Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-*.f64 99.8

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/l/ N/A

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

   10. *-commutative N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

3. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

3. Applied rewrites 99.8%

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

Alternative 3: 86.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \frac{eh}{\tan t}\\ t_3 := \frac{t\_2}{ew}\\ t_4 := \cosh \sinh^{-1} t\_3\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{+35}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_3, eh, t\_1\right)}{t\_4}\right|\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+31}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot ew\right) \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{t\_2 \cdot \cos t}{ew}, eh, t\_1\right)}{t\_4}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew))
        (t_2 (/ eh (tan t)))
        (t_3 (/ t_2 ew))
        (t_4 (cosh (asinh t_3))))
   (if (<= t -9.2e+35)
     (fabs (/ (fma (* (cos t) t_3) eh t_1) t_4))
     (if (<= t 2.65e+31)
       (fabs
        (+
         (* (* (cos (atan (/ eh (* ew (tan t))))) ew) (sin t))
         (* eh (sin (atan (/ (/ eh ew) (tan t)))))))
       (fabs (/ (fma (/ (* t_2 (cos t)) ew) eh t_1) t_4))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = eh / tan(t);
	double t_3 = t_2 / ew;
	double t_4 = cosh(asinh(t_3));
	double tmp;
	if (t <= -9.2e+35) {
		tmp = fabs((fma((cos(t) * t_3), eh, t_1) / t_4));
	} else if (t <= 2.65e+31) {
		tmp = fabs((((cos(atan((eh / (ew * tan(t))))) * ew) * sin(t)) + (eh * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((fma(((t_2 * cos(t)) / ew), eh, t_1) / t_4));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(eh / tan(t))
	t_3 = Float64(t_2 / ew)
	t_4 = cosh(asinh(t_3))
	tmp = 0.0
	if (t <= -9.2e+35)
		tmp = abs(Float64(fma(Float64(cos(t) * t_3), eh, t_1) / t_4));
	elseif (t <= 2.65e+31)
		tmp = abs(Float64(Float64(Float64(cos(atan(Float64(eh / Float64(ew * tan(t))))) * ew) * sin(t)) + Float64(eh * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(fma(Float64(Float64(t_2 * cos(t)) / ew), eh, t_1) / t_4));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / ew), $MachinePrecision]}, Block[{t$95$4 = N[Cosh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -9.2e+35], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$3), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.65e+31], N[Abs[N[(N[(N[(N[Cos[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(t$95$2 * N[Cos[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / t$95$4), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.1999999999999993e35

   1. Initial program 99.6%

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

   3. Applied rewrites 73.9%

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

   if -9.1999999999999993e35 < t < 2.6500000000000002e31

   1. Initial program 99.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 99.9

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/l/ N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

   3. Applied rewrites 99.9%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 99.9

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. unpow1 N/A

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

      2. metadata-eval N/A

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

      3. pow-unpow N/A

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

      4. pow-to-exp N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-log.f64 N/A

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

      8. lower-pow.f64 94.7

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

   7. Applied rewrites 94.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 96.0%

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

   if 2.6500000000000002e31 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 74.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 74.9

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

   5. Applied rewrites 74.9%

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

Alternative 4: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;-1 \cdot t\_1\\ \mathbf{elif}\;eh \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew} \cdot \cos t}{\tan t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -4.3e+80)
     (* -1.0 t_1)
     (if (<= eh 3.9e+27)
       (fabs
        (/
         (fma (/ (* (/ eh ew) (cos t)) (tan t)) eh (* (sin t) ew))
         (cosh (asinh (/ (/ eh (tan t)) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -4.3e+80) {
		tmp = -1.0 * t_1;
	} else if (eh <= 3.9e+27) {
		tmp = fabs((fma((((eh / ew) * cos(t)) / tan(t)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -4.3e+80)
		tmp = Float64(-1.0 * t_1);
	elseif (eh <= 3.9e+27)
		tmp = abs(Float64(fma(Float64(Float64(Float64(eh / ew) * cos(t)) / tan(t)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -4.3e+80], N[(-1.0 * t$95$1), $MachinePrecision], If[LessEqual[eh, 3.9e+27], N[Abs[N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.30000000000000004e80

   1. Initial program 99.8%

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

   3. Applied rewrites 13.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 7.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \cos t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\cos t}\right) \]

      3. lift-cos.f64 67.1

         \[\leadsto -1 \cdot \left(eh \cdot \cos t\right) \]

