Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.6s

Alternatives: 15

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (* (/ (tan t) ew) eh))))
   (fabs
    (fma (* (cos t) ew) (/ 1.0 (cosh t_1)) (* (* (tanh t_1) eh) (sin t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh(((tan(t) / ew) * eh));
	return fabs(fma((cos(t) * ew), (1.0 / cosh(t_1)), ((tanh(t_1) * eh) * sin(t))));
}

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(Float64(tan(t) / ew) * eh))
	return abs(fma(Float64(cos(t) * ew), Float64(1.0 / cosh(t_1)), Float64(Float64(tanh(t_1) * eh) * sin(t))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Tanh[t$95$1], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in ew around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 92.3%

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

6. Applied rewrites 92.3%

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

8. Applied rewrites 99.7%

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

10. Applied rewrites 99.8%

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

Alternative 3: 92.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := \cos t \cdot ew\\ t_4 := eh \cdot \sin t\\ t_5 := t\_1 \cdot \cos t\_2 - t\_4 \cdot \sin t\_2\\ t_6 := \tan^{-1} \left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)\\ t_7 := \frac{\tan t}{ew}\\ t_8 := t\_7 \cdot eh\\ t_9 := \sinh^{-1} \left(t\_7 \cdot \left(-eh\right)\right)\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+164}:\\ \;\;\;\;\left|t\_1 \cdot \cos t\_6 - t\_4 \cdot \sin t\_6\right|\\ \mathbf{elif}\;t\_5 \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\frac{\left|\left({t\_8}^{2} - -1\right) \cdot t\_3\right|}{\cosh \sinh^{-1} t\_8}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_3}{\cosh t\_9} - \tanh t\_9 \cdot \left(\sin t \cdot eh\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (* (cos t) ew))
        (t_4 (* eh (sin t)))
        (t_5 (- (* t_1 (cos t_2)) (* t_4 (sin t_2))))
        (t_6 (atan (/ (* -1.0 (* eh t)) ew)))
        (t_7 (/ (tan t) ew))
        (t_8 (* t_7 eh))
        (t_9 (asinh (* t_7 (- eh)))))
   (if (<= t_5 -1e+164)
     (fabs (- (* t_1 (cos t_6)) (* t_4 (sin t_6))))
     (if (<= t_5 -1e-279)
       (/ (fabs (* (- (pow t_8 2.0) -1.0) t_3)) (cosh (asinh t_8)))
       (- (/ t_3 (cosh t_9)) (* (tanh t_9) (* (sin t) eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = cos(t) * ew;
	double t_4 = eh * sin(t);
	double t_5 = (t_1 * cos(t_2)) - (t_4 * sin(t_2));
	double t_6 = atan(((-1.0 * (eh * t)) / ew));
	double t_7 = tan(t) / ew;
	double t_8 = t_7 * eh;
	double t_9 = asinh((t_7 * -eh));
	double tmp;
	if (t_5 <= -1e+164) {
