Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.5s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 99.1%

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

Alternative 4: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := \left(-eh\right) \cdot \frac{\tan t}{ew}\\ t_4 := \cos \tan^{-1} t\_3\\ \mathbf{if}\;eh \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\left|ew \cdot \cos t\_2 - t\_1 \cdot \sin t\_2\right|\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{-205}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot ew, t\_4, \frac{{t\_1}^{2}}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, t\_4, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} t\_3\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (* (- eh) (/ (tan t) ew)))
        (t_4 (cos (atan t_3))))
   (if (<= eh -1e-104)
     (fabs (- (* ew (cos t_2)) (* t_1 (sin t_2))))
     (if (<= eh 3.7e-205)
       (fabs (fma (* (cos t) ew) t_4 (/ (pow t_1 2.0) (* ew (cos t)))))
       (fabs (fma ew t_4 (* (- (* (sin t) eh)) (tanh (asinh t_3)))))))))

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 = -eh * (tan(t) / ew);
	double t_4 = cos(atan(t_3));
	double tmp;
	if (eh <= -1e-104) {
		tmp = fabs(((ew * cos(t_2)) - (t_1 * sin(t_2))));
	} else if (eh <= 3.7e-205) {
		tmp = fabs(fma((cos(t) * ew), t_4, (pow(t_1, 2.0) / (ew * cos(t)))));
	} else {
		tmp = fabs(fma(ew, t_4, (-(sin(t) * eh) * tanh(asinh(t_3)))));
	}
	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(Float64(-eh) * Float64(tan(t) / ew))
	t_4 = cos(atan(t_3))
	tmp = 0.0
	if (eh <= -1e-104)
		tmp = abs(Float64(Float64(ew * cos(t_2)) - Float64(t_1 * sin(t_2))));
	elseif (eh <= 3.7e-205)
		tmp = abs(fma(Float64(cos(t) * ew), t_4, Float64((t_1 ^ 2.0) / Float64(ew * cos(t)))));
	else
		tmp = abs(fma(ew, t_4, Float64(Float64(-Float64(sin(t) * eh)) * tanh(asinh(t_3)))));
	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[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1e-104], N[Abs[N[(N[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 3.7e-205], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * t$95$4 + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t$95$4 + N[((-N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]) * N[Tanh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -9.99999999999999927e-105

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 86.1%

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

   if -9.99999999999999927e-105 < eh < 3.7000000000000001e-205

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. pow-prod-down N/A

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

      3. lower-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if 3.7000000000000001e-205 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 82.4%

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

Alternative 5: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := ew \cdot \cos t\_2\\ \mathbf{if}\;eh \leq -1 \cdot 10^{-104}:\\ \;\;\;\;\left|t\_3 - t\_1 \cdot \sin t\_2\right|\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{-205}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right), \frac{{t\_1}^{2}}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_3 - t\_1 \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \end{array} \end{array} \]

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 -1e-104)
     (fabs (- t_3 (* t_1 (sin t_2))))
     (if (<= eh 3.7e-205)
       (fabs
        (fma
         (* (cos t) ew)
         (cos (atan (* (- eh) (/ (tan t) ew))))
         (/ (pow t_1 2.0) (* ew (cos t)))))
       (fabs (- t_3 (* t_1 (sin (atan (/ (* (- eh) t) ew))))))))))

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 <= -1e-104) {
		tmp = fabs((t_3 - (t_1 * sin(t_2))));
	} else if (eh <= 3.7e-205) {
		tmp = fabs(fma((cos(t) * ew), cos(atan((-eh * (tan(t) / ew)))), (pow(t_1, 2.0) / (ew * cos(t)))));
	} else {
		tmp = fabs((t_3 - (t_1 * sin(atan(((-eh * t) / ew))))));
	}
	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 <= -1e-104)
		tmp = abs(Float64(t_3 - Float64(t_1 * sin(t_2))));
	elseif (eh <= 3.7e-205)
		tmp = abs(fma(Float64(cos(t) * ew), cos(atan(Float64(Float64(-eh) * Float64(tan(t) / ew)))), Float64((t_1 ^ 2.0) / Float64(ew * cos(t)))));
	else
		tmp = abs(Float64(t_3 - Float64(t_1 * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	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, -1e-104], N[Abs[N[(t$95$3 - N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 3.7e-205], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 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]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -9.99999999999999927e-105

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 86.1%

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

   if -9.99999999999999927e-105 < eh < 3.7000000000000001e-205

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. pow-prod-down N/A

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

      3. lower-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 91.3

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

   6. Applied rewrites 91.3%

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

   if 3.7000000000000001e-205 < 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. Taylor expanded in t around 0

