Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.3s

Alternatives: 13

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((eh * tan(t)) / -ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * cos(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 * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * cos(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[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 98.6

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

7. Applied rewrites 98.6%

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

8. Final simplification 98.6%

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

Alternative 3: 98.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -2.7 \cdot 10^{-36} \lor \neg \left(eh \leq 4.05 \cdot 10^{-16}\right):\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t \cdot \cos \tan^{-1} t\_1}{eh}, \tanh \left(\frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(t\_1 \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= eh -2.7e-36) (not (<= eh 4.05e-16)))
     (fabs
      (*
       eh
       (fma
        ew
        (/ (* (cos t) (cos (atan t_1))) eh)
        (* (tanh (/ (* eh t) ew)) (sin t)))))
     (fabs
      (/
       (fma (cos t) ew (* (* t_1 eh) (sin t)))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((eh <= -2.7e-36) || !(eh <= 4.05e-16)) {
		tmp = fabs((eh * fma(ew, ((cos(t) * cos(atan(t_1))) / eh), (tanh(((eh * t) / ew)) * sin(t)))));
	} else {
		tmp = fabs((fma(cos(t), ew, ((t_1 * eh) * sin(t))) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((eh <= -2.7e-36) || !(eh <= 4.05e-16))
		tmp = abs(Float64(eh * fma(ew, Float64(Float64(cos(t) * cos(atan(t_1))) / eh), Float64(tanh(Float64(Float64(eh * t) / ew)) * sin(t)))));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(t_1 * eh) * sin(t))) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -2.7e-36], N[Not[LessEqual[eh, 4.05e-16]], $MachinePrecision]], N[Abs[N[(eh * N[(ew * N[(N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] + N[(N[Tanh[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.70000000000000007e-36 or 4.05000000000000024e-16 < eh

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 98.0%

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

   if -2.70000000000000007e-36 < eh < 4.05000000000000024e-16

   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 97.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 98.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.9

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

   6. Applied rewrites 98.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 99.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.7 \cdot 10^{-36} \lor \neg \left(eh \leq 4.05 \cdot 10^{-16}\right):\\ \;\;\;\;\left|eh \cdot \mathsf{fma}\left(ew, \frac{\cos t \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}{eh}, \tanh \left(\frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -1.55 \cdot 10^{+109} \lor \neg \left(eh \leq 3.5 \cdot 10^{+68}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot t\_1\right) \cdot eh\right) \cdot \sin t\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= eh -1.55e+109) (not (<= eh 3.5e+68)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (* (* (cos (atan (/ (* eh t) ew))) (cos t)) ew)))
     (fabs
      (/
       (fma (cos t) ew (* (* (* eh t_1) eh) (sin t)))
       (cosh (asinh (* t_1 eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((eh <= -1.55e+109) || !(eh <= 3.5e+68)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((eh * t) / ew))) * cos(t)) * ew)));
	} else {
		tmp = fabs((fma(cos(t), ew, (((eh * t_1) * eh) * sin(t))) / cosh(asinh((t_1 * eh)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((eh <= -1.55e+109) || !(eh <= 3.5e+68))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t)) * ew)));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(Float64(eh * t_1) * eh) * sin(t))) / cosh(asinh(Float64(t_1 * eh)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -1.55e+109], N[Not[LessEqual[eh, 3.5e+68]], $MachinePrecision]], 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[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * t$95$1), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.54999999999999996e109 or 3.49999999999999977e68 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 92.1

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

   10. Applied rewrites 92.1%

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

   if -1.54999999999999996e109 < eh < 3.49999999999999977e68

   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 95.7%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.7

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 96.9

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.9

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

   6. Applied rewrites 96.9%

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

4. Final simplification 94.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.55 \cdot 10^{+109} \lor \neg \left(eh \leq 3.5 \cdot 10^{+68}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -0.45 \lor \neg \left(eh \leq 1.65 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(t\_1 \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= eh -0.45) (not (<= eh 1.65e+66)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (* (* (cos (atan (/ (* eh t) ew))) (cos t)) ew)))
     (fabs
      (/
       (fma (cos t) ew (* (* t_1 eh) (sin t)))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((eh <= -0.45) || !(eh <= 1.65e+66)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((eh * t) / ew))) * cos(t)) * ew)));
	} else {
		tmp = fabs((fma(cos(t), ew, ((t_1 * eh) * sin(t))) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((eh <= -0.45) || !(eh <= 1.65e+66))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t)) * ew)));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(t_1 * eh) * sin(t))) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -0.45], N[Not[LessEqual[eh, 1.65e+66]], $MachinePrecision]], 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[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -0.450000000000000011 or 1.6500000000000001e66 < eh

