Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 21.8s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

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

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

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

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

   7. lift-*.f64 N/A

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

   8. *-lft-identity N/A

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

   9. remove-double-neg N/A

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

   10. associate-*l* N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := \frac{\frac{eh}{\tan t}}{ew}\\ t_3 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_4 := \left(ew \cdot \sin t\right) \cdot \cos t\_3\\ \mathbf{if}\;\left|t\_1 \cdot \sin t\_3 + t\_4\right| \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_2, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_2}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) + t\_4\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (/ (/ eh (tan t)) ew))
        (t_3 (atan (/ (/ eh ew) (tan t))))
        (t_4 (* (* ew (sin t)) (cos t_3))))
   (if (<= (fabs (+ (* t_1 (sin t_3)) t_4)) 5e+65)
     (fabs (/ (fma (* (cos t) t_2) eh (* (sin t) ew)) (cosh (asinh t_2))))
     (fabs (+ (* t_1 (sin (atan (/ (/ eh ew) t)))) t_4)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = (eh / tan(t)) / ew;
	double t_3 = atan(((eh / ew) / tan(t)));
	double t_4 = (ew * sin(t)) * cos(t_3);
	double tmp;
	if (fabs(((t_1 * sin(t_3)) + t_4)) <= 5e+65) {
		tmp = fabs((fma((cos(t) * t_2), eh, (sin(t) * ew)) / cosh(asinh(t_2))));
	} else {
		tmp = fabs(((t_1 * sin(atan(((eh / ew) / t)))) + t_4));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(Float64(eh / tan(t)) / ew)
	t_3 = atan(Float64(Float64(eh / ew) / tan(t)))
	t_4 = Float64(Float64(ew * sin(t)) * cos(t_3))
	tmp = 0.0
	if (abs(Float64(Float64(t_1 * sin(t_3)) + t_4)) <= 5e+65)
		tmp = abs(Float64(fma(Float64(cos(t) * t_2), eh, Float64(sin(t) * ew)) / cosh(asinh(t_2))));
	else
		tmp = abs(Float64(Float64(t_1 * sin(atan(Float64(Float64(eh / ew) / t)))) + t_4));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision], 5e+65], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$2), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$4), $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))))))) < 4.99999999999999973e65

   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 90.3%

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

   if 4.99999999999999973e65 < (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. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 92.9

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

   5. Applied rewrites 92.9%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 5 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -3.8 \cdot 10^{+107} \lor \neg \left(eh \leq 3.3 \cdot 10^{+106}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)) (t_2 (* eh (cos t))))
   (if (or (<= eh -3.8e+107) (not (<= eh 3.3e+106)))
     (fabs (* t_2 (sin (atan (/ t_2 (* ew (sin t)))))))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double t_2 = eh * cos(t);
	double tmp;
	if ((eh <= -3.8e+107) || !(eh <= 3.3e+106)) {
		tmp = fabs((t_2 * sin(atan((t_2 / (ew * sin(t)))))));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -3.8e+107) || !(eh <= 3.3e+106))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -3.8e+107], N[Not[LessEqual[eh, 3.3e+106]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.7999999999999998e107 or 3.30000000000000008e106 < 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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 91.8