		tmp = fabs(((t_1 * cos(t_6)) - (t_4 * sin(t_6))));
	} else if (t_5 <= -1e-279) {
		tmp = fabs(((pow(t_8, 2.0) - -1.0) * t_3)) / cosh(asinh(t_8));
	} else {
		tmp = (t_3 / cosh(t_9)) - (tanh(t_9) * (sin(t) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	t_3 = math.cos(t) * ew
	t_4 = eh * math.sin(t)
	t_5 = (t_1 * math.cos(t_2)) - (t_4 * math.sin(t_2))
	t_6 = math.atan(((-1.0 * (eh * t)) / ew))
	t_7 = math.tan(t) / ew
	t_8 = t_7 * eh
	t_9 = math.asinh((t_7 * -eh))
	tmp = 0
	if t_5 <= -1e+164:
		tmp = math.fabs(((t_1 * math.cos(t_6)) - (t_4 * math.sin(t_6))))
	elif t_5 <= -1e-279:
		tmp = math.fabs(((math.pow(t_8, 2.0) - -1.0) * t_3)) / math.cosh(math.asinh(t_8))
	else:
		tmp = (t_3 / math.cosh(t_9)) - (math.tanh(t_9) * (math.sin(t) * eh))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(cos(t) * ew)
	t_4 = Float64(eh * sin(t))
	t_5 = Float64(Float64(t_1 * cos(t_2)) - Float64(t_4 * sin(t_2)))
	t_6 = atan(Float64(Float64(-1.0 * Float64(eh * t)) / ew))
	t_7 = Float64(tan(t) / ew)
	t_8 = Float64(t_7 * eh)
	t_9 = asinh(Float64(t_7 * Float64(-eh)))
	tmp = 0.0
	if (t_5 <= -1e+164)
		tmp = abs(Float64(Float64(t_1 * cos(t_6)) - Float64(t_4 * sin(t_6))));
	elseif (t_5 <= -1e-279)
		tmp = Float64(abs(Float64(Float64((t_8 ^ 2.0) - -1.0) * t_3)) / cosh(asinh(t_8)));
	else
		tmp = Float64(Float64(t_3 / cosh(t_9)) - Float64(tanh(t_9) * Float64(sin(t) * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((-eh * tan(t)) / ew));
	t_3 = cos(t) * ew;
	t_4 = eh * sin(t);
	t_5 = (t_1 * cos(t_2)) - (t_4 * sin(t_2));
	t_6 = atan(((-1.0 * (eh * t)) / ew));
	t_7 = tan(t) / ew;
	t_8 = t_7 * eh;
	t_9 = asinh((t_7 * -eh));
	tmp = 0.0;
	if (t_5 <= -1e+164)
		tmp = abs(((t_1 * cos(t_6)) - (t_4 * sin(t_6))));
	elseif (t_5 <= -1e-279)
		tmp = abs((((t_8 ^ 2.0) - -1.0) * t_3)) / cosh(asinh(t_8));
	else
		tmp = (t_3 / cosh(t_9)) - (tanh(t_9) * (sin(t) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$4 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[ArcTan[N[(N[(-1.0 * N[(eh * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$7 * eh), $MachinePrecision]}, Block[{t$95$9 = N[ArcSinh[N[(t$95$7 * (-eh)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -1e+164], N[Abs[N[(N[(t$95$1 * N[Cos[t$95$6], $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[Sin[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, -1e-279], N[(N[Abs[N[(N[(N[Power[t$95$8, 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$3), $MachinePrecision]], $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$8], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 / N[Cosh[t$95$9], $MachinePrecision]), $MachinePrecision] - N[(N[Tanh[t$95$9], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1e164

