Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.0s

Alternatives: 12

Speedup: N/A×

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(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] + 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]), $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. Add Preprocessing
3. 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

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 94.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\\ t_2 := ew \cdot \sin t\\ t_3 := eh \cdot \cos t\\ t_4 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|t\_3 \cdot \sin t\_4 + t\_2 \cdot \cos t\_4\right| \leq 5 \cdot 10^{+50}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\cos t}{ew} \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), t\_1 \cdot \sin t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + t\_2 \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (cos (atan (/ (/ eh ew) t))))
        (t_2 (* ew (sin t)))
        (t_3 (* eh (cos t)))
        (t_4 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (fabs (+ (* t_3 (sin t_4)) (* t_2 (cos t_4)))) 5e+50)
     (fabs
      (*
       (fma
        (* (/ (cos t) ew) eh)
        (tanh (asinh (/ (/ eh (tan t)) ew)))
        (* t_1 (sin t)))
       ew))
     (fabs (+ (* t_3 (sin (atan (/ eh (* ew t))))) (* t_2 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(atan(((eh / ew) / t)));
	double t_2 = ew * sin(t);
	double t_3 = eh * cos(t);
	double t_4 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs(((t_3 * sin(t_4)) + (t_2 * cos(t_4)))) <= 5e+50) {
		tmp = fabs((fma(((cos(t) / ew) * eh), tanh(asinh(((eh / tan(t)) / ew))), (t_1 * sin(t))) * ew));
	} else {
		tmp = fabs(((t_3 * sin(atan((eh / (ew * t))))) + (t_2 * t_1)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = cos(atan(Float64(Float64(eh / ew) / t)))
	t_2 = Float64(ew * sin(t))
	t_3 = Float64(eh * cos(t))
	t_4 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(t_3 * sin(t_4)) + Float64(t_2 * cos(t_4)))) <= 5e+50)
		tmp = abs(Float64(fma(Float64(Float64(cos(t) / ew) * eh), tanh(asinh(Float64(Float64(eh / tan(t)) / ew))), Float64(t_1 * sin(t))) * ew));
	else
		tmp = abs(Float64(Float64(t_3 * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(t_2 * t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$3 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+50], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(t$95$1 * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$3 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 5e50

   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. Add Preprocessing

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.4%

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

   if 5e50 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   1. Initial program 99.7%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   2. Add Preprocessing
   3. 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 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 93.7%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 5 \cdot 10^{+50}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\cos t}{ew} \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot \sin t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ t_3 := \frac{\frac{eh}{\tan t}}{ew}\\ t_4 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|t\_2 \cdot \sin t\_4 + t\_1 \cdot \cos t\_4\right| \leq 4 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_3, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_3}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + t\_1 \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (* eh (cos t)))
        (t_3 (/ (/ eh (tan t)) ew))
        (t_4 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (fabs (+ (* t_2 (sin t_4)) (* t_1 (cos t_4)))) 4e-45)
     (fabs (/ (fma (* (cos t) t_3) eh (* (sin t) ew)) (cosh (asinh t_3))))
     (fabs
      (+
       (* t_2 (sin (atan (/ eh (* ew t)))))
       (* t_1 (cos (atan (/ (/ eh ew) t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = eh * cos(t);
	double t_3 = (eh / tan(t)) / ew;
	double t_4 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs(((t_2 * sin(t_4)) + (t_1 * cos(t_4)))) <= 4e-45) {
		tmp = fabs((fma((cos(t) * t_3), eh, (sin(t) * ew)) / cosh(asinh(t_3))));
	} else {
		tmp = fabs(((t_2 * sin(atan((eh / (ew * t))))) + (t_1 * cos(atan(((eh / ew) / t))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(eh * cos(t))
	t_3 = Float64(Float64(eh / tan(t)) / ew)
	t_4 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(t_2 * sin(t_4)) + Float64(t_1 * cos(t_4)))) <= 4e-45)
		tmp = abs(Float64(fma(Float64(cos(t) * t_3), eh, Float64(sin(t) * ew)) / cosh(asinh(t_3))));
	else
		tmp = abs(Float64(Float64(t_2 * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(t_1 * cos(atan(Float64(Float64(eh / ew) / t))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$2 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 4e-45], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$3), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$2 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 3.99999999999999994e-45

