Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.8s

Alternatives: 19

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

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

FPCore

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

C

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

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(t) * ew) * cos(atan(((tan(t) / ew) * eh))))))
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(t) * ew) * Math.cos(Math.atan(((Math.tan(t) / ew) * eh))))));
}

Python

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

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(t) * ew) * cos(atan(Float64(Float64(tan(t) / ew) * eh))))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. lower-*.f64 99.8

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. atan-neg-rev N/A

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

   9. lower-atan.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. remove-double-neg N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 54.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right|\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_3 := \left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2 \cdot 10^{-112}:\\ \;\;\;\;e^{\log \left(\left(ew \cdot ew\right) \cdot {\cos t}^{2}\right) \cdot 0.5}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (cos
            (atan
             (*
              (/ (sin t) (- ew))
              (/
               eh
               (+
                1.0
                (*
                 (* t t)
                 (-
                  (*
                   (* t t)
                   (+ 0.041666666666666664 (* -0.001388888888888889 (* t t))))
                  0.5)))))))
           ew)))
        (t_2 (atan (/ (* eh (tan t)) (- ew))))
        (t_3 (- (* (* ew (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2)))))
   (if (<= t_3 -2e+158)
     t_1
     (if (<= t_3 -2e-112)
       (exp (* (log (* (* ew ew) (pow (cos t) 2.0))) 0.5))
       (if (<= t_3 -5e-270) t_1 (* (cos t) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	double t_2 = atan(((eh * tan(t)) / -ew));
	double t_3 = ((ew * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double tmp;
	if (t_3 <= -2e+158) {
		tmp = t_1;
	} else if (t_3 <= -2e-112) {
		tmp = exp((log(((ew * ew) * pow(cos(t), 2.0))) * 0.5));
	} else if (t_3 <= -5e-270) {
		tmp = t_1;
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0d0 + ((t * t) * (((t * t) * (0.041666666666666664d0 + ((-0.001388888888888889d0) * (t * t)))) - 0.5d0))))))) * ew))
    t_2 = atan(((eh * tan(t)) / -ew))
    t_3 = ((ew * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2))
    if (t_3 <= (-2d+158)) then
        tmp = t_1
    else if (t_3 <= (-2d-112)) then
        tmp = exp((log(((ew * ew) * (cos(t) ** 2.0d0))) * 0.5d0))
    else if (t_3 <= (-5d-270)) then
        tmp = t_1
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.cos(Math.atan(((Math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double t_3 = ((ew * Math.cos(t)) * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2));
	double tmp;
	if (t_3 <= -2e+158) {
		tmp = t_1;
	} else if (t_3 <= -2e-112) {
		tmp = Math.exp((Math.log(((ew * ew) * Math.pow(Math.cos(t), 2.0))) * 0.5));
	} else if (t_3 <= -5e-270) {
		tmp = t_1;
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(math.atan(((math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew))
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	t_3 = ((ew * math.cos(t)) * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))
	tmp = 0
	if t_3 <= -2e+158:
		tmp = t_1
	elif t_3 <= -2e-112:
		tmp = math.exp((math.log(((ew * ew) * math.pow(math.cos(t), 2.0))) * 0.5))
	elif t_3 <= -5e-270:
		tmp = t_1
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(Float64(t * t) * Float64(0.041666666666666664 + Float64(-0.001388888888888889 * Float64(t * t)))) - 0.5))))))) * ew))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_3 = Float64(Float64(Float64(ew * cos(t)) * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	tmp = 0.0
	if (t_3 <= -2e+158)
		tmp = t_1;
	elseif (t_3 <= -2e-112)
		tmp = exp(Float64(log(Float64(Float64(ew * ew) * (cos(t) ^ 2.0))) * 0.5));
	elseif (t_3 <= -5e-270)
		tmp = t_1;
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	t_2 = atan(((eh * tan(t)) / -ew));
	t_3 = ((ew * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	tmp = 0.0;
	if (t_3 <= -2e+158)
		tmp = t_1;
	elseif (t_3 <= -2e-112)
		tmp = exp((log(((ew * ew) * (cos(t) ^ 2.0))) * 0.5));
	elseif (t_3 <= -5e-270)
		tmp = t_1;
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.001388888888888889 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e+158], t$95$1, If[LessEqual[t$95$3, -2e-112], N[Exp[N[(N[Log[N[(N[(ew * ew), $MachinePrecision] * N[Power[N[Cos[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, -5e-270], t$95$1, N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.99999999999999991e158 or -1.9999999999999999e-112 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {t}^{2}\right) - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right| \]