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if -1e164 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.00000000000000006e-279

   1. Initial program 99.8%

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

   3. Applied rewrites 82.5%

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

   5. Applied rewrites 77.4%

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

   if -1.00000000000000006e-279 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.5

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

   3. Applied rewrites 50.5%

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

Alternative 4: 91.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := t \cdot \frac{eh}{ew}\\ \mathbf{if}\;ew \leq 2.1 \cdot 10^{-204}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{\mathsf{fma}\left(t\_2 \cdot t, \frac{eh}{ew}, 1\right)}}, ew, \left(\tanh \sinh^{-1} t\_2 \cdot t\_1\right) \cdot 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \left(\frac{\tanh \sinh^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot t\_1}{ew} + \frac{\cos t}{1}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (* t (/ eh ew))))
   (if (<= ew 2.1e-204)
     (fabs
      (fma
       (/ (cos t) (sqrt (fma (* t_2 t) (/ eh ew) 1.0)))
       ew
       (* (* (tanh (asinh t_2)) t_1) 1.0)))
     (fabs
      (*
       ew
       (+
        (/ (* (tanh (asinh (* (/ eh ew) (tan t)))) t_1) ew)
        (/ (cos t) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = t * (eh / ew);
	double tmp;
	if (ew <= 2.1e-204) {
		tmp = fabs(fma((cos(t) / sqrt(fma((t_2 * t), (eh / ew), 1.0))), ew, ((tanh(asinh(t_2)) * t_1) * 1.0)));
	} else {
		tmp = fabs((ew * (((tanh(asinh(((eh / ew) * tan(t)))) * t_1) / ew) + (cos(t) / 1.0))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(t * Float64(eh / ew))
	tmp = 0.0
	if (ew <= 2.1e-204)
		tmp = abs(fma(Float64(cos(t) / sqrt(fma(Float64(t_2 * t), Float64(eh / ew), 1.0))), ew, Float64(Float64(tanh(asinh(t_2)) * t_1) * 1.0)));
	else
		tmp = abs(Float64(ew * Float64(Float64(Float64(tanh(asinh(Float64(Float64(eh / ew) * tan(t)))) * t_1) / ew) + Float64(cos(t) / 1.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, 2.1e-204], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(N[(t$95$2 * t), $MachinePrecision] * N[(eh / ew), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[(N[(N[(N[Tanh[N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] / ew), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < 2.10000000000000009e-204

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.0%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

   14. Applied rewrites 89.5%

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

   if 2.10000000000000009e-204 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

   9. Applied rewrites 91.1%

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

Alternative 5: 89.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ t_4 := t \cdot \frac{eh}{ew}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{\mathsf{fma}\left(t\_4 \cdot t, \frac{eh}{ew}, 1\right)}}, ew, \left(\tanh \sinh^{-1} t\_4 \cdot \left(\sin t \cdot eh\right)\right) \cdot 1\right)\right|\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-279}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(eh, -\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \sin t, \frac{\cos t \cdot ew}{1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))))
        (t_4 (* t (/ eh ew))))
   (if (<= t_3 -5e+50)
     (fabs
      (fma
       (/ (cos t) (sqrt (fma (* t_4 t) (/ eh ew) 1.0)))
       ew
       (* (* (tanh (asinh t_4)) (* (sin t) eh)) 1.0)))
     (if (<= t_3 -1e-279)
       (fabs t_1)
       (fma
        eh
        (- (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) (sin t)))
        (/ (* (cos t) ew) 1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = (t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double t_4 = t * (eh / ew);
	double tmp;
	if (t_3 <= -5e+50) {
		tmp = fabs(fma((cos(t) / sqrt(fma((t_4 * t), (eh / ew), 1.0))), ew, ((tanh(asinh(t_4)) * (sin(t) * eh)) * 1.0)));
	} else if (t_3 <= -1e-279) {
		tmp = fabs(t_1);
	} else {
		tmp = fma(eh, -(tanh(asinh(((tan(t) / ew) * -eh))) * sin(t)), ((cos(t) * ew) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	t_4 = Float64(t * Float64(eh / ew))
	tmp = 0.0
	if (t_3 <= -5e+50)
		tmp = abs(fma(Float64(cos(t) / sqrt(fma(Float64(t_4 * t), Float64(eh / ew), 1.0))), ew, Float64(Float64(tanh(asinh(t_4)) * Float64(sin(t) * eh)) * 1.0)));
	elseif (t_3 <= -1e-279)
		tmp = abs(t_1);
	else
		tmp = fma(eh, Float64(-Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * sin(t))), Float64(Float64(cos(t) * ew) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+50], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 * t), $MachinePrecision] * N[(eh / ew), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Tanh[N[ArcSinh[t$95$4], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, -1e-279], N[Abs[t$95$1], $MachinePrecision], N[(eh * (-N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]) + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.0%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

   14. Applied rewrites 89.5%

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

   if -5e50 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.00000000000000006e-279

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

   if -1.00000000000000006e-279 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.5

