Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.8s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (*
    (* ew (cos t))
    (/ 1.0 (sqrt (+ 1.0 (pow (* (- eh) (/ (tan t) ew)) 2.0)))))
   (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) * (1.0 / sqrt((1.0 + pow((-eh * (tan(t) / ew)), 2.0))))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((ew * cos(t)) * (1.0d0 / sqrt((1.0d0 + ((-eh * (tan(t) / ew)) ** 2.0d0))))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) * (1.0 / Math.sqrt((1.0 + Math.pow((-eh * (Math.tan(t) / ew)), 2.0))))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((-eh * Math.tan(t)) / ew))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) * (1.0 / math.sqrt((1.0 + math.pow((-eh * (math.tan(t) / ew)), 2.0))))) - ((eh * math.sin(t)) * math.sin(math.atan(((-eh * math.tan(t)) / ew))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) * Float64(1.0 / sqrt(Float64(1.0 + (Float64(Float64(-eh) * Float64(tan(t) / ew)) ^ 2.0))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) * (1.0 / sqrt((1.0 + ((-eh * (tan(t) / ew)) ^ 2.0))))) - ((eh * sin(t)) * sin(atan(((-eh * tan(t)) / ew))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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-278)
     (fabs 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-278) {
		tmp = fabs(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-278)) then
        tmp = abs(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-278) {
		tmp = Math.abs(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-278:
		tmp = math.fabs(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-278)
		tmp = abs(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-278)
		tmp = abs(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-278], N[Abs[ew], $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))))) < -9.99999999999999938e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 41.6%

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

   if -9.99999999999999938e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 43.4%

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

      1. lift-fabs.f64 N/A

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

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

         \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

   9. Applied rewrites 25.0%

      \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

   10. Taylor expanded in eh around 0

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

       1. lower-*.f64 N/A

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

       2. lift-cos.f64 61.6

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

   12. Applied rewrites 61.6%

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

Alternative 3: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := ew \cdot \cos t\_2\\ \mathbf{if}\;eh \leq -6 \cdot 10^{+162}:\\ \;\;\;\;\left|t\_3 - t\_1 \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t}{ew}, -\cos \tan^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3 - t\_1 \cdot \sin t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (* ew (cos t_2))))
   (if (<= eh -6e+162)
     (fabs (- t_3 (* t_1 (sin (atan (/ (* (- eh) t) ew))))))
     (if (<= eh 3.2e+30)
       (fabs
        (-
         (*
          (fma
           eh
           (/ (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)) ew)
           (- (* (cos (atan (- (* (/ eh ew) (tan t))))) (cos t))))
          ew)))
       (fabs (- t_3 (* t_1 (sin t_2))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = ew * cos(t_2);
	double tmp;
	if (eh <= -6e+162) {
		tmp = fabs((t_3 - (t_1 * sin(atan(((-eh * t) / ew))))));
	} else if (eh <= 3.2e+30) {
		tmp = fabs(-(fma(eh, ((tanh((-1.0 * ((eh * t) / ew))) * sin(t)) / ew), -(cos(atan(-((eh / ew) * tan(t)))) * cos(t))) * ew));
	} else {
		tmp = fabs((t_3 - (t_1 * sin(t_2))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(ew * cos(t_2))
	tmp = 0.0
	if (eh <= -6e+162)
		tmp = abs(Float64(t_3 - Float64(t_1 * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	elseif (eh <= 3.2e+30)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t)) / ew), Float64(-Float64(cos(atan(Float64(-Float64(Float64(eh / ew) * tan(t))))) * cos(t)))) * ew)));
	else
		tmp = abs(Float64(t_3 - Float64(t_1 * sin(t_2))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[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[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -6e+162], N[Abs[N[(t$95$3 - N[(t$95$1 * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 3.2e+30], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[(N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[(t$95$3 - N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -5.9999999999999996e162

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 94.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 94.7%

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

   if -5.9999999999999996e162 < eh < 3.19999999999999973e30

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

   if 3.19999999999999973e30 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 91.0%