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

   4. Applied rewrites 82.4%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 82.4%

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

Alternative 6: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \left|ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - t\_1 \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -1 \cdot 10^{-104}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{-205}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right), \frac{{t\_1}^{2}}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2
         (fabs
          (-
           (* ew (cos (atan (/ (* (- eh) (tan t)) ew))))
           (* t_1 (sin (atan (/ (* (- eh) t) ew))))))))
   (if (<= eh -1e-104)
     t_2
     (if (<= eh 3.7e-205)
       (fabs
        (fma
         (* (cos t) ew)
         (cos (atan (* (- eh) (/ (tan t) ew))))
         (/ (pow t_1 2.0) (* ew (cos t)))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = fabs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - (t_1 * sin(atan(((-eh * t) / ew))))));
	double tmp;
	if (eh <= -1e-104) {
		tmp = t_2;
	} else if (eh <= 3.7e-205) {
		tmp = fabs(fma((cos(t) * ew), cos(atan((-eh * (tan(t) / ew)))), (pow(t_1, 2.0) / (ew * cos(t)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = abs(Float64(Float64(ew * cos(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew)))) - Float64(t_1 * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))))
	tmp = 0.0
	if (eh <= -1e-104)
		tmp = t_2;
	elseif (eh <= 3.7e-205)
		tmp = abs(fma(Float64(cos(t) * ew), cos(atan(Float64(Float64(-eh) * Float64(tan(t) / ew)))), Float64((t_1 ^ 2.0) / Float64(ew * cos(t)))));
	else
		tmp = 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[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1e-104], t$95$2, If[LessEqual[eh, 3.7e-205], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Power[t$95$1, 2.0], $MachinePrecision] / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.99999999999999927e-105 or 3.7000000000000001e-205 < 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. Taylor expanded in t around 0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 84.0%

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

   if -9.99999999999999927e-105 < eh < 3.7000000000000001e-205

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. pow-prod-down N/A

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

      3. lower-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 91.3

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

   6. Applied rewrites 91.3%

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

Alternative 7: 85.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{if}\;eh \leq -1 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{-205}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -1e-104)
     t_1
     (if (<= eh 3.7e-205)
       (fabs (* (* (cos t) ew) (cos (atan (- (* (/ eh ew) (tan 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 <= -1e-104) {
		tmp = t_1;
	} else if (eh <= 3.7e-205) {
		tmp = fabs(((cos(t) * ew) * cos(atan(-((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(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))))
    if (eh <= (-1d-104)) then
        tmp = t_1
    else if (eh <= 3.7d-205) then
        tmp = abs(((cos(t) * ew) * cos(atan(-((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(((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 <= -1e-104) {
		tmp = t_1;
	} else if (eh <= 3.7e-205) {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(-((eh / ew) * Math.tan(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 <= -1e-104:
		tmp = t_1
	elif eh <= 3.7e-205:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(-((eh / ew) * math.tan(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 <= -1e-104)
		tmp = t_1;
	elseif (eh <= 3.7e-205)
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(-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(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * sin(t)) * sin(atan(((-eh * t) / ew))))));
	tmp = 0.0;
	if (eh <= -1e-104)
		tmp = t_1;
	elseif (eh <= 3.7e-205)
		tmp = abs(((cos(t) * ew) * cos(atan(-((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[(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, -1e-104], t$95$1, If[LessEqual[eh, 3.7e-205], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.99999999999999927e-105 or 3.7000000000000001e-205 < 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. Taylor expanded in t around 0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 84.0%