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

   8. Applied rewrites 91.8%

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

   if -3.7999999999999998e107 < eh < 3.30000000000000008e106

   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 89.8%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+107} \lor \neg \left(eh \leq 3.3 \cdot 10^{+106}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -3.8 \cdot 10^{+107} \lor \neg \left(eh \leq 3.3 \cdot 10^{+106}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{\frac{eh}{ew} \cdot eh}{\tan t} \cdot \cos t\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (or (<= eh -3.8e+107) (not (<= eh 3.3e+106)))
     (fabs (* t_1 (sin (atan (/ t_1 (* ew (sin t)))))))
     (fabs
      (/
       (fma (sin t) ew (* (/ (* (/ eh ew) eh) (tan t)) (cos t)))
       (cosh (asinh (/ (/ eh (tan t)) ew))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if ((eh <= -3.8e+107) || !(eh <= 3.3e+106)) {
		tmp = fabs((t_1 * sin(atan((t_1 / (ew * sin(t)))))));
	} else {
		tmp = fabs((fma(sin(t), ew, ((((eh / ew) * eh) / tan(t)) * cos(t))) / cosh(asinh(((eh / tan(t)) / ew)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -3.8e+107) || !(eh <= 3.3e+106))
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(sin(t), ew, Float64(Float64(Float64(Float64(eh / ew) * eh) / tan(t)) * cos(t))) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -3.8e+107], N[Not[LessEqual[eh, 3.3e+106]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew + N[(N[(N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $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 < -3.7999999999999998e107 or 3.30000000000000008e106 < 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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 91.8

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

   8. Applied rewrites 91.8%

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

   if -3.7999999999999998e107 < eh < 3.30000000000000008e106

   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

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. cos-atan N/A

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

      7. associate-*l/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. sin-atan N/A

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

   4. Applied rewrites 86.4%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.8 \cdot 10^{+107} \lor \neg \left(eh \leq 3.3 \cdot 10^{+106}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, ew, \frac{\frac{eh}{ew} \cdot eh}{\tan t} \cdot \cos t\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ t_2 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -6000000 \lor \neg \left(eh \leq 2.7 \cdot 10^{-36}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)) (t_2 (* eh (cos t))))
   (if (or (<= eh -6000000.0) (not (<= eh 2.7e-36)))
     (fabs (* t_2 (sin (atan (/ t_2 (* ew (sin t)))))))
     (fabs
      (/
       (fma (* (cos t) t_1) eh (* (sin t) ew))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double t_2 = eh * cos(t);
	double tmp;
	if ((eh <= -6000000.0) || !(eh <= 2.7e-36)) {
		tmp = fabs((t_2 * sin(atan((t_2 / (ew * sin(t)))))));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	t_2 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -6000000.0) || !(eh <= 2.7e-36))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -6000000.0], N[Not[LessEqual[eh, 2.7e-36]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6e6 or 2.70000000000000007e-36 < 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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 50.5

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

   5. Applied rewrites 50.5%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 83.8

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

   8. Applied rewrites 83.8%

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

   if -6e6 < eh < 2.70000000000000007e-36

   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 95.0%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right| \]
   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 91.2%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6000000 \lor \neg \left(eh \leq 2.7 \cdot 10^{-36}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -7 \cdot 10^{-81} \lor \neg \left(eh \leq 8 \cdot 10^{-36}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (or (<= eh -7e-81) (not (<= eh 8e-36)))
     (fabs (* t_1 (sin (atan (/ t_1 (* ew (sin 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 t_1 = eh * cos(t);
	double tmp;
	if ((eh <= -7e-81) || !(eh <= 8e-36)) {
		tmp = fabs((t_1 * sin(atan((t_1 / (ew * sin(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)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -7e-81) || !(eh <= 8e-36))
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / Float64(ew * sin(t)))))));
	else
		tmp = abs(Float64(fma(Float64(eh / Float64(ew * t)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -7e-81], N[Not[LessEqual[eh, 8e-36]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.99999999999999973e-81 or 7.9999999999999995e-36 < 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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 82.4

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

   8. Applied rewrites 82.4%

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

   if -6.99999999999999973e-81 < eh < 7.9999999999999995e-36

   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 95.9%

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

   5. 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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 83.5