   if -1.99999999999999991e158 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.9999999999999999e-112

   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 61.6%

      \[\leadsto \color{blue}{e^{\log \left({\left(\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right)}^{2}\right) \cdot 0.5}} \]

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

         \[\leadsto e^{\log \color{blue}{\left({ew}^{2} \cdot {\cos t}^{2}\right)} \cdot \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto e^{\log \left(\color{blue}{\left(ew \cdot ew\right)} \cdot {\cos t}^{2}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto e^{\log \left(\color{blue}{\left(ew \cdot ew\right)} \cdot {\cos t}^{2}\right) \cdot \frac{1}{2}} \]

      4. lower-pow.f64 N/A

         \[\leadsto e^{\log \left(\left(ew \cdot ew\right) \cdot \color{blue}{{\cos t}^{2}}\right) \cdot \frac{1}{2}} \]

      5. lower-cos.f64 51.9

         \[\leadsto e^{\log \left(\left(ew \cdot ew\right) \cdot {\color{blue}{\cos t}}^{2}\right) \cdot 0.5} \]

   7. Applied rewrites 51.9%

      \[\leadsto e^{\log \color{blue}{\left(\left(ew \cdot ew\right) \cdot {\cos t}^{2}\right)} \cdot 0.5} \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{+158}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-112}:\\ \;\;\;\;e^{\log \left(\left(ew \cdot ew\right) \cdot {\cos t}^{2}\right) \cdot 0.5}\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs
      (*
       (cos
        (atan
         (*
          (/ (sin t) (- ew))
          (/
           eh
           (+
            1.0
            (*
             (* t t)
             (-
              (*
               (* t t)
               (+ 0.041666666666666664 (* -0.001388888888888889 (* t t))))
              0.5)))))))
       ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0d0 + ((t * t) * (((t * t) * (0.041666666666666664d0 + ((-0.001388888888888889d0) * (t * t)))) - 0.5d0))))))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.fabs((math.cos(math.atan(((math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(Float64(t * t) * Float64(0.041666666666666664 + Float64(-0.001388888888888889 * Float64(t * t)))) - 0.5))))))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * (((t * t) * (0.041666666666666664 + (-0.001388888888888889 * (t * t)))) - 0.5))))))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(0.041666666666666664 + N[(-0.001388888888888889 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {t}^{2}\right) - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 38.0%

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right| \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(\left(t \cdot t\right) \cdot \left(0.041666666666666664 + -0.001388888888888889 \cdot \left(t \cdot t\right)\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs
      (*
       (cos
        (atan
         (*
          (/ (sin t) (- ew))
          (/
           eh
           (+ 1.0 (* (* t t) (- (* 0.041666666666666664 (* t t)) 0.5)))))))
       ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5))))))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0d0 + ((t * t) * ((0.041666666666666664d0 * (t * t)) - 0.5d0))))))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5))))))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.fabs((math.cos(math.atan(((math.sin(t) / -ew) * (eh / (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5))))))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.041666666666666664 * Float64(t * t)) - 0.5))))))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + ((t * t) * ((0.041666666666666664 * (t * t)) - 0.5))))))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 38.0%