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

Alternative 4: 95.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -6 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.2 \cdot 10^{+30}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t}{ew}, -\cos \tan^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (-
           (* ew (cos (atan (/ (* (- eh) (tan t)) ew))))
           (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))))))
   (if (<= eh -6e+162)
     t_1
     (if (<= eh 3.2e+30)
       (fabs
        (-
         (*
          (fma
           eh
           (/ (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)) ew)
           (- (* (cos (atan (- (* (/ eh ew) (tan t))))) (cos t))))
          ew)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -6e+162) {
		tmp = t_1;
	} else if (eh <= 3.2e+30) {
		tmp = fabs(-(fma(eh, ((tanh((-1.0 * ((eh * t) / ew))) * sin(t)) / ew), -(cos(atan(-((eh / ew) * tan(t)))) * cos(t))) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(ew * cos(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))))
	tmp = 0.0
	if (eh <= -6e+162)
		tmp = t_1;
	elseif (eh <= 3.2e+30)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t)) / ew), Float64(-Float64(cos(atan(Float64(-Float64(Float64(eh / ew) * tan(t))))) * cos(t)))) * ew)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6e+162], t$95$1, If[LessEqual[eh, 3.2e+30], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[(N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.9999999999999996e162 or 3.19999999999999973e30 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 92.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 92.3%

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

   if -5.9999999999999996e162 < eh < 3.19999999999999973e30

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

Alternative 5: 87.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -3 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.8 \cdot 10^{-12}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (-
           (* ew (cos (atan (/ (* (- eh) (tan t)) ew))))
           (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))))))
   (if (<= eh -3e-134) t_1 (if (<= eh 2.8e-12) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -3e-134) {
		tmp = t_1;
	} else if (eh <= 2.8e-12) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))))
    if (eh <= (-3d-134)) then
        tmp = t_1
    else if (eh <= 2.8d-12) then
        tmp = abs((ew * cos(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((ew * Math.cos(Math.atan(((-eh * Math.tan(t)) / ew)))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -3e-134) {
		tmp = t_1;
	} else if (eh <= 2.8e-12) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((ew * math.cos(math.atan(((-eh * math.tan(t)) / ew)))) - ((eh * math.sin(t)) * math.sin(math.atan(((-eh * t) / ew))))))
	tmp = 0
	if eh <= -3e-134:
		tmp = t_1
	elif eh <= 2.8e-12:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(ew * cos(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew)))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))))
	tmp = 0.0
	if (eh <= -3e-134)
		tmp = t_1;
	elseif (eh <= 2.8e-12)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	tmp = 0.0;
	if (eh <= -3e-134)
		tmp = t_1;
	elseif (eh <= 2.8e-12)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3e-134], t$95$1, If[LessEqual[eh, 2.8e-12], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3e-134 or 2.8000000000000002e-12 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 87.7%