   10. Applied rewrites 67.1%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \cos t\right)} \]

   if -4.30000000000000004e80 < eh < 3.8999999999999999e27

   1. Initial program 99.8%

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

   3. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/l/ N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 89.5

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

   5. Applied rewrites 89.5%

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

   if 3.8999999999999999e27 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 17.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 10.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 65.2

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{eh \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 79.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;-1 \cdot t\_2\\ \mathbf{elif}\;eh \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)) (t_2 (* eh (cos t))))
   (if (<= eh -4.3e+80)
     (* -1.0 t_2)
     (if (<= eh 3.9e+27)
       (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double t_2 = eh * cos(t);
	double tmp;
	if (eh <= -4.3e+80) {
		tmp = -1.0 * t_2;
	} else if (eh <= 3.9e+27) {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -4.3e+80)
		tmp = Float64(-1.0 * t_2);
	elseif (eh <= 3.9e+27)
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -4.3e+80], N[(-1.0 * t$95$2), $MachinePrecision], If[LessEqual[eh, 3.9e+27], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.30000000000000004e80

   1. Initial program 99.8%

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

   3. Applied rewrites 13.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 7.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \cos t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\cos t}\right) \]

      3. lift-cos.f64 67.1

         \[\leadsto -1 \cdot \left(eh \cdot \cos t\right) \]

   10. Applied rewrites 67.1%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \cos t\right)} \]

   if -4.30000000000000004e80 < eh < 3.8999999999999999e27

   1. Initial program 99.8%

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

   3. Applied rewrites 89.5%

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

   if 3.8999999999999999e27 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 17.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 10.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 65.2

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{eh \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{+80}:\\ \;\;\;\;-1 \cdot t\_1\\ \mathbf{elif}\;eh \leq 3.9 \cdot 10^{+27}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{\tan t \cdot ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -4.3e+80)
     (* -1.0 t_1)
     (if (<= eh 3.9e+27)
       (fabs
        (/
         (fma (/ (* (cos t) eh) (* (tan t) ew)) eh (* (sin t) ew))
         (cosh (asinh (/ (/ eh (tan t)) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -4.3e+80) {
		tmp = -1.0 * t_1;
	} else if (eh <= 3.9e+27) {
		tmp = fabs((fma(((cos(t) * eh) / (tan(t) * ew)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -4.3e+80)
		tmp = Float64(-1.0 * t_1);
	elseif (eh <= 3.9e+27)
		tmp = abs(Float64(fma(Float64(Float64(cos(t) * eh) / Float64(tan(t) * ew)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -4.3e+80], N[(-1.0 * t$95$1), $MachinePrecision], If[LessEqual[eh, 3.9e+27], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.30000000000000004e80

   1. Initial program 99.8%

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

   3. Applied rewrites 13.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 12.9

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

   5. Applied rewrites 12.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 7.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 7.4

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

   7. Applied rewrites 7.4%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \cos t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\cos t}\right) \]

      3. lift-cos.f64 67.1

         \[\leadsto -1 \cdot \left(eh \cdot \cos t\right) \]

   10. Applied rewrites 67.1%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \cos t\right)} \]

   if -4.30000000000000004e80 < eh < 3.8999999999999999e27

   1. Initial program 99.8%

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

   3. Applied rewrites 89.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-/.f64 83.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.9

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   if 3.8999999999999999e27 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 17.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 15.5

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

   5. Applied rewrites 15.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 10.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 10.2

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

   7. Applied rewrites 10.2%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 65.2

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 65.2%

       \[\leadsto \color{blue}{eh \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 71.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -1.24 \cdot 10^{+64}:\\ \;\;\;\;-1 \cdot t\_1\\ \mathbf{elif}\;eh \leq 2.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\mathsf{fma}\left(t, 1, 0\right)}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -1.24e+64)
     (* -1.0 t_1)
     (if (<= eh 2.1)
       (fabs
        (/
         (fma (* (cos t) (/ (/ eh (fma t 1.0 0.0)) ew)) eh (* (sin t) ew))
         (cosh (asinh (/ (/ eh (tan t)) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -1.24e+64) {
		tmp = -1.0 * t_1;
	} else if (eh <= 2.1) {
		tmp = fabs((fma((cos(t) * ((eh / fma(t, 1.0, 0.0)) / ew)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -1.24e+64)
		tmp = Float64(-1.0 * t_1);
	elseif (eh <= 2.1)
		tmp = abs(Float64(fma(Float64(cos(t) * Float64(Float64(eh / fma(t, 1.0, 0.0)) / ew)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -1.24e+64], N[(-1.0 * t$95$1), $MachinePrecision], If[LessEqual[eh, 2.1], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * N[(N[(eh / N[(t * 1.0 + 0.0), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.24e64