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)}\right) \cdot ew\right| \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + \left(t \cdot t\right) \cdot \left(0.041666666666666664 \cdot \left(t \cdot t\right) - 0.5\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + -0.5 \cdot \left(t \cdot t\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs
      (*
       (cos (atan (* (/ (sin t) (- ew)) (/ eh (+ 1.0 (* -0.5 (* t t)))))))
       ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + (-0.5 * (t * t))))))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0d0 + ((-0.5d0) * (t * t))))))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.sin(t) / -ew) * (eh / (1.0 + (-0.5 * (t * t))))))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.fabs((math.cos(math.atan(((math.sin(t) / -ew) * (eh / (1.0 + (-0.5 * (t * t))))))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / Float64(1.0 + Float64(-0.5 * Float64(t * t))))))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs((cos(atan(((sin(t) / -ew) * (eh / (1.0 + (-0.5 * (t * t))))))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 38.0%

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\sin t}{ew}\right) \cdot \frac{eh}{1 + -0.5 \cdot \left(t \cdot t\right)}\right) \cdot ew\right| \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1 + -0.5 \cdot \left(t \cdot t\right)}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs (* (cos (atan (* (/ (sin t) (- ew)) (/ eh 1.0)))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan(((sin(t) / -ew) * (eh / 1.0)))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = abs((cos(atan(((sin(t) / -ew) * (eh / 1.0d0)))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.abs((Math.cos(Math.atan(((Math.sin(t) / -ew) * (eh / 1.0)))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.fabs((math.cos(math.atan(((math.sin(t) / -ew) * (eh / 1.0)))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / 1.0)))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs((cos(atan(((sin(t) / -ew) * (eh / 1.0)))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 37.9%

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

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{1}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs (* (cos (atan (* eh (/ (tan t) ew)))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan((eh * (tan(t) / ew)))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = abs((cos(atan((eh * (tan(t) / ew)))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.abs((Math.cos(Math.atan((eh * (Math.tan(t) / ew)))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.fabs((math.cos(math.atan((eh * (math.tan(t) / ew)))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(eh * Float64(tan(t) / ew)))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs((cos(atan((eh * (tan(t) / ew)))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   7. Applied rewrites 37.9%

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

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\left(-t\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{eh}{ew} \cdot 0.3333333333333333, \frac{eh}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (fabs
      (*
       (cos
        (atan
         (* (- t) (fma (* t t) (* (/ eh ew) 0.3333333333333333) (/ eh ew)))))
       ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = fabs((cos(atan((-t * fma((t * t), ((eh / ew) * 0.3333333333333333), (eh / ew))))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = abs(Float64(cos(atan(Float64(Float64(-t) * fma(Float64(t * t), Float64(Float64(eh / ew) * 0.3333333333333333), Float64(eh / ew))))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Abs[N[(N[Cos[N[ArcTan[N[((-t) * N[(N[(t * t), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 37.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 34.7%

      \[\leadsto \left|\cos \tan^{-1} \left(t \cdot \left(-1 \cdot \mathsf{fma}\left(t \cdot t, \frac{eh}{ew} \cdot 0.3333333333333333, \frac{eh}{ew}\right)\right)\right) \cdot ew\right| \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\left(-t\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{eh}{ew} \cdot 0.3333333333333333, \frac{eh}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-270}:\\ \;\;\;\;e^{\log \left(ew \cdot ew\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -5e-270)
     (exp (* (log (* ew ew)) 0.5))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270) {
		tmp = exp((log((ew * ew)) * 0.5));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-5d-270)) then
        tmp = exp((log((ew * ew)) * 0.5d0))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -5e-270) {
		tmp = Math.exp((Math.log((ew * ew)) * 0.5));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -5e-270:
		tmp = math.exp((math.log((ew * ew)) * 0.5))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = exp(Float64(log(Float64(ew * ew)) * 0.5));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -5e-270)
		tmp = exp((log((ew * ew)) * 0.5));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -5e-270], N[Exp[N[(N[Log[N[(ew * ew), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999998e-270

   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 39.0%

      \[\leadsto \color{blue}{e^{\log \left({\left(\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right)}^{2}\right) \cdot 0.5}} \]

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto e^{\log \color{blue}{\left(ew \cdot ew\right)} \cdot \frac{1}{2}} \]

      2. lower-*.f64 22.9

         \[\leadsto e^{\log \color{blue}{\left(ew \cdot ew\right)} \cdot 0.5} \]