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

   if -3e-134 < eh < 2.8000000000000002e-12

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 6: 81.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\\ t_2 := \left|-\mathsf{fma}\left(eh, \frac{t\_1}{ew}, -\cos \left(-\tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right) \cdot ew\right|\\ t_3 := \left|\left(-eh\right) \cdot t\_1\right|\\ \mathbf{if}\;eh \leq -3.8 \cdot 10^{+163}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq -3 \cdot 10^{-134}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2.8 \cdot 10^{-12}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;eh \leq 4.6 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)))
        (t_2
         (fabs
          (-
           (*
            (fma eh (/ t_1 ew) (- (cos (- (atan (* (/ eh ew) (tan t)))))))
            ew))))
        (t_3 (fabs (* (- eh) t_1))))
   (if (<= eh -3.8e+163)
     t_3
     (if (<= eh -3e-134)
       t_2
       (if (<= eh 2.8e-12)
         (fabs (* ew (cos t)))
         (if (<= eh 4.6e+145) t_2 t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tanh((-1.0 * ((eh * t) / ew))) * sin(t);
	double t_2 = fabs(-(fma(eh, (t_1 / ew), -cos(-atan(((eh / ew) * tan(t))))) * ew));
	double t_3 = fabs((-eh * t_1));
	double tmp;
	if (eh <= -3.8e+163) {
		tmp = t_3;
	} else if (eh <= -3e-134) {
		tmp = t_2;
	} else if (eh <= 2.8e-12) {
		tmp = fabs((ew * cos(t)));
	} else if (eh <= 4.6e+145) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t))
	t_2 = abs(Float64(-Float64(fma(eh, Float64(t_1 / ew), Float64(-cos(Float64(-atan(Float64(Float64(eh / ew) * tan(t))))))) * ew)))
	t_3 = abs(Float64(Float64(-eh) * t_1))
	tmp = 0.0
	if (eh <= -3.8e+163)
		tmp = t_3;
	elseif (eh <= -3e-134)
		tmp = t_2;
	elseif (eh <= 2.8e-12)
		tmp = abs(Float64(ew * cos(t)));
	elseif (eh <= 4.6e+145)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[(-N[(N[(eh * N[(t$95$1 / ew), $MachinePrecision] + (-N[Cos[(-N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[((-eh) * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.8e+163], t$95$3, If[LessEqual[eh, -3e-134], t$95$2, If[LessEqual[eh, 2.8e-12], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.6e+145], t$95$2, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.80000000000000008e163 or 4.6e145 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 75.6

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

   8. Applied rewrites 75.6%

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

   if -3.80000000000000008e163 < eh < -3e-134 or 2.8000000000000002e-12 < eh < 4.6e145

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

   5. Applied rewrites 95.4%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 94.3

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

   8. Applied rewrites 94.3%

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

   9. Taylor expanded in t around 0

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

       1. atan-neg N/A

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

       2. times-frac N/A

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

       3. tan-quot N/A

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

       4. atan-neg N/A

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

       5. lower-cos.f64 N/A

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

       6. atan-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. lower-atan.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift-tan.f64 N/A

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

       11. lift-*.f64 78.2

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

   11. Applied rewrites 78.2%

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

   if -3e-134 < eh < 2.8000000000000002e-12

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 87.3

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

   7. Applied rewrites 87.3%

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

Alternative 7: 75.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;t \leq -7.2 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -0.0015:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;t \leq 0.0017:\\ \;\;\;\;\left|ew + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (- eh) (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t))))))
   (if (<= t -7.2e+112)
     t_1
     (if (<= t -0.0015)
       (fabs (* ew (cos t)))
       (if (<= t 0.0017)
         (fabs
          (+
           ew
           (* -1.0 (* eh (* t (sin (atan (* -1.0 (* (/ eh ew) (tan t))))))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	double tmp;
	if (t <= -7.2e+112) {
		tmp = t_1;
	} else if (t <= -0.0015) {
		tmp = fabs((ew * cos(t)));
	} else if (t <= 0.0017) {
		tmp = fabs((ew + (-1.0 * (eh * (t * sin(atan((-1.0 * ((eh / ew) * tan(t))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((-eh * (tanh(((-1.0d0) * ((eh * t) / ew))) * sin(t))))
    if (t <= (-7.2d+112)) then
        tmp = t_1
    else if (t <= (-0.0015d0)) then
        tmp = abs((ew * cos(t)))
    else if (t <= 0.0017d0) then
        tmp = abs((ew + ((-1.0d0) * (eh * (t * sin(atan(((-1.0d0) * ((eh / ew) * tan(t))))))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((-eh * (Math.tanh((-1.0 * ((eh * t) / ew))) * Math.sin(t))));
	double tmp;
	if (t <= -7.2e+112) {
		tmp = t_1;
	} else if (t <= -0.0015) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else if (t <= 0.0017) {
		tmp = Math.abs((ew + (-1.0 * (eh * (t * Math.sin(Math.atan((-1.0 * ((eh / ew) * Math.tan(t))))))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((-eh * (math.tanh((-1.0 * ((eh * t) / ew))) * math.sin(t))))
	tmp = 0
	if t <= -7.2e+112:
		tmp = t_1
	elif t <= -0.0015:
		tmp = math.fabs((ew * math.cos(t)))
	elif t <= 0.0017:
		tmp = math.fabs((ew + (-1.0 * (eh * (t * math.sin(math.atan((-1.0 * ((eh / ew) * math.tan(t))))))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(-eh) * Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t))))
	tmp = 0.0
	if (t <= -7.2e+112)
		tmp = t_1;
	elseif (t <= -0.0015)
		tmp = abs(Float64(ew * cos(t)));
	elseif (t <= 0.0017)
		tmp = abs(Float64(ew + Float64(-1.0 * Float64(eh * Float64(t * sin(atan(Float64(-1.0 * Float64(Float64(eh / ew) * tan(t))))))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	tmp = 0.0;
	if (t <= -7.2e+112)
		tmp = t_1;
	elseif (t <= -0.0015)
		tmp = abs((ew * cos(t)));
	elseif (t <= 0.0017)
		tmp = abs((ew + (-1.0 * (eh * (t * sin(atan((-1.0 * ((eh / ew) * tan(t))))))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[((-eh) * N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.2e+112], t$95$1, If[LessEqual[t, -0.0015], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 0.0017], N[Abs[N[(ew + N[(-1.0 * N[(eh * N[(t * N[Sin[N[ArcTan[N[(-1.0 * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.20000000000000001e112 or 0.00169999999999999991 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 52.1