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 90.0

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

   10. Applied rewrites 90.0%

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

   if -0.450000000000000011 < eh < 1.6500000000000001e66

   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 97.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 97.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 98.6

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 98.6

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

   6. Applied rewrites 98.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 99.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 99.0

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

   8. Applied rewrites 99.0%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -0.45 \lor \neg \left(eh \leq 1.65 \cdot 10^{+66}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ t_2 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -3.8 \cdot 10^{+157} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{\frac{t\_2}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(t\_1 \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))) (t_2 (* (- eh) (sin t))))
   (if (or (<= eh -3.8e+157) (not (<= eh 1.45e+69)))
     (fabs (* t_2 (sin (atan (/ (/ t_2 ew) (cos t))))))
     (fabs
      (/
       (fma (cos t) ew (* (* t_1 eh) (sin t)))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double t_2 = -eh * sin(t);
	double tmp;
	if ((eh <= -3.8e+157) || !(eh <= 1.45e+69)) {
		tmp = fabs((t_2 * sin(atan(((t_2 / ew) / cos(t))))));
	} else {
		tmp = fabs((fma(cos(t), ew, ((t_1 * eh) * sin(t))) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	t_2 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -3.8e+157) || !(eh <= 1.45e+69))
		tmp = abs(Float64(t_2 * sin(atan(Float64(Float64(t_2 / ew) / cos(t))))));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(t_1 * eh) * sin(t))) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -3.8e+157], N[Not[LessEqual[eh, 1.45e+69]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.8000000000000001e157 or 1.4499999999999999e69 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

   10. Taylor expanded in eh around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. lower-atan.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-sin.f64 N/A

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

       16. lower-cos.f64 80.2

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

   12. Applied rewrites 80.2%

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

   if -3.8000000000000001e157 < eh < 1.4499999999999999e69

   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 94.0%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.0

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 95.2

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.2

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

   6. Applied rewrites 95.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 93.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 93.0

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

   8. Applied rewrites 93.0%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+157} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\left(-eh\right) \cdot \sin t}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))))
   (if (or (<= eh -6.8e+120) (not (<= eh 1.45e+69)))
     (fabs (* t_1 (sin (atan (/ (/ t_1 ew) (cos t))))))
     (fabs
      (/ (fma (cos t) ew (* (* (* eh (/ (tan t) ew)) eh) (sin t))) 1.0)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double tmp;
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69)) {
		tmp = fabs((t_1 * sin(atan(((t_1 / ew) / cos(t))))));
	} else {
		tmp = fabs((fma(cos(t), ew, (((eh * (tan(t) / ew)) * eh) * sin(t))) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69))
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(t_1 / ew) / cos(t))))));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(Float64(eh * Float64(tan(t) / ew)) * eh) * sin(t))) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -6.8e+120], N[Not[LessEqual[eh, 1.45e+69]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.79999999999999998e120 or 1.4499999999999999e69 < eh

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.6%

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

   10. Taylor expanded in eh around inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 N/A

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

       3. associate-*r* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-sin.f64 N/A

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

       8. lower-atan.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 N/A

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

       11. associate-/r* N/A

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

       12. lower-/.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-*.f64 N/A

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

       15. lower-sin.f64 N/A

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

       16. lower-cos.f64 78.2

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

   12. Applied rewrites 78.2%

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

   if -6.79999999999999998e120 < eh < 1.4499999999999999e69

   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 95.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 96.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 79.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\left(-eh\right) \cdot \sin t}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))))
   (if (or (<= eh -6.8e+120) (not (<= eh 1.45e+69)))
     (fabs (* (* t_1 eh) (sin (atan (* (/ t_1 ew) (/ eh (cos t)))))))
     (fabs
      (/ (fma (cos t) ew (* (* (* eh (/ (tan t) ew)) eh) (sin t))) 1.0)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double tmp;
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69)) {
		tmp = fabs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))));
	} else {
		tmp = fabs((fma(cos(t), ew, (((eh * (tan(t) / ew)) * eh) * sin(t))) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	tmp = 0.0
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69))
		tmp = abs(Float64(Float64(t_1 * eh) * sin(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t)))))));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(Float64(eh * Float64(tan(t) / ew)) * eh) * sin(t))) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, If[Or[LessEqual[eh, -6.8e+120], N[Not[LessEqual[eh, 1.45e+69]], $MachinePrecision]], N[Abs[N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.79999999999999998e120 or 1.4499999999999999e69 < eh