   7. Applied rewrites 22.9%

      \[\leadsto e^{\log \color{blue}{\left(ew \cdot ew\right)} \cdot 0.5} \]

   if -4.9999999999999998e-270 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 76.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 33.0

          \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

   7. Applied rewrites 33.0%

      \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 61.6

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

   10. Applied rewrites 61.6%

       \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -5 \cdot 10^{-270}:\\ \;\;\;\;e^{\log \left(ew \cdot ew\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

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(t) * ew) * cos(atan(((tan(t) / ew) * eh))))))
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(t) * ew) * Math.cos(Math.atan(((Math.tan(t) / ew) * eh))))));
}

Python

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

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(t) * ew) * cos(atan(Float64(Float64(tan(t) / ew) * eh))))))
end

MATLAB

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

Derivation

1. Initial program 99.8%

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

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

   3. lower-*.f64 99.8

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. atan-neg-rev N/A

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

   9. lower-atan.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. remove-double-neg N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Alternative 11: 94.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -1.35 \cdot 10^{+69} \lor \neg \left(eh \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew} \cdot t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot eh, \sin t, ew \cdot \cos t\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (or (<= eh -1.35e+69) (not (<= eh 2.2e-6)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (*
        (* (cos t) ew)
        (cos (atan (* (/ (fma 0.3333333333333333 (* (* t t) eh) eh) ew) t))))))
     (fabs (/ (fma (* t_1 eh) (sin t) (* ew (cos t))) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if ((eh <= -1.35e+69) || !(eh <= 2.2e-6)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(t) * ew) * cos(atan(((fma(0.3333333333333333, ((t * t) * eh), eh) / ew) * t))))));
	} else {
		tmp = fabs((fma((t_1 * eh), sin(t), (ew * cos(t))) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if ((eh <= -1.35e+69) || !(eh <= 2.2e-6))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(t) * ew) * cos(atan(Float64(Float64(fma(0.3333333333333333, Float64(Float64(t * t) * eh), eh) / ew) * t))))));
	else
		tmp = abs(Float64(fma(Float64(t_1 * eh), sin(t), Float64(ew * cos(t))) / cosh(asinh(t_1))));
	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, -1.35e+69], N[Not[LessEqual[eh, 2.2e-6]], $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[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[(0.3333333333333333 * N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.3499999999999999e69 or 2.2000000000000001e-6 < 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. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. atan-neg-rev N/A

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

      9. lower-atan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 93.9

          \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\mathsf{fma}\left(0.3333333333333333, \color{blue}{\left(t \cdot t\right)} \cdot eh, eh\right)}{ew} \cdot t\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-\frac{eh \cdot t}{ew}\right)\right| \]

   10. Applied rewrites 93.9%

       \[\leadsto \left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew} \cdot t\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(-\frac{eh \cdot t}{ew}\right)\right| \]