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

   7. Applied rewrites 83.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7 \cdot 10^{-81} \lor \neg \left(eh \leq 8 \cdot 10^{-36}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right)\\ \mathbf{if}\;eh \leq -7 \cdot 10^{-81}:\\ \;\;\;\;\left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot t\_1\right) \cdot ew\right|\\ \mathbf{elif}\;eh \leq 8 \cdot 10^{-36}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh}{ew \cdot t}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \mathbf{elif}\;eh \leq 4.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{eh}{ew} \cdot -0.3333333333333333, t \cdot t, \frac{eh}{ew}\right)}{t}\right)}{ew} \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (tanh
          (asinh (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) t) ew)))))
   (if (<= eh -7e-81)
     (fabs (* (* (* (/ (cos t) ew) eh) t_1) ew))
     (if (<= eh 8e-36)
       (fabs
        (/
         (fma (/ eh (* ew t)) eh (* (sin t) ew))
         (cosh (asinh (/ (/ eh (tan t)) ew)))))
       (if (<= eh 4.5e+123)
         (fabs
          (*
           (/
            (*
             (* eh (cos t))
             (sin
              (atan
               (/
                (fma (* (/ eh ew) -0.3333333333333333) (* t t) (/ eh ew))
                t))))
            ew)
           ew))
         (fabs (* t_1 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tanh(asinh(((fma(((t * t) * eh), -0.3333333333333333, eh) / t) / ew)));
	double tmp;
	if (eh <= -7e-81) {
		tmp = fabs(((((cos(t) / ew) * eh) * t_1) * ew));
	} else if (eh <= 8e-36) {
		tmp = fabs((fma((eh / (ew * t)), eh, (sin(t) * ew)) / cosh(asinh(((eh / tan(t)) / ew)))));
	} else if (eh <= 4.5e+123) {
		tmp = fabs(((((eh * cos(t)) * sin(atan((fma(((eh / ew) * -0.3333333333333333), (t * t), (eh / ew)) / t)))) / ew) * ew));
	} else {
		tmp = fabs((t_1 * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = tanh(asinh(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / t) / ew)))
	tmp = 0.0
	if (eh <= -7e-81)
		tmp = abs(Float64(Float64(Float64(Float64(cos(t) / ew) * eh) * t_1) * ew));
	elseif (eh <= 8e-36)
		tmp = abs(Float64(fma(Float64(eh / Float64(ew * t)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(Float64(eh / tan(t)) / ew)))));
	elseif (eh <= 4.5e+123)
		tmp = abs(Float64(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(fma(Float64(Float64(eh / ew) * -0.3333333333333333), Float64(t * t), Float64(eh / ew)) / t)))) / ew) * ew));
	else
		tmp = abs(Float64(t_1 * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Tanh[N[ArcSinh[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7e-81], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 8e-36], N[Abs[N[(N[(N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.5e+123], N[Abs[N[(N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -6.99999999999999973e-81

   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. 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 66.6%

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

   10. Applied rewrites 66.5%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\frac{eh + \frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{t}}{ew}\right)\right) \cdot ew\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 66.6%

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

   if -6.99999999999999973e-81 < eh < 7.9999999999999995e-36

   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 95.9%

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

   5. 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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 83.5

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

   7. Applied rewrites 83.5%

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

   if 7.9999999999999995e-36 < eh < 4.49999999999999983e123

   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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 76.5%