   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. Add Preprocessing

   4. Applied rewrites 93.6%

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

   if 3.99999999999999994e-45 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   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. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 93.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 4 \cdot 10^{-45}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= t 1.45e+234)
     (fabs
      (+
       (* t_1 (sin (atan (/ eh (* ew t)))))
       (* (* ew (sin t)) (cos (atan (/ (/ eh ew) t))))))
     (fabs t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (t <= 1.45e+234) {
		tmp = fabs(((t_1 * sin(atan((eh / (ew * t))))) + ((ew * sin(t)) * cos(atan(((eh / ew) / t))))));
	} else {
		tmp = fabs(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 (t <= 1.45d+234) then
        tmp = abs(((t_1 * sin(atan((eh / (ew * t))))) + ((ew * sin(t)) * cos(atan(((eh / ew) / t))))))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if (t <= 1.45e+234) {
		tmp = Math.abs(((t_1 * Math.sin(Math.atan((eh / (ew * t))))) + ((ew * Math.sin(t)) * Math.cos(Math.atan(((eh / ew) / t))))));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if t <= 1.45e+234:
		tmp = math.fabs(((t_1 * math.sin(math.atan((eh / (ew * t))))) + ((ew * math.sin(t)) * math.cos(math.atan(((eh / ew) / t))))))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (t <= 1.45e+234)
		tmp = abs(Float64(Float64(t_1 * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * sin(t)) * cos(atan(Float64(Float64(eh / ew) / t))))));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if (t <= 1.45e+234)
		tmp = abs(((t_1 * sin(atan((eh / (ew * t))))) + ((ew * sin(t)) * cos(atan(((eh / ew) / t))))));
	else
		tmp = abs(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[t, 1.45e+234], N[Abs[N[(N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.44999999999999993e234

   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. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 92.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 92.2%

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

   if 1.44999999999999993e234 < t

   1. Initial program 99.4%

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

   4. Applied rewrites 48.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 99.4%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \tan t \cdot ew\\ t_3 := eh \cdot \cos t\\ \mathbf{if}\;t \leq -9 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{t\_2}, eh, t\_1\right)}{\sqrt{1 + {\left(\frac{eh}{t\_2}\right)}^{2}}}\right|\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\left|t\_3 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{\cos t}{ew} \cdot eh}{\tan t}, eh, t\_1\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)) (t_2 (* (tan t) ew)) (t_3 (* eh (cos t))))
   (if (<= t -9e+28)
     (fabs
      (/
       (fma (/ (* (cos t) eh) t_2) eh t_1)
       (sqrt (+ 1.0 (pow (/ eh t_2) 2.0)))))
     (if (<= t 8.2e+45)
       (fabs
        (+
         (* t_3 (sin (atan (/ eh (* ew t)))))
         (* (* ew t) (cos (atan (/ (/ eh ew) t))))))
       (if (<= t 1.45e+234)
         (fabs
          (/
           (fma (/ (* (/ (cos t) ew) eh) (tan t)) eh t_1)
           (sqrt (+ 1.0 (pow (/ (/ eh (tan t)) ew) 2.0)))))
         (fabs t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = tan(t) * ew;
	double t_3 = eh * cos(t);
	double tmp;
	if (t <= -9e+28) {
		tmp = fabs((fma(((cos(t) * eh) / t_2), eh, t_1) / sqrt((1.0 + pow((eh / t_2), 2.0)))));
	} else if (t <= 8.2e+45) {
		tmp = fabs(((t_3 * sin(atan((eh / (ew * t))))) + ((ew * t) * cos(atan(((eh / ew) / t))))));
	} else if (t <= 1.45e+234) {
		tmp = fabs((fma((((cos(t) / ew) * eh) / tan(t)), eh, t_1) / sqrt((1.0 + pow(((eh / tan(t)) / ew), 2.0)))));
	} else {
		tmp = fabs(t_3);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(tan(t) * ew)
	t_3 = Float64(eh * cos(t))
	tmp = 0.0
	if (t <= -9e+28)
		tmp = abs(Float64(fma(Float64(Float64(cos(t) * eh) / t_2), eh, t_1) / sqrt(Float64(1.0 + (Float64(eh / t_2) ^ 2.0)))));
	elseif (t <= 8.2e+45)
		tmp = abs(Float64(Float64(t_3 * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * t) * cos(atan(Float64(Float64(eh / ew) / t))))));
	elseif (t <= 1.45e+234)
		tmp = abs(Float64(fma(Float64(Float64(Float64(cos(t) / ew) * eh) / tan(t)), eh, t_1) / sqrt(Float64(1.0 + (Float64(Float64(eh / tan(t)) / ew) ^ 2.0)))));
	else
		tmp = abs(t_3);
	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[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9e+28], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / t$95$2), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(eh / t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 8.2e+45], N[Abs[N[(N[(t$95$3 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.45e+234], N[Abs[N[(N[(N[(N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.9999999999999994e28