   if -1.3499999999999999e69 < eh < 2.2000000000000001e-6

   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 50.6%

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

   6. Applied rewrites 98.2%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.35 \cdot 10^{+69} \lor \neg \left(eh \leq 2.2 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\mathsf{fma}\left(0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew} \cdot t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, ew \cdot \cos t\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 87.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := eh \cdot \frac{\tan t}{ew}\\ t_3 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -3.3 \cdot 10^{+82} \lor \neg \left(eh \leq 1.45 \cdot 10^{+96}\right):\\ \;\;\;\;\left|t\_3 \cdot \sin \tan^{-1} \left(\frac{t\_3}{t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_2 \cdot eh, \sin t, t\_1\right)}{\cosh \sinh^{-1} t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (* eh (/ (tan t) ew)))
        (t_3 (* (- eh) (sin t))))
   (if (or (<= eh -3.3e+82) (not (<= eh 1.45e+96)))
     (fabs (* t_3 (sin (atan (/ t_3 t_1)))))
     (fabs (/ (fma (* t_2 eh) (sin t) t_1) (cosh (asinh t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = eh * (tan(t) / ew);
	double t_3 = -eh * sin(t);
	double tmp;
	if ((eh <= -3.3e+82) || !(eh <= 1.45e+96)) {
		tmp = fabs((t_3 * sin(atan((t_3 / t_1)))));
	} else {
		tmp = fabs((fma((t_2 * eh), sin(t), t_1) / cosh(asinh(t_2))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(eh * Float64(tan(t) / ew))
	t_3 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -3.3e+82) || !(eh <= 1.45e+96))
		tmp = abs(Float64(t_3 * sin(atan(Float64(t_3 / t_1)))));
	else
		tmp = abs(Float64(fma(Float64(t_2 * eh), sin(t), t_1) / cosh(asinh(t_2))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -3.3e+82], N[Not[LessEqual[eh, 1.45e+96]], $MachinePrecision]], N[Abs[N[(t$95$3 * N[Sin[N[ArcTan[N[(t$95$3 / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$2 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$1), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.2999999999999998e82 or 1.44999999999999989e96 < 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. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. atan-neg-rev N/A

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

      9. lower-atan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   6. 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. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 76.3

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

   7. Applied rewrites 76.3%

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

   if -3.2999999999999998e82 < eh < 1.44999999999999989e96

   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 49.8%

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

   6. Applied rewrites 96.2%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+82} \lor \neg \left(eh \leq 1.45 \cdot 10^{+96}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, ew \cdot \cos t\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 85.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -3.3 \cdot 10^{+82} \lor \neg \left(eh \leq 1.45 \cdot 10^{+96}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* (- eh) (sin t))))
   (if (or (<= eh -3.3e+82) (not (<= eh 1.45e+96)))
     (fabs (* t_2 (sin (atan (/ t_2 (* ew (cos t)))))))
     (fabs
      (/
       (fma (sin t) (* t_1 (* eh eh)) (* (cos t) ew))
       (cosh (asinh (* t_1 eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = -eh * sin(t);
	double tmp;
	if ((eh <= -3.3e+82) || !(eh <= 1.45e+96)) {
		tmp = fabs((t_2 * sin(atan((t_2 / (ew * cos(t)))))));
	} else {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), (cos(t) * ew)) / cosh(asinh((t_1 * eh)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -3.3e+82) || !(eh <= 1.45e+96))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * cos(t)))))));
	else
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), Float64(cos(t) * ew)) / 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]}, Block[{t$95$2 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -3.3e+82], N[Not[LessEqual[eh, 1.45e+96]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $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 < -3.2999999999999998e82 or 1.44999999999999989e96 < 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. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. atan-neg-rev N/A

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

      9. lower-atan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   6. 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. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 76.3