   1. Initial program 99.7%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. 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. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. distribute-lft-neg-in N/A

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

   5. Applied rewrites 78.2%

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

   if -6.79999999999999998e120 < eh < 1.4499999999999999e69

   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 95.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 96.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 79.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-\sin t\right) \cdot eh}{\cos t \cdot ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -6.8e+120) (not (<= eh 1.45e+69)))
   (fabs
    (* (- eh) (* (sin (atan (/ (* (- (sin t)) eh) (* (cos t) ew)))) (sin t))))
   (fabs (/ (fma (cos t) ew (* (* (* eh (/ (tan t) ew)) eh) (sin t))) 1.0))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69)) {
		tmp = fabs((-eh * (sin(atan(((-sin(t) * eh) / (cos(t) * ew)))) * sin(t))));
	} else {
		tmp = fabs((fma(cos(t), ew, (((eh * (tan(t) / ew)) * eh) * sin(t))) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -6.8e+120) || !(eh <= 1.45e+69))
		tmp = abs(Float64(Float64(-eh) * Float64(sin(atan(Float64(Float64(Float64(-sin(t)) * eh) / Float64(cos(t) * ew)))) * sin(t))));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(Float64(eh * Float64(tan(t) / ew)) * eh) * sin(t))) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -6.8e+120], N[Not[LessEqual[eh, 1.45e+69]], $MachinePrecision]], N[Abs[N[((-eh) * N[(N[Sin[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] / N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.79999999999999998e120 or 1.4499999999999999e69 < eh

   1. Initial program 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 98.5

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

   7. Applied rewrites 98.5%

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

   8. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   10. Applied rewrites 78.1%

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

   if -6.79999999999999998e120 < eh < 1.4499999999999999e69

   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 95.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 95.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 96.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 96.4

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

   6. Applied rewrites 96.4%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 79.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.8 \cdot 10^{+120} \lor \neg \left(eh \leq 1.45 \cdot 10^{+69}\right):\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\sin \tan^{-1} \left(\frac{\left(-\sin t\right) \cdot eh}{\cos t \cdot ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{1}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 65.0%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 65.0

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*r* N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 75.1

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 75.1

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

6. Applied rewrites 75.1%

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

7. Taylor expanded in eh around 0

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

9. Applied rewrites 57.7%

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

Alternative 11: 62.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 * cos(t)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 65.0%

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

5. Taylor expanded in eh around 0

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 57.1

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

7. Applied rewrites 57.1%

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

Alternative 12: 39.5% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\left(-0.5 \cdot ew\right) \cdot t, t, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* (* -0.5 ew) t) t ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma(((-0.5 * ew) * t), t, ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(Float64(-0.5 * ew) * t), t, ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(-0.5 * ew), $MachinePrecision] * t), $MachinePrecision] * t + ew), $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 65.0%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 30.6

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right) - 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}, ew\right)\right| \]

7. Applied rewrites 30.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right) - 0.5 \cdot \frac{eh \cdot eh}{ew}, ew\right)}\right| \]

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.6%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, -0.5 \cdot \color{blue}{ew}, ew\right)\right| \]
11. Step-by-step derivation

12. Applied rewrites 35.6%

    \[\leadsto \left|\mathsf{fma}\left(\left(-0.5 \cdot ew\right) \cdot t, \color{blue}{t}, ew\right)\right| \]
13. Add Preprocessing

Alternative 13: 39.4% accurate, 45.4× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(t \cdot t, -0.5 \cdot ew, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (fma (* t t) (* -0.5 ew) ew)))

C

double code(double eh, double ew, double t) {
	return fabs(fma((t * t), (-0.5 * ew), ew));
}

Julia

function code(eh, ew, t)
	return abs(fma(Float64(t * t), Float64(-0.5 * ew), ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(t * t), $MachinePrecision] * N[(-0.5 * ew), $MachinePrecision] + ew), $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 65.0%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 30.6

       \[\leadsto \left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right) - 0.5 \cdot \frac{\color{blue}{eh \cdot eh}}{ew}, ew\right)\right| \]

7. Applied rewrites 30.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right) - 0.5 \cdot \frac{eh \cdot eh}{ew}, ew\right)}\right| \]

8. Taylor expanded in eh around 0

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

10. Applied rewrites 35.6%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, -0.5 \cdot \color{blue}{ew}, ew\right)\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024350 
(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)))))))