   1. Initial program 99.8%

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

   3. Applied rewrites 15.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 13.7

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

   5. Applied rewrites 13.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 8.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 8.6

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

   7. Applied rewrites 8.6%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \cos t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\cos t}\right) \]

      3. lift-cos.f64 66.0

         \[\leadsto -1 \cdot \left(eh \cdot \cos t\right) \]

   10. Applied rewrites 66.0%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \cos t\right)} \]

   if -1.24e64 < eh < 2.10000000000000009

   1. Initial program 99.8%

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

   3. Applied rewrites 90.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{\sin \mathsf{PI}\left(\right) \cdot \sin \mathsf{PI}\left(\right)}{{\cos \mathsf{PI}\left(\right)}^{2}}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      2. sin-PI N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{0 \cdot \sin \mathsf{PI}\left(\right)}{{\cos \mathsf{PI}\left(\right)}^{2}}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      3. sin-PI N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{0 \cdot 0}{{\cos \mathsf{PI}\left(\right)}^{2}}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{0}{{\cos \mathsf{PI}\left(\right)}^{2}}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      5. cos-PI N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{0}{{-1}^{2}}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      6. metadata-eval N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot \frac{0}{1}\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      7. metadata-eval N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - -1 \cdot 0\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot \left(1 - 0\right) + \frac{\sin \mathsf{PI}\left(\right)}{\cos \mathsf{PI}\left(\right)}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      9. metadata-eval N/A

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

      10. cos-PI N/A

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

      11. sin-PI N/A

          \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot 1 + \frac{0}{-1}}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      12. metadata-eval N/A

          \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{t \cdot 1 + 0}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right| \]

      13. sin-PI N/A

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

   8. Applied rewrites 77.7%

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

   if 2.10000000000000009 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 19.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 17.2

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

   5. Applied rewrites 17.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 12.3

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 12.3

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

   7. Applied rewrites 12.3%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 63.4

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 63.4%

       \[\leadsto \color{blue}{eh \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 62.8% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -4.2 \cdot 10^{+61}:\\ \;\;\;\;-1 \cdot t\_1\\ \mathbf{elif}\;eh \leq 0.0058:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -4.2e+61)
     (* -1.0 t_1)
     (if (<= eh 0.0058) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -4.2e+61) {
		tmp = -1.0 * t_1;
	} else if (eh <= 0.0058) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = 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) :: tmp
    t_1 = eh * cos(t)
    if (eh <= (-4.2d+61)) then
        tmp = (-1.0d0) * t_1
    else if (eh <= 0.0058d0) then
        tmp = abs((ew * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if (eh <= -4.2e+61) {
		tmp = -1.0 * t_1;
	} else if (eh <= 0.0058) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if eh <= -4.2e+61:
		tmp = -1.0 * t_1
	elif eh <= 0.0058:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -4.2e+61)
		tmp = Float64(-1.0 * t_1);
	elseif (eh <= 0.0058)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if (eh <= -4.2e+61)
		tmp = -1.0 * t_1;
	elseif (eh <= 0.0058)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -4.2e+61], N[(-1.0 * t$95$1), $MachinePrecision], If[LessEqual[eh, 0.0058], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.2000000000000002e61

   1. Initial program 99.8%

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

   3. Applied rewrites 15.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 13.8

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

   5. Applied rewrites 13.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 8.8

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 8.8

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

   7. Applied rewrites 8.8%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \cos t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\cos t}\right) \]

      3. lift-cos.f64 65.9

         \[\leadsto -1 \cdot \left(eh \cdot \cos t\right) \]

   10. Applied rewrites 65.9%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \cos t\right)} \]

   if -4.2000000000000002e61 < eh < 0.0058

   1. Initial program 99.8%

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

   3. Applied rewrites 90.5%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 61.3

         \[\leadsto \left|ew \cdot \sin t\right| \]

   8. Applied rewrites 61.3%

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

   if 0.0058 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 19.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 17.4

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

   5. Applied rewrites 17.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 12.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 12.6