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

   7. Applied rewrites 76.3%

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

   if -3.2999999999999998e82 < eh < 1.44999999999999989e96

   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 93.4%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+82} \lor \neg \left(eh \leq 1.45 \cdot 10^{+96}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -2.1 \cdot 10^{+77} \lor \neg \left(eh \leq 3.2 \cdot 10^{+18}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{t\_1}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (* (- eh) (sin t))))
   (if (or (<= eh -2.1e+77) (not (<= eh 3.2e+18)))
     (fabs (* t_2 (sin (atan (/ t_2 t_1)))))
     (fabs (* t_1 (cos (atan (/ (* eh (sin t)) t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = -eh * sin(t);
	double tmp;
	if ((eh <= -2.1e+77) || !(eh <= 3.2e+18)) {
		tmp = fabs((t_2 * sin(atan((t_2 / t_1)))));
	} else {
		tmp = fabs((t_1 * cos(atan(((eh * sin(t)) / t_1)))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = -eh * sin(t)
    if ((eh <= (-2.1d+77)) .or. (.not. (eh <= 3.2d+18))) then
        tmp = abs((t_2 * sin(atan((t_2 / t_1)))))
    else
        tmp = abs((t_1 * cos(atan(((eh * sin(t)) / t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = -eh * Math.sin(t);
	double tmp;
	if ((eh <= -2.1e+77) || !(eh <= 3.2e+18)) {
		tmp = Math.abs((t_2 * Math.sin(Math.atan((t_2 / t_1)))));
	} else {
		tmp = Math.abs((t_1 * Math.cos(Math.atan(((eh * Math.sin(t)) / t_1)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = -eh * math.sin(t)
	tmp = 0
	if (eh <= -2.1e+77) or not (eh <= 3.2e+18):
		tmp = math.fabs((t_2 * math.sin(math.atan((t_2 / t_1)))))
	else:
		tmp = math.fabs((t_1 * math.cos(math.atan(((eh * math.sin(t)) / t_1)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -2.1e+77) || !(eh <= 3.2e+18))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / t_1)))));
	else
		tmp = abs(Float64(t_1 * cos(atan(Float64(Float64(eh * sin(t)) / t_1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = -eh * sin(t);
	tmp = 0.0;
	if ((eh <= -2.1e+77) || ~((eh <= 3.2e+18)))
		tmp = abs((t_2 * sin(atan((t_2 / t_1)))));
	else
		tmp = abs((t_1 * cos(atan(((eh * sin(t)) / t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -2.1e+77], N[Not[LessEqual[eh, 3.2e+18]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.0999999999999999e77 or 3.2e18 < 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. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. atan-neg-rev N/A

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

      9. lower-atan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   6. 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. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if -2.0999999999999999e77 < eh < 3.2e18

   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. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. atan-neg-rev N/A

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

      9. lower-atan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. associate-/l* N/A

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

      13. lift-neg.f64 N/A

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

      14. distribute-lft-neg-out N/A

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

      15. remove-double-neg N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 80.0

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

   7. Applied rewrites 80.0%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.1 \cdot 10^{+77} \lor \neg \left(eh \leq 3.2 \cdot 10^{+18}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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

   3. lower-*.f64 99.8

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. atan-neg-rev N/A

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

   9. lower-atan.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. remove-double-neg N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-cos.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-cos.f64 58.1

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

7. Applied rewrites 58.1%

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

Alternative 16: 56.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left({\cos t}^{2}\right)\\ \mathbf{if}\;ew \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\left(t\_1 + -2 \cdot \log \left(\frac{-1}{ew}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(t\_1 - -2 \cdot \log ew\right) \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (log (pow (cos t) 2.0))))
   (if (<= ew -4e-310)
     (exp (* (+ t_1 (* -2.0 (log (/ -1.0 ew)))) 0.5))
     (exp (* (- t_1 (* -2.0 (log ew))) 0.5)))))

C

double code(double eh, double ew, double t) {
	double t_1 = log(pow(cos(t), 2.0));
	double tmp;
	if (ew <= -4e-310) {
		tmp = exp(((t_1 + (-2.0 * log((-1.0 / ew)))) * 0.5));
	} else {
		tmp = exp(((t_1 - (-2.0 * log(ew))) * 0.5));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = log((cos(t) ** 2.0d0))
    if (ew <= (-4d-310)) then
        tmp = exp(((t_1 + ((-2.0d0) * log(((-1.0d0) / ew)))) * 0.5d0))
    else
        tmp = exp(((t_1 - ((-2.0d0) * log(ew))) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.log(Math.pow(Math.cos(t), 2.0));
	double tmp;
	if (ew <= -4e-310) {
		tmp = Math.exp(((t_1 + (-2.0 * Math.log((-1.0 / ew)))) * 0.5));
	} else {
		tmp = Math.exp(((t_1 - (-2.0 * Math.log(ew))) * 0.5));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.log(math.pow(math.cos(t), 2.0))
	tmp = 0
	if ew <= -4e-310:
		tmp = math.exp(((t_1 + (-2.0 * math.log((-1.0 / ew)))) * 0.5))
	else:
		tmp = math.exp(((t_1 - (-2.0 * math.log(ew))) * 0.5))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = log((cos(t) ^ 2.0))
	tmp = 0.0
	if (ew <= -4e-310)
		tmp = exp(Float64(Float64(t_1 + Float64(-2.0 * log(Float64(-1.0 / ew)))) * 0.5));
	else
		tmp = exp(Float64(Float64(t_1 - Float64(-2.0 * log(ew))) * 0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = log((cos(t) ^ 2.0));
	tmp = 0.0;
	if (ew <= -4e-310)
		tmp = exp(((t_1 + (-2.0 * log((-1.0 / ew)))) * 0.5));
	else
		tmp = exp(((t_1 - (-2.0 * log(ew))) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Log[N[Power[N[Cos[t], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -4e-310], N[Exp[N[(N[(t$95$1 + N[(-2.0 * N[Log[N[(-1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(t$95$1 - N[(-2.0 * N[Log[ew], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.999999999999988e-310