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

   3. Applied rewrites 50.5%

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

   4. Taylor expanded in eh around 0

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

   6. Applied rewrites 49.9%

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

Alternative 6: 88.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{eh}{ew}\\ \mathbf{if}\;ew \leq 5 \cdot 10^{+68}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\cos t}{\sqrt{\mathsf{fma}\left(t\_1 \cdot t, \frac{eh}{ew}, 1\right)}}, ew, \left(\tanh \sinh^{-1} t\_1 \cdot \left(\sin t \cdot eh\right)\right) \cdot 1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* t (/ eh ew))))
   (if (<= ew 5e+68)
     (fabs
      (fma
       (/ (cos t) (sqrt (fma (* t_1 t) (/ eh ew) 1.0)))
       ew
       (* (* (tanh (asinh t_1)) (* (sin t) eh)) 1.0)))
     (fabs (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = t * (eh / ew);
	double tmp;
	if (ew <= 5e+68) {
		tmp = fabs(fma((cos(t) / sqrt(fma((t_1 * t), (eh / ew), 1.0))), ew, ((tanh(asinh(t_1)) * (sin(t) * eh)) * 1.0)));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(t * Float64(eh / ew))
	tmp = 0.0
	if (ew <= 5e+68)
		tmp = abs(fma(Float64(cos(t) / sqrt(fma(Float64(t_1 * t), Float64(eh / ew), 1.0))), ew, Float64(Float64(tanh(asinh(t_1)) * Float64(sin(t) * eh)) * 1.0)));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, 5e+68], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] / N[Sqrt[N[(N[(t$95$1 * t), $MachinePrecision] * N[(eh / ew), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < 5.0000000000000004e68

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.0%

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

       1. lift-*.f64 N/A

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

       2. lift-+.f64 N/A

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

       3. +-commutative N/A

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

       4. distribute-rgt-in N/A

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

   14. Applied rewrites 89.5%

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

   if 5.0000000000000004e68 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 7: 69.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 0.8:\\ \;\;\;\;\mathsf{fma}\left(eh, -\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \sin t, ew + {t}^{2} \cdot \left(-0.5 \cdot ew\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 0.8)
   (fma
    eh
    (- (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) (sin t)))
    (+ ew (* (pow t 2.0) (* -0.5 ew))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 0.8) {
		tmp = fma(eh, -(tanh(asinh(((tan(t) / ew) * -eh))) * sin(t)), (ew + (pow(t, 2.0) * (-0.5 * ew))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 0.8)
		tmp = fma(eh, Float64(-Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * sin(t))), Float64(ew + Float64((t ^ 2.0) * Float64(-0.5 * ew))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 0.8], N[(eh * (-N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]) + N[(ew + N[(N[Power[t, 2.0], $MachinePrecision] * N[(-0.5 * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 0.80000000000000004

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.5

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

      5. lift--.f64 N/A

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

      6. sub-flip N/A

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

   3. Applied rewrites 50.5%

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

   4. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-pow.f64 18.0

         \[\leadsto \mathsf{fma}\left(eh, -\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \sin t, ew + {t}^{2} \cdot \left(-0.5 \cdot ew - 0.5 \cdot \frac{{eh}^{2}}{ew}\right)\right) \]

   6. Applied rewrites 18.0%

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

   7. Taylor expanded in eh around 0

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

      1. lower-*.f64 30.0

         \[\leadsto \mathsf{fma}\left(eh, -\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \sin t, ew + {t}^{2} \cdot \left(-0.5 \cdot ew\right)\right) \]