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

   8. Applied rewrites 52.1%

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

   if -7.20000000000000001e112 < t < -0.0015

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 51.5

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

   7. Applied rewrites 51.5%

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

   if -0.0015 < t < 0.00169999999999999991

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. times-frac N/A

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

      8. tan-quot N/A

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

   7. Applied rewrites 98.3%

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

Alternative 8: 73.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.32 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.42 \cdot 10^{+30}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.32e-83)
     t_1
     (if (<= ew 1.42e+30)
       (fabs (* (- eh) (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.32e-83) {
		tmp = t_1;
	} else if (ew <= 1.42e+30) {
		tmp = fabs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-1.32d-83)) then
        tmp = t_1
    else if (ew <= 1.42d+30) then
        tmp = abs((-eh * (tanh(((-1.0d0) * ((eh * t) / ew))) * sin(t))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -1.32e-83) {
		tmp = t_1;
	} else if (ew <= 1.42e+30) {
		tmp = Math.abs((-eh * (Math.tanh((-1.0 * ((eh * t) / ew))) * Math.sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -1.32e-83:
		tmp = t_1
	elif ew <= 1.42e+30:
		tmp = math.fabs((-eh * (math.tanh((-1.0 * ((eh * t) / ew))) * math.sin(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.32e-83)
		tmp = t_1;
	elseif (ew <= 1.42e+30)
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -1.32e-83)
		tmp = t_1;
	elseif (ew <= 1.42e+30)
		tmp = abs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.32e-83], t$95$1, If[LessEqual[ew, 1.42e+30], N[Abs[N[((-eh) * N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.31999999999999994e-83 or 1.41999999999999991e30 < ew

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. cos-atan N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 81.3

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

   7. Applied rewrites 81.3%

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

   if -1.31999999999999994e-83 < ew < 1.41999999999999991e30

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 64.7

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

   8. Applied rewrites 64.7%

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

Alternative 9: 61.6% accurate, 8.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 61.6

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

7. Applied rewrites 61.6%

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

Alternative 10: 42.5% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 42.5%

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

Alternative 11: 22.6% accurate, 862.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. cos-atan N/A

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

   8. lower-/.f64 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-+.f64 N/A

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

   11. pow2 N/A

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

   12. lower-pow.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 42.5%

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

   1. lift-fabs.f64 N/A

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

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

      \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

9. Applied rewrites 25.2%

   \[\leadsto \color{blue}{\sqrt{ew \cdot ew}} \]

10. Taylor expanded in t around 0

    \[\leadsto \color{blue}{ew} \]
11. Step-by-step derivation

12. Applied rewrites 22.6%

    \[\leadsto \color{blue}{ew} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025088 
(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)))))))