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

   if 4.49999999999999983e123 < eh

   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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.0%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right) \cdot eh\right| \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 63.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right)\\ \mathbf{if}\;eh \leq -8.8 \cdot 10^{-113}:\\ \;\;\;\;\left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot t\_1\right) \cdot ew\right|\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 4.5 \cdot 10^{+123}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{eh}{ew} \cdot -0.3333333333333333, t \cdot t, \frac{eh}{ew}\right)}{t}\right)}{ew} \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (tanh
          (asinh (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) t) ew)))))
   (if (<= eh -8.8e-113)
     (fabs (* (* (* (/ (cos t) ew) eh) t_1) ew))
     (if (<= eh 2e-36)
       (fabs (* ew (sin t)))
       (if (<= eh 4.5e+123)
         (fabs
          (*
           (/
            (*
             (* eh (cos t))
             (sin
              (atan
               (/
                (fma (* (/ eh ew) -0.3333333333333333) (* t t) (/ eh ew))
                t))))
            ew)
           ew))
         (fabs (* t_1 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tanh(asinh(((fma(((t * t) * eh), -0.3333333333333333, eh) / t) / ew)));
	double tmp;
	if (eh <= -8.8e-113) {
		tmp = fabs(((((cos(t) / ew) * eh) * t_1) * ew));
	} else if (eh <= 2e-36) {
		tmp = fabs((ew * sin(t)));
	} else if (eh <= 4.5e+123) {
		tmp = fabs(((((eh * cos(t)) * sin(atan((fma(((eh / ew) * -0.3333333333333333), (t * t), (eh / ew)) / t)))) / ew) * ew));
	} else {
		tmp = fabs((t_1 * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = tanh(asinh(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / t) / ew)))
	tmp = 0.0
	if (eh <= -8.8e-113)
		tmp = abs(Float64(Float64(Float64(Float64(cos(t) / ew) * eh) * t_1) * ew));
	elseif (eh <= 2e-36)
		tmp = abs(Float64(ew * sin(t)));
	elseif (eh <= 4.5e+123)
		tmp = abs(Float64(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(fma(Float64(Float64(eh / ew) * -0.3333333333333333), Float64(t * t), Float64(eh / ew)) / t)))) / ew) * ew));
	else
		tmp = abs(Float64(t_1 * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Tanh[N[ArcSinh[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8.8e-113], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 2e-36], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.5e+123], N[Abs[N[(N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -8.80000000000000016e-113

   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. 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 66.0%

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

   10. Applied rewrites 65.9%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\frac{eh + \frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{t}}{ew}\right)\right) \cdot ew\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 66.2%

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

   if -8.80000000000000016e-113 < eh < 1.9999999999999999e-36

   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 96.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 1.9999999999999999e-36 < eh < 4.49999999999999983e123

   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

   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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 76.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 76.5%

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

   if 4.49999999999999983e123 < eh

   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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.0%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right) \cdot eh\right| \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right)\\ t_2 := \left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot t\_1\right) \cdot ew\right|\\ \mathbf{if}\;eh \leq -8.8 \cdot 10^{-113}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{-36}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 4.5 \cdot 10^{+123}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (tanh
          (asinh (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) t) ew))))
        (t_2 (fabs (* (* (* (/ (cos t) ew) eh) t_1) ew))))
   (if (<= eh -8.8e-113)
     t_2
     (if (<= eh 2e-36)
       (fabs (* ew (sin t)))
       (if (<= eh 4.5e+123) t_2 (fabs (* t_1 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tanh(asinh(((fma(((t * t) * eh), -0.3333333333333333, eh) / t) / ew)));
	double t_2 = fabs(((((cos(t) / ew) * eh) * t_1) * ew));
	double tmp;
	if (eh <= -8.8e-113) {
		tmp = t_2;
	} else if (eh <= 2e-36) {
		tmp = fabs((ew * sin(t)));
	} else if (eh <= 4.5e+123) {
		tmp = t_2;
	} else {
		tmp = fabs((t_1 * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = tanh(asinh(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / t) / ew)))
	t_2 = abs(Float64(Float64(Float64(Float64(cos(t) / ew) * eh) * t_1) * ew))
	tmp = 0.0
	if (eh <= -8.8e-113)
		tmp = t_2;
	elseif (eh <= 2e-36)
		tmp = abs(Float64(ew * sin(t)));
	elseif (eh <= 4.5e+123)
		tmp = t_2;
	else
		tmp = abs(Float64(t_1 * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Tanh[N[ArcSinh[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8.8e-113], t$95$2, If[LessEqual[eh, 2e-36], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.5e+123], t$95$2, N[Abs[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.80000000000000016e-113 or 1.9999999999999999e-36 < eh < 4.49999999999999983e123