   9. Applied rewrites 30.0%

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

   if 0.80000000000000004 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 8: 65.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)\\ \mathbf{if}\;t \leq 0.8:\\ \;\;\;\;\left|ew \cdot \left(\frac{\tanh t\_1 \cdot \left(\sin t \cdot eh\right)}{ew} + \frac{1 + -0.5 \cdot {t}^{2}}{\cosh t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (* (/ eh ew) t))))
   (if (<= t 0.8)
     (fabs
      (*
       ew
       (+
        (/ (* (tanh t_1) (* (sin t) eh)) ew)
        (/ (+ 1.0 (* -0.5 (pow t 2.0))) (cosh t_1)))))
     (fabs (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh(((eh / ew) * t));
	double tmp;
	if (t <= 0.8) {
		tmp = fabs((ew * (((tanh(t_1) * (sin(t) * eh)) / ew) + ((1.0 + (-0.5 * pow(t, 2.0))) / cosh(t_1)))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.asinh(((eh / ew) * t))
	tmp = 0
	if t <= 0.8:
		tmp = math.fabs((ew * (((math.tanh(t_1) * (math.sin(t) * eh)) / ew) + ((1.0 + (-0.5 * math.pow(t, 2.0))) / math.cosh(t_1)))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(Float64(eh / ew) * t))
	tmp = 0.0
	if (t <= 0.8)
		tmp = abs(Float64(ew * Float64(Float64(Float64(tanh(t_1) * Float64(sin(t) * eh)) / ew) + Float64(Float64(1.0 + Float64(-0.5 * (t ^ 2.0))) / cosh(t_1)))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = asinh(((eh / ew) * t));
	tmp = 0.0;
	if (t <= 0.8)
		tmp = abs((ew * (((tanh(t_1) * (sin(t) * eh)) / ew) + ((1.0 + (-0.5 * (t ^ 2.0))) / cosh(t_1)))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 0.8], N[Abs[N[(ew * N[(N[(N[(N[Tanh[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 0.80000000000000004

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.0%

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

   13. Taylor expanded in t around 0

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

       1. lower-+.f64 N/A

          \[\leadsto \left|ew \cdot \left(\frac{\tanh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\sin t \cdot eh\right)}{ew} + \frac{1 + \frac{-1}{2} \cdot {t}^{2}}{\cosh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)}\right)\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|ew \cdot \left(\frac{\tanh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\sin t \cdot eh\right)}{ew} + \frac{1 + \frac{-1}{2} \cdot {t}^{2}}{\cosh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)}\right)\right| \]

       3. lower-pow.f64 57.5

          \[\leadsto \left|ew \cdot \left(\frac{\tanh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\sin t \cdot eh\right)}{ew} + \frac{1 + -0.5 \cdot {t}^{2}}{\cosh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)}\right)\right| \]

   15. Applied rewrites 57.5%

       \[\leadsto \left|ew \cdot \left(\frac{\tanh \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot \left(\sin t \cdot eh\right)}{ew} + \frac{1 + -0.5 \cdot {t}^{2}}{\cosh \color{blue}{\sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)}}\right)\right| \]

   if 0.80000000000000004 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 9: 62.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sinh^{-1} \left(\frac{eh}{ew} \cdot t\right)\\ \mathbf{if}\;t \leq 3.2 \cdot 10^{+14}:\\ \;\;\;\;\left|ew \cdot \left(\frac{\tanh t\_1 \cdot \left(t \cdot eh\right)}{ew} + \frac{\cos t}{\cosh t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (* (/ eh ew) t))))
   (if (<= t 3.2e+14)
     (fabs (* ew (+ (/ (* (tanh t_1) (* t eh)) ew) (/ (cos t) (cosh t_1)))))
     (fabs (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh(((eh / ew) * t));
	double tmp;
	if (t <= 3.2e+14) {
		tmp = fabs((ew * (((tanh(t_1) * (t * eh)) / ew) + (cos(t) / cosh(t_1)))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.asinh(((eh / ew) * t))
	tmp = 0
	if t <= 3.2e+14:
		tmp = math.fabs((ew * (((math.tanh(t_1) * (t * eh)) / ew) + (math.cos(t) / math.cosh(t_1)))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(Float64(eh / ew) * t))
	tmp = 0.0
	if (t <= 3.2e+14)
		tmp = abs(Float64(ew * Float64(Float64(Float64(tanh(t_1) * Float64(t * eh)) / ew) + Float64(cos(t) / cosh(t_1)))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = asinh(((eh / ew) * t));
	tmp = 0.0;
	if (t <= 3.2e+14)
		tmp = abs((ew * (((tanh(t_1) * (t * eh)) / ew) + (cos(t) / cosh(t_1)))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 3.2e+14], N[Abs[N[(ew * N[(N[(N[(N[Tanh[t$95$1], $MachinePrecision] * N[(t * eh), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 3.2e14

