(t))));
	} else {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(((-eh / ew) * Math.tan(t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.9e+116) or not (eh <= 6.5e+53):
		tmp = math.fabs((eh * (math.tanh(((-eh * t) / ew)) * -math.sin(t))))
	else:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(((-eh / ew) * math.tan(t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.9e+116) || !(eh <= 6.5e+53))
		tmp = abs(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))));
	else
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(Float64(Float64(-eh) / ew) * tan(t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.9e+116) || ~((eh <= 6.5e+53)))
		tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	else
		tmp = abs(((cos(t) * ew) * cos(atan(((-eh / ew) * tan(t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.9e+116], N[Not[LessEqual[eh, 6.5e+53]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8999999999999999e116 or 6.50000000000000017e53 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if -1.8999999999999999e116 < eh < 6.50000000000000017e53

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 78.0

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

   5. Applied rewrites 78.0%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.9 \cdot 10^{+116} \lor \neg \left(eh \leq 6.5 \cdot 10^{+53}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+91} \lor \neg \left(eh \leq 8.2 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{-eh}{ew} \cdot \tan t\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -4.5e+91) (not (<= eh 8.2e+52)))
   (fabs (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))))
   (fabs
    (*
     (* (cos t) ew)
     (/ 1.0 (sqrt (+ 1.0 (pow (* (/ (- eh) ew) (tan t)) 2.0))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.5e+91) || !(eh <= 8.2e+52)) {
		tmp = fabs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	} else {
		tmp = fabs(((cos(t) * ew) * (1.0 / sqrt((1.0 + pow(((-eh / ew) * tan(t)), 2.0))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-4.5d+91)) .or. (.not. (eh <= 8.2d+52))) then
        tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))))
    else
        tmp = abs(((cos(t) * ew) * (1.0d0 / sqrt((1.0d0 + (((-eh / ew) * tan(t)) ** 2.0d0))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.5e+91) || !(eh <= 8.2e+52)) {
		tmp = Math.abs((eh * (Math.tanh(((-eh * t) / ew)) * -Math.sin(t))));
	} else {
		tmp = Math.abs(((Math.cos(t) * ew) * (1.0 / Math.sqrt((1.0 + Math.pow(((-eh / ew) * Math.tan(t)), 2.0))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -4.5e+91) or not (eh <= 8.2e+52):
		tmp = math.fabs((eh * (math.tanh(((-eh * t) / ew)) * -math.sin(t))))
	else:
		tmp = math.fabs(((math.cos(t) * ew) * (1.0 / math.sqrt((1.0 + math.pow(((-eh / ew) * math.tan(t)), 2.0))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -4.5e+91) || !(eh <= 8.2e+52))
		tmp = abs(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))));
	else
		tmp = abs(Float64(Float64(cos(t) * ew) * Float64(1.0 / sqrt(Float64(1.0 + (Float64(Float64(Float64(-eh) / ew) * tan(t)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -4.5e+91) || ~((eh <= 8.2e+52)))
		tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	else
		tmp = abs(((cos(t) * ew) * (1.0 / sqrt((1.0 + (((-eh / ew) * tan(t)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -4.5e+91], N[Not[LessEqual[eh, 8.2e+52]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.5e91 or 8.1999999999999999e52 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 66.5

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

   8. Applied rewrites 66.5%

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

   if -4.5e91 < eh < 8.1999999999999999e52

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 78.9

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

   5. Applied rewrites 78.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

         \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + \left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right) \cdot \left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right)}}\right| \]

      10. lower-+.f64 N/A

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + \left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right) \cdot \left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right)}}\right| \]

      11. pow2 N/A

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right)}^{2}}}\right| \]

      12. lower-pow.f64 N/A

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \tan t\right)\right)}^{2}}}\right| \]