   1. Initial program 99.7%

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

   4. Applied rewrites 73.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 68.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-tan.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lower-*.f64 68.0

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

   8. Applied rewrites 68.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 68.2

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

   10. Applied rewrites 68.2%

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

   if -8.9999999999999994e28 < t < 8.20000000000000025e45

   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. Add Preprocessing
   3. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 95.7%

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

   if 8.20000000000000025e45 < t < 1.44999999999999993e234

   1. Initial program 99.4%

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

   4. Applied rewrites 83.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 79.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. *-commutative N/A

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

      6. associate-/l/ N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l* N/A

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

      11. lift-*.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lower-/.f64 79.9

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

      14. lift-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l/ N/A

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

      17. lift-/.f64 N/A

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

      18. lower-*.f64 79.9

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

   8. Applied rewrites 79.9%

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

   if 1.44999999999999993e234 < t

   1. Initial program 99.4%

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

   4. Applied rewrites 48.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 99.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{\tan t \cdot ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{\tan t \cdot ew}\right)}^{2}}}\right|\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{\cos t}{ew} \cdot eh}{\tan t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan t \cdot ew\\ t_2 := \left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{t\_1}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{t\_1}\right)}^{2}}}\right|\\ t_3 := eh \cdot \cos t\\ \mathbf{if}\;t \leq -9 \cdot 10^{+28}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\left|t\_3 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) ew))
        (t_2
         (fabs
          (/
           (fma (/ (* (cos t) eh) t_1) eh (* (sin t) ew))
           (sqrt (+ 1.0 (pow (/ eh t_1) 2.0))))))
        (t_3 (* eh (cos t))))
   (if (<= t -9e+28)
     t_2
     (if (<= t 8.2e+45)
       (fabs
        (+
         (* t_3 (sin (atan (/ eh (* ew t)))))
         (* (* ew t) (cos (atan (/ (/ eh ew) t))))))
       (if (<= t 1.45e+234) t_2 (fabs t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * ew;
	double t_2 = fabs((fma(((cos(t) * eh) / t_1), eh, (sin(t) * ew)) / sqrt((1.0 + pow((eh / t_1), 2.0)))));
	double t_3 = eh * cos(t);
	double tmp;
	if (t <= -9e+28) {
		tmp = t_2;
	} else if (t <= 8.2e+45) {
		tmp = fabs(((t_3 * sin(atan((eh / (ew * t))))) + ((ew * t) * cos(atan(((eh / ew) / t))))));
	} else if (t <= 1.45e+234) {
		tmp = t_2;
	} else {
		tmp = fabs(t_3);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * ew)
	t_2 = abs(Float64(fma(Float64(Float64(cos(t) * eh) / t_1), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(eh / t_1) ^ 2.0)))))
	t_3 = Float64(eh * cos(t))
	tmp = 0.0
	if (t <= -9e+28)
		tmp = t_2;
	elseif (t <= 8.2e+45)
		tmp = abs(Float64(Float64(t_3 * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * t) * cos(atan(Float64(Float64(eh / ew) / t))))));
	elseif (t <= 1.45e+234)
		tmp = t_2;
	else
		tmp = abs(t_3);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(eh / t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9e+28], t$95$2, If[LessEqual[t, 8.2e+45], N[Abs[N[(N[(t$95$3 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.45e+234], t$95$2, N[Abs[t$95$3], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.9999999999999994e28 or 8.20000000000000025e45 < t < 1.44999999999999993e234