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in t around 0

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

   9. Applied rewrites 91.5%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 82.0%

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

   13. Taylor expanded in t around 0

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

   15. Applied rewrites 64.5%

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

   if 3.2e14 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 10: 52.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{ew} \cdot \left(-eh\right)\\ \mathbf{if}\;t \leq 1.6:\\ \;\;\;\;\frac{\cos t \cdot ew - t\_1 \cdot \left(eh \cdot t\right)}{\cosh \sinh^{-1} t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ t ew) (- eh))))
   (if (<= t 1.6)
     (/ (- (* (cos t) ew) (* t_1 (* eh t))) (cosh (asinh t_1)))
     (fabs (* ew (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (t / ew) * -eh;
	double tmp;
	if (t <= 1.6) {
		tmp = ((cos(t) * ew) - (t_1 * (eh * t))) / cosh(asinh(t_1));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (t / ew) * -eh
	tmp = 0
	if t <= 1.6:
		tmp = ((math.cos(t) * ew) - (t_1 * (eh * t))) / math.cosh(math.asinh(t_1))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(t / ew) * Float64(-eh))
	tmp = 0.0
	if (t <= 1.6)
		tmp = Float64(Float64(Float64(cos(t) * ew) - Float64(t_1 * Float64(eh * t))) / cosh(asinh(t_1)));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (t / ew) * -eh;
	tmp = 0.0;
	if (t <= 1.6)
		tmp = ((cos(t) * ew) - (t_1 * (eh * t))) / cosh(asinh(t_1));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(t / ew), $MachinePrecision] * (-eh)), $MachinePrecision]}, If[LessEqual[t, 1.6], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - N[(t$95$1 * N[(eh * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.6000000000000001

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.5

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 33.1

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

   6. Applied rewrites 33.1%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 32.9

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

   9. Applied rewrites 32.9%

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

   10. Taylor expanded in t around 0

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

       1. lower-*.f64 31.1

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

   12. Applied rewrites 31.1%

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

   if 1.6000000000000001 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

   4. Applied rewrites 92.3%

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

   6. Applied rewrites 92.3%

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

   7. Taylor expanded in eh around 0

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

      1. lower-cos.f64 62.7

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

   9. Applied rewrites 62.7%

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

Alternative 11: 42.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in ew around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

4. Applied rewrites 92.3%

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

6. Applied rewrites 92.3%

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

7. Taylor expanded in eh around 0

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

   1. lower-cos.f64 62.7

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

9. Applied rewrites 62.7%

   \[\leadsto \left|ew \cdot \cos t\right| \]
10. Add Preprocessing

Alternative 12: 42.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq 10^{-224}:\\ \;\;\;\;ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) 1e-224)
     (* ew (/ (/ 1.0 ew) (fabs (/ 1.0 ew))))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 1e-224) {
		tmp = ew * ((1.0 / ew) / fabs((1.0 / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((-eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 1d-224) then
        tmp = ew * ((1.0d0 / ew) / abs((1.0d0 / ew)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= 1e-224) {
		tmp = ew * ((1.0 / ew) / Math.abs((1.0 / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= 1e-224:
		tmp = ew * ((1.0 / ew) / math.fabs((1.0 / ew)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= 1e-224)
		tmp = Float64(ew * Float64(Float64(1.0 / ew) / abs(Float64(1.0 / ew))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 1e-224)
		tmp = ew * ((1.0 / ew) / abs((1.0 / ew)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-224], N[(ew * N[(N[(1.0 / ew), $MachinePrecision] / N[Abs[N[(1.0 / ew), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1e-224