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

   7. Applied rewrites 12.6%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 63.4

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 63.4%

       \[\leadsto \color{blue}{eh \cdot \cos t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 53.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -1.7 \cdot 10^{+244}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 0.0058:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -1.7e+244) t_1 (if (<= eh 0.0058) (fabs (* ew (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -1.7e+244) {
		tmp = t_1;
	} else if (eh <= 0.0058) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = 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) :: tmp
    t_1 = eh * cos(t)
    if (eh <= (-1.7d+244)) then
        tmp = t_1
    else if (eh <= 0.0058d0) then
        tmp = abs((ew * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if (eh <= -1.7e+244) {
		tmp = t_1;
	} else if (eh <= 0.0058) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if eh <= -1.7e+244:
		tmp = t_1
	elif eh <= 0.0058:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -1.7e+244)
		tmp = t_1;
	elseif (eh <= 0.0058)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if (eh <= -1.7e+244)
		tmp = t_1;
	elseif (eh <= 0.0058)
		tmp = abs((ew * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -1.7e+244], t$95$1, If[LessEqual[eh, 0.0058], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.70000000000000005e244 or 0.0058 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 16.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 15.0

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

   5. Applied rewrites 15.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 10.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 56.9

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 56.9%

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

   if -1.70000000000000005e244 < eh < 0.0058

   1. Initial program 99.8%

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

   3. Applied rewrites 78.7%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 52.3

         \[\leadsto \left|ew \cdot \sin t\right| \]

   8. Applied rewrites 52.3%

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

Alternative 10: 38.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -7 \cdot 10^{+205}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.4 \cdot 10^{-172}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -7e+205) t_1 (if (<= eh 2.4e-172) (* ew (sin t)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -7e+205) {
		tmp = t_1;
	} else if (eh <= 2.4e-172) {
		tmp = ew * sin(t);
	} else {
		tmp = 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) :: tmp
    t_1 = eh * cos(t)
    if (eh <= (-7d+205)) then
        tmp = t_1
    else if (eh <= 2.4d-172) then
        tmp = ew * sin(t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if (eh <= -7e+205) {
		tmp = t_1;
	} else if (eh <= 2.4e-172) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if eh <= -7e+205:
		tmp = t_1
	elif eh <= 2.4e-172:
		tmp = ew * math.sin(t)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -7e+205)
		tmp = t_1;
	elseif (eh <= 2.4e-172)
		tmp = Float64(ew * sin(t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if (eh <= -7e+205)
		tmp = t_1;
	elseif (eh <= 2.4e-172)
		tmp = ew * sin(t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -7e+205], t$95$1, If[LessEqual[eh, 2.4e-172], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.9999999999999996e205 or 2.4000000000000001e-172 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 24.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 21.7

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

   5. Applied rewrites 21.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 18.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 18.1

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

   7. Applied rewrites 18.1%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 50.3

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 50.3%

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

   if -6.9999999999999996e205 < eh < 2.4000000000000001e-172

   1. Initial program 99.8%

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

   3. Applied rewrites 39.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower--.f64 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval 35.8

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

   5. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. lift-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 31.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 31.6

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

   7. Applied rewrites 31.6%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 27.9

         \[\leadsto ew \cdot \sin t \]

   10. Applied rewrites 27.9%

       \[\leadsto \color{blue}{ew \cdot \sin t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 32.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 32.3%

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

   1. lift-cosh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. cosh-asinh N/A

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

   4. +-commutative N/A

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

   5. lower-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

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

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 29.1

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

5. Applied rewrites 29.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 25.2

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 25.2

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

7. Applied rewrites 25.2%

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

8. Taylor expanded in eh around inf

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

   1. lower-*.f64 N/A

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

   2. lift-cos.f64 32.1

      \[\leadsto eh \cdot \cos t \]

10. Applied rewrites 32.1%

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

Alternative 12: 22.0% accurate, 870.0× speedup

Math

\[\begin{array}{l} \\ eh \end{array} \]

FPCore

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

C

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

Java

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

Python

def code(eh, ew, t):
	return eh

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := eh

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 32.3%

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

   1. lift-cosh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. cosh-asinh N/A

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

   4. +-commutative N/A

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

   5. lower-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. metadata-eval N/A

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

   8. fp-cancel-sign-sub-inv N/A

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

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

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

   10. metadata-eval N/A

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

   11. lower--.f64 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f64 N/A

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

   14. metadata-eval 29.1

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

5. Applied rewrites 29.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. lift-/.f64 N/A

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

   5. frac-times N/A

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

   6. lift-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 25.2

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 25.2

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

7. Applied rewrites 25.2%

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

8. Taylor expanded in t around 0

   \[\leadsto \color{blue}{eh} \]
9. Step-by-step derivation

10. Applied rewrites 22.0%

    \[\leadsto \color{blue}{eh} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))