   1. Initial program 99.6%

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

   4. Applied rewrites 77.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lift-tan.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lower-*.f64 73.1

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

   8. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-*.f64 N/A

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

      5. lower-/.f64 73.2

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

   10. Applied rewrites 73.2%

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

   if -8.9999999999999994e28 < t < 8.20000000000000025e45

   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. Add Preprocessing
   3. 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 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 95.7%

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

   if 1.44999999999999993e234 < t

   1. Initial program 99.4%

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

   4. Applied rewrites 48.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 38.4%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 99.4%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+28}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{\tan t \cdot ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{\tan t \cdot ew}\right)}^{2}}}\right|\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+45}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right|\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+234}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{\tan t \cdot ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{\tan t \cdot ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.2 \cdot 10^{+45} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.2e+45) (not (<= eh 3.1e-129)))
   (fabs (* eh (cos t)))
   (fabs
    (/
     (fma (/ (/ eh ew) t) eh (* (sin t) ew))
     (cosh (asinh (/ (/ eh (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.2e+45) || !(eh <= 3.1e-129)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.2e+45) || !(eh <= 3.1e-129))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.2e+45], N[Not[LessEqual[eh, 3.1e-129]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $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]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.2000000000000003e45 or 3.1e-129 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 37.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 32.5%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 83.9%

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

   if -3.2000000000000003e45 < eh < 3.1e-129

   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. Add Preprocessing

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 85.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.2 \cdot 10^{+45} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+44} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.3e+44) (not (<= eh 3.1e-129)))
   (fabs (* eh (cos t)))
   (fabs
    (/
     (fma (/ (/ eh ew) t) eh (* (sin t) ew))
     (sqrt (+ 1.0 (pow (/ (/ eh (tan t)) ew) 2.0)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.3e+44) || !(eh <= 3.1e-129)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma(((eh / ew) / t), eh, (sin(t) * ew)) / sqrt((1.0 + pow(((eh / tan(t)) / ew), 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.3e+44) || !(eh <= 3.1e-129))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(Float64(Float64(eh / ew) / t), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(Float64(eh / tan(t)) / ew) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.3e+44], N[Not[LessEqual[eh, 3.1e-129]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.30000000000000013e44 or 3.1e-129 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 37.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 32.5%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 83.9%

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

   if -3.30000000000000013e44 < eh < 3.1e-129

   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. Add Preprocessing

   4. Applied rewrites 94.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 86.8%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 80.5%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+44} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew}}{t}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{+43} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -4.1e+43) (not (<= eh 3.1e-129)))
   (fabs (* eh (cos t)))
   (fabs (* ew (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.1e+43) || !(eh <= 3.1e-129)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-4.1d+43)) .or. (.not. (eh <= 3.1d-129))) then
        tmp = abs((eh * cos(t)))
    else
        tmp = abs((ew * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.1e+43) || !(eh <= 3.1e-129)) {
		tmp = Math.abs((eh * Math.cos(t)));
	} else {
		tmp = Math.abs((ew * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -4.1e+43) or not (eh <= 3.1e-129):
		tmp = math.fabs((eh * math.cos(t)))
	else:
		tmp = math.fabs((ew * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -4.1e+43) || !(eh <= 3.1e-129))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -4.1e+43) || ~((eh <= 3.1e-129)))
		tmp = abs((eh * cos(t)));
	else
		tmp = abs((ew * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -4.1e+43], N[Not[LessEqual[eh, 3.1e-129]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.1e43 or 3.1e-129 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 37.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 32.5%