   1. Initial program 99.8%

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

   3. Applied rewrites 82.5%

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

   4. Taylor expanded in t around 0

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   5. Step-by-step derivation

      1. lower-/.f64 42.8

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{ew}}\right|} \]

   6. Applied rewrites 42.8%

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\left|\frac{1}{ew}\right|}} \]

      2. rgt-mult-inverse N/A

         \[\leadsto \frac{\color{blue}{ew \cdot \frac{1}{ew}}}{\left|\frac{1}{ew}\right|} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{ew \cdot \color{blue}{\frac{1}{ew}}}{\left|\frac{1}{ew}\right|} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

      6. lower-/.f64 42.9

         \[\leadsto ew \cdot \color{blue}{\frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

   8. Applied rewrites 42.9%

      \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

   if 1e-224 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

      2. rem-sqrt-square-rev N/A

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 50.5

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

      5. lift--.f64 N/A

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

   3. Applied rewrites 42.3%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 32.3

         \[\leadsto ew \cdot \cos t \]

   6. Applied rewrites 32.3%

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

Alternative 13: 42.8% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 82.5%

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

4. Taylor expanded in t around 0

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
5. Step-by-step derivation

   1. lower-/.f64 42.8

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{ew}}\right|} \]

6. Applied rewrites 42.8%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left|\frac{1}{ew}\right|}} \]

   2. rgt-mult-inverse N/A

      \[\leadsto \frac{\color{blue}{ew \cdot \frac{1}{ew}}}{\left|\frac{1}{ew}\right|} \]

   3. lift-/.f64 N/A

      \[\leadsto \frac{ew \cdot \color{blue}{\frac{1}{ew}}}{\left|\frac{1}{ew}\right|} \]

   4. associate-/l* N/A

      \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

   5. lower-*.f64 N/A

      \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

   6. lower-/.f64 42.9

      \[\leadsto ew \cdot \color{blue}{\frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]

8. Applied rewrites 42.9%

   \[\leadsto \color{blue}{ew \cdot \frac{\frac{1}{ew}}{\left|\frac{1}{ew}\right|}} \]
9. Add Preprocessing

Alternative 14: 40.7% accurate, 20.8× speedup

Math

\[\begin{array}{l} \\ \frac{ew}{ew \cdot \left|\frac{1}{ew}\right|} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 82.5%

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

4. Taylor expanded in t around 0

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
5. Step-by-step derivation

   1. lower-/.f64 42.8

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{ew}}\right|} \]

6. Applied rewrites 42.8%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\left|\frac{1}{ew}\right|}} \]

   2. rgt-mult-inverse N/A

      \[\leadsto \frac{\color{blue}{ew \cdot \frac{1}{ew}}}{\left|\frac{1}{ew}\right|} \]

   3. mult-flip-rev N/A

      \[\leadsto \frac{\color{blue}{\frac{ew}{ew}}}{\left|\frac{1}{ew}\right|} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{ew}{ew \cdot \left|\frac{1}{ew}\right|}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{ew}{ew \cdot \left|\frac{1}{ew}\right|}} \]

   6. lower-*.f64 42.8

      \[\leadsto \frac{ew}{\color{blue}{ew \cdot \left|\frac{1}{ew}\right|}} \]

8. Applied rewrites 42.8%

   \[\leadsto \color{blue}{\frac{ew}{ew \cdot \left|\frac{1}{ew}\right|}} \]
9. Add Preprocessing

Alternative 15: 40.1% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\left|\frac{1}{ew}\right|} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 82.5%

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

4. Taylor expanded in t around 0

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
5. Step-by-step derivation

   1. lower-/.f64 42.8

      \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{ew}}\right|} \]

6. Applied rewrites 42.8%

   \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))