   7. Applied rewrites 78.8%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+91} \lor \neg \left(eh \leq 8.2 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\frac{-eh}{ew} \cdot \tan t\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+114} \lor \neg \left(eh \leq 8 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7.5e+114) (not (<= eh 8e+52)))
   (fabs (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))))
   (fabs (* (* (cos t) ew) (cos (atan (* (/ (- eh) ew) t)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.5e+114) || !(eh <= 8e+52)) {
		tmp = fabs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	} else {
		tmp = fabs(((cos(t) * ew) * cos(atan(((-eh / ew) * t)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-7.5d+114)) .or. (.not. (eh <= 8d+52))) then
        tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))))
    else
        tmp = abs(((cos(t) * ew) * cos(atan(((-eh / ew) * t)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.5e+114) || !(eh <= 8e+52)) {
		tmp = Math.abs((eh * (Math.tanh(((-eh * t) / ew)) * -Math.sin(t))));
	} else {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(((-eh / ew) * t)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -7.5e+114) or not (eh <= 8e+52):
		tmp = math.fabs((eh * (math.tanh(((-eh * t) / ew)) * -math.sin(t))))
	else:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(((-eh / ew) * t)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7.5e+114) || !(eh <= 8e+52))
		tmp = abs(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))));
	else
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(Float64(Float64(-eh) / ew) * t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -7.5e+114) || ~((eh <= 8e+52)))
		tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	else
		tmp = abs(((cos(t) * ew) * cos(atan(((-eh / ew) * t)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7.5e+114], N[Not[LessEqual[eh, 8e+52]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.5000000000000001e114 or 7.9999999999999999e52 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 67.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 67.6

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

   8. Applied rewrites 67.6%

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

   if -7.5000000000000001e114 < eh < 7.9999999999999999e52

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 67.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.5 \cdot 10^{+114} \lor \neg \left(eh \leq 8 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 57.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{+88} \lor \neg \left(eh \leq 7.5 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7e+88) (not (<= eh 7.5e+52)))
   (fabs (* eh (* (tanh (/ (* (- eh) t) ew)) (- (sin t)))))
   (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7e+88) || !(eh <= 7.5e+52)) {
		tmp = fabs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-7d+88)) .or. (.not. (eh <= 7.5d+52))) then
        tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7e+88) || !(eh <= 7.5e+52)) {
		tmp = Math.abs((eh * (Math.tanh(((-eh * t) / ew)) * -Math.sin(t))));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -7e+88) or not (eh <= 7.5e+52):
		tmp = math.fabs((eh * (math.tanh(((-eh * t) / ew)) * -math.sin(t))))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7e+88) || !(eh <= 7.5e+52))
		tmp = abs(Float64(eh * Float64(tanh(Float64(Float64(Float64(-eh) * t) / ew)) * Float64(-sin(t)))));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -7e+88) || ~((eh <= 7.5e+52)))
		tmp = abs((eh * (tanh(((-eh * t) / ew)) * -sin(t))));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7e+88], N[Not[LessEqual[eh, 7.5e+52]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Tanh[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.9999999999999995e88 or 7.49999999999999995e52 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 66.4

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

   8. Applied rewrites 66.4%

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

   if -6.9999999999999995e88 < eh < 7.49999999999999995e52

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 50.9

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

   8. Applied rewrites 50.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 50.2

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

   10. Applied rewrites 50.2%

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

   11. Taylor expanded in eh around 0

       \[\leadsto \left|ew\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 52.5%

       \[\leadsto \left|ew\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{+88} \lor \neg \left(eh \leq 7.5 \cdot 10^{+52}\right):\\ \;\;\;\;\left|eh \cdot \left(\tanh \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(-\sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.5% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 43.3%

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

6. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 42.1

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

8. Applied rewrites 42.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 41.1

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

10. Applied rewrites 41.1%

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

11. Taylor expanded in eh around 0

    \[\leadsto \left|ew\right| \]
12. Step-by-step derivation

13. Applied rewrites 43.5%

    \[\leadsto \left|ew\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025086 
(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)))))))