   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

   4. Applied rewrites 36.8%

      \[\leadsto \color{blue}{e^{\log \left({\left(\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right)}^{2}\right) \cdot 0.5}} \]

   5. Taylor expanded in ew around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-/.f64 53.4

         \[\leadsto e^{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \log \color{blue}{\left(\frac{-1}{ew}\right)}\right) \cdot 0.5} \]

   7. Applied rewrites 53.4%

      \[\leadsto e^{\color{blue}{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \log \left(\frac{-1}{ew}\right)\right)} \cdot 0.5} \]

   if -3.999999999999988e-310 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 40.0%

      \[\leadsto \color{blue}{e^{\log \left({\left(\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right)}^{2}\right) \cdot 0.5}} \]

   5. Taylor expanded in ew around inf

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

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. log-rec N/A

         \[\leadsto e^{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \color{blue}{\left(\mathsf{neg}\left(\log ew\right)\right)}\right) \cdot \frac{1}{2}} \]

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

         \[\leadsto e^{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \color{blue}{\left(\mathsf{neg}\left(\log ew\right)\right)}\right) \cdot \frac{1}{2}} \]

      9. lower-neg.f64 N/A

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

      10. lower-log.f64 53.5

          \[\leadsto e^{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \left(-\color{blue}{\log ew}\right)\right) \cdot 0.5} \]

   7. Applied rewrites 53.5%

      \[\leadsto e^{\color{blue}{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \left(-\log ew\right)\right)} \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\left(\log \left({\cos t}^{2}\right) + -2 \cdot \log \left(\frac{-1}{ew}\right)\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left({\cos t}^{2}\right) - -2 \cdot \log ew\right) \cdot 0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 31.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return 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 = cos(t) * ew
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 38.1%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 17.2

       \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

7. Applied rewrites 17.2%

   \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

8. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 31.0

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

10. Applied rewrites 31.0%

    \[\leadsto \color{blue}{\cos t \cdot ew} \]
11. Add Preprocessing

Alternative 18: 20.2% accurate, 50.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 38.1%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 17.2

       \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

7. Applied rewrites 17.2%

   \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

8. Taylor expanded in eh around 0

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

10. Applied rewrites 19.3%

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

12. Applied rewrites 19.4%

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

Alternative 19: 3.3% accurate, 53.9× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return -0.5 * (ew * (t * 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 = (-0.5d0) * (ew * (t * t))
end function

Java

public static double code(double eh, double ew, double t) {
	return -0.5 * (ew * (t * t));
}

Python

def code(eh, ew, t):
	return -0.5 * (ew * (t * t))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = -0.5 * (ew * (t * t));
end

Wolfram

code[eh_, ew_, t_] := N[(-0.5 * N[(ew * N[(t * 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 38.1%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 17.2

       \[\leadsto ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{\color{blue}{eh \cdot eh}}{ew} \cdot -0.5\right) \]

7. Applied rewrites 17.2%

   \[\leadsto \color{blue}{ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5\right)} \]

8. Taylor expanded in eh around 0

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

10. Applied rewrites 19.3%

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

11. Taylor expanded in t around inf

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

13. Applied rewrites 3.4%

    \[\leadsto -0.5 \cdot \left(ew \cdot \color{blue}{\left(t \cdot t\right)}\right) \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024352 
(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)))))))