   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. 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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 69.5%

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

   10. Applied rewrites 69.4%

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

   11. Taylor expanded in t around 0

       \[\leadsto \left|\left(\left(\frac{\cos t}{ew} \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\frac{eh + \frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{t}}{ew}\right)\right) \cdot ew\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 69.7%

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

   if -8.80000000000000016e-113 < eh < 1.9999999999999999e-36

   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 96.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 75.3

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

   7. Applied rewrites 75.3%

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

   if 4.49999999999999983e123 < eh

   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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 69.9

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 69.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 70.0%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{t}}{ew}\right) \cdot eh\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 61.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-63} \lor \neg \left(t \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.3e-63) (not (<= t 1.9e-40)))
   (fabs (/ (fma (* (cos t) (/ (/ eh (tan t)) ew)) eh (* (sin t) ew)) 1.0))
   (fabs (* (tanh (asinh (/ (/ eh t) ew))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.3e-63) || !(t <= 1.9e-40)) {
		tmp = fabs((fma((cos(t) * ((eh / tan(t)) / ew)), eh, (sin(t) * ew)) / 1.0));
	} else {
		tmp = fabs((tanh(asinh(((eh / t) / ew))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.3e-63) || !(t <= 1.9e-40))
		tmp = abs(Float64(fma(Float64(cos(t) * Float64(Float64(eh / tan(t)) / ew)), eh, Float64(sin(t) * ew)) / 1.0));
	else
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / t) / ew))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.3e-63], N[Not[LessEqual[t, 1.9e-40]], $MachinePrecision]], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3000000000000001e-63 or 1.8999999999999999e-40 < 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 81.2%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 52.1%

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

   if -1.3000000000000001e-63 < t < 1.8999999999999999e-40

   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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.5%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-63} \lor \neg \left(t \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 60.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-63} \lor \neg \left(t \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.3e-63) (not (<= t 1.9e-40)))
   (fabs (* ew (sin t)))
   (fabs (* (tanh (asinh (/ (/ eh t) ew))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.3e-63) || !(t <= 1.9e-40)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((tanh(asinh(((eh / t) / ew))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -1.3e-63) or not (t <= 1.9e-40):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((math.tanh(math.asinh(((eh / t) / ew))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.3e-63) || !(t <= 1.9e-40))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(tanh(asinh(Float64(Float64(eh / t) / ew))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -1.3e-63) || ~((t <= 1.9e-40)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((tanh(asinh(((eh / t) / ew))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.3e-63], N[Not[LessEqual[t, 1.9e-40]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3000000000000001e-63 or 1.8999999999999999e-40 < 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 81.2%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 51.6

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

   7. Applied rewrites 51.6%

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

   if -1.3000000000000001e-63 < t < 1.8999999999999999e-40

   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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 78.5%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-63} \lor \neg \left(t \leq 1.9 \cdot 10^{-40}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\frac{eh}{t}}{ew}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.0% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.6

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

7. Applied rewrites 41.6%

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

Alternative 13: 19.0% accurate, 66.9× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \left(1 \cdot t\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew (* 1.0 t))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * (1.0 * t)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * (1.0d0 * t)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * (1.0 * t)));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * (1.0 * t)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * Float64(1.0 * t)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * (1.0 * t)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(1.0 * t), $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

4. Applied rewrites 69.1%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.6

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

7. Applied rewrites 41.6%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 16.5%

    \[\leadsto \left|ew \cdot \left(\mathsf{fma}\left(-0.16666666666666666, t \cdot t, 1\right) \cdot \color{blue}{t}\right)\right| \]

11. Taylor expanded in t around 0

    \[\leadsto \left|ew \cdot \left(1 \cdot t\right)\right| \]
12. Step-by-step derivation

13. Applied rewrites 17.1%

    \[\leadsto \left|ew \cdot \left(1 \cdot t\right)\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024353 
(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))))))))