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

   7. Taylor expanded in eh around inf

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

   9. Applied rewrites 83.9%

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

   if -4.1e43 < eh < 3.1e-129

   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. Add Preprocessing

   4. Applied rewrites 94.8%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 71.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.1 \cdot 10^{+43} \lor \neg \left(eh \leq 3.1 \cdot 10^{-129}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-42} \lor \neg \left(t \leq 5.5 \cdot 10^{-34}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -2e-42) (not (<= t 5.5e-34))) (fabs (* ew (sin t))) (fabs eh)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2e-42) || !(t <= 5.5e-34)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2d-42)) .or. (.not. (t <= 5.5d-34))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2e-42) || !(t <= 5.5e-34)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -2e-42) or not (t <= 5.5e-34):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -2e-42) || !(t <= 5.5e-34))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -2e-42) || ~((t <= 5.5e-34)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -2e-42], N[Not[LessEqual[t, 5.5e-34]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.00000000000000008e-42 or 5.50000000000000014e-34 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 74.3%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 54.3%

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

   if -2.00000000000000008e-42 < t < 5.50000000000000014e-34

   1. Initial program 100.0%

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

   4. Applied rewrites 49.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 43.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 74.8%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-42} \lor \neg \left(t \leq 5.5 \cdot 10^{-34}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 44.8% accurate, 43.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.5 \cdot 10^{-70} \lor \neg \left(eh \leq 6.9 \cdot 10^{-130}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.5e-70) (not (<= eh 6.9e-130))) (fabs eh) (fabs (* ew t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.5e-70) || !(eh <= 6.9e-130)) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-2.5d-70)) .or. (.not. (eh <= 6.9d-130))) then
        tmp = abs(eh)
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.5e-70) || !(eh <= 6.9e-130)) {
		tmp = Math.abs(eh);
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.5e-70) or not (eh <= 6.9e-130):
		tmp = math.fabs(eh)
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.5e-70) || !(eh <= 6.9e-130))
		tmp = abs(eh);
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.5e-70) || ~((eh <= 6.9e-130)))
		tmp = abs(eh);
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.5e-70], N[Not[LessEqual[eh, 6.9e-130]], $MachinePrecision]], N[Abs[eh], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.4999999999999999e-70 or 6.90000000000000035e-130 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 42.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 37.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 54.5%

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

   if -2.4999999999999999e-70 < eh < 6.90000000000000035e-130

   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. Add Preprocessing

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 75.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|ew \cdot t\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 40.8%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.5 \cdot 10^{-70} \lor \neg \left(eh \leq 6.9 \cdot 10^{-130}\right):\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.2% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $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. Add Preprocessing

4. Applied rewrites 62.4%

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

   1. lift-cosh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. cosh-asinh N/A

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

   4. +-commutative N/A

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

   5. rem-square-sqrt N/A

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

   6. +-commutative N/A

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

   7. cosh-asinh N/A

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

   8. lift-asinh.f64 N/A

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

   9. lift-cosh.f64 N/A

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

   10. +-commutative N/A

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

   11. cosh-asinh N/A

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

   12. lift-asinh.f64 N/A

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

   13. lift-cosh.f64 N/A

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

6. Applied rewrites 55.8%

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

7. Taylor expanded in t around 0

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

9. Applied rewrites 43.2%

   \[\leadsto \left|\color{blue}{eh}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025026 
(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))))))))