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

   if -9.99999999999999927e-105 < eh < 3.7000000000000001e-205

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 91.2

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

   4. Applied rewrites 91.2%

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

Alternative 8: 74.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ t_2 := \left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right)\right|\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+250}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -5 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -0.004:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-5}:\\ \;\;\;\;\left|ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 18000000000:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (- eh) (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)))))
        (t_2 (fabs (* (* (cos t) ew) (cos (atan (- (* (/ eh ew) (tan t)))))))))
   (if (<= t -7.5e+250)
     t_2
     (if (<= t -5e+123)
       t_1
       (if (<= t -0.004)
         t_2
         (if (<= t 9.8e-5)
           (fabs
            (-
             (* ew (cos (atan (/ (* (- eh) (tan t)) ew))))
             (* (* eh t) (sin (atan (/ (* (- eh) t) ew))))))
           (if (<= t 18000000000.0) t_1 t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	double t_2 = fabs(((cos(t) * ew) * cos(atan(-((eh / ew) * tan(t))))));
	double tmp;
	if (t <= -7.5e+250) {
		tmp = t_2;
	} else if (t <= -5e+123) {
		tmp = t_1;
	} else if (t <= -0.004) {
		tmp = t_2;
	} else if (t <= 9.8e-5) {
		tmp = fabs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * t) * sin(atan(((-eh * t) / ew))))));
	} else if (t <= 18000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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 = abs((-eh * (tanh(((-1.0d0) * ((eh * t) / ew))) * sin(t))))
    t_2 = abs(((cos(t) * ew) * cos(atan(-((eh / ew) * tan(t))))))
    if (t <= (-7.5d+250)) then
        tmp = t_2
    else if (t <= (-5d+123)) then
        tmp = t_1
    else if (t <= (-0.004d0)) then
        tmp = t_2
    else if (t <= 9.8d-5) then
        tmp = abs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * t) * sin(atan(((-eh * t) / ew))))))
    else if (t <= 18000000000.0d0) then
        tmp = t_1
    else
        tmp = t_2
    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 t_2 = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(-((eh / ew) * Math.tan(t))))));
	double tmp;
	if (t <= -7.5e+250) {
		tmp = t_2;
	} else if (t <= -5e+123) {
		tmp = t_1;
	} else if (t <= -0.004) {
		tmp = t_2;
	} else if (t <= 9.8e-5) {
		tmp = Math.abs(((ew * Math.cos(Math.atan(((-eh * Math.tan(t)) / ew)))) - ((eh * t) * Math.sin(Math.atan(((-eh * t) / ew))))));
	} else if (t <= 18000000000.0) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((-eh * (math.tanh((-1.0 * ((eh * t) / ew))) * math.sin(t))))
	t_2 = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(-((eh / ew) * math.tan(t))))))
	tmp = 0
	if t <= -7.5e+250:
		tmp = t_2
	elif t <= -5e+123:
		tmp = t_1
	elif t <= -0.004:
		tmp = t_2
	elif t <= 9.8e-5:
		tmp = math.fabs(((ew * math.cos(math.atan(((-eh * math.tan(t)) / ew)))) - ((eh * t) * math.sin(math.atan(((-eh * t) / ew))))))
	elif t <= 18000000000.0:
		tmp = t_1
	else:
		tmp = t_2
	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))))
	t_2 = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(-Float64(Float64(eh / ew) * tan(t)))))))
	tmp = 0.0
	if (t <= -7.5e+250)
		tmp = t_2;
	elseif (t <= -5e+123)
		tmp = t_1;
	elseif (t <= -0.004)
		tmp = t_2;
	elseif (t <= 9.8e-5)
		tmp = abs(Float64(Float64(ew * cos(atan(Float64(Float64(Float64(-eh) * tan(t)) / ew)))) - Float64(Float64(eh * t) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	elseif (t <= 18000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((-eh * (tanh((-1.0 * ((eh * t) / ew))) * sin(t))));
	t_2 = abs(((cos(t) * ew) * cos(atan(-((eh / ew) * tan(t))))));
	tmp = 0.0;
	if (t <= -7.5e+250)
		tmp = t_2;
	elseif (t <= -5e+123)
		tmp = t_1;
	elseif (t <= -0.004)
		tmp = t_2;
	elseif (t <= 9.8e-5)
		tmp = abs(((ew * cos(atan(((-eh * tan(t)) / ew)))) - ((eh * t) * sin(atan(((-eh * t) / ew))))));
	elseif (t <= 18000000000.0)
		tmp = t_1;
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.5e+250], t$95$2, If[LessEqual[t, -5e+123], t$95$1, If[LessEqual[t, -0.004], t$95$2, If[LessEqual[t, 9.8e-5], 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 * t), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 18000000000.0], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4999999999999997e250 or -4.99999999999999974e123 < t < -0.0040000000000000001 or 1.8e10 < 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. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 51.5

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

   4. Applied rewrites 51.5%

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

   if -7.4999999999999997e250 < t < -4.99999999999999974e123 or 9.8e-5 < t < 1.8e10

   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. Taylor expanded in eh around inf

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

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right| \]
   6. 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 51.0

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

   7. Applied rewrites 51.0%

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

   if -0.0040000000000000001 < t < 9.8e-5

   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. Taylor expanded in t around 0

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 99.6%

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

Alternative 9: 73.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;eh \leq -9.5 \cdot 10^{+51}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.6 \cdot 10^{+152}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (- eh) (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t))))))
   (if (<= eh -9.5e+51)
     t_1
     (if (<= eh 3.6e+152)
       (fabs (* (* (cos t) ew) (cos (atan (- (* (/ 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 (eh <= -9.5e+51) {
		tmp = t_1;
	} else if (eh <= 3.6e+152) {
		tmp = fabs(((cos(t) * ew) * cos(atan(-((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 (eh <= (-9.5d+51)) then
        tmp = t_1
    else if (eh <= 3.6d+152) then
        tmp = abs(((cos(t) * ew) * cos(atan(-((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 (eh <= -9.5e+51) {
		tmp = t_1;
	} else if (eh <= 3.6e+152) {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(-((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 eh <= -9.5e+51:
		tmp = t_1
	elif eh <= 3.6e+152:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(-((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 (eh <= -9.5e+51)
		tmp = t_1;
	elseif (eh <= 3.6e+152)
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(-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 (eh <= -9.5e+51)
		tmp = t_1;
	elseif (eh <= 3.6e+152)
		tmp = abs(((cos(t) * ew) * cos(atan(-((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[eh, -9.5e+51], t$95$1, If[LessEqual[eh, 3.6e+152], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.4999999999999999e51 or 3.5999999999999999e152 < 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. Taylor expanded in eh around inf

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

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

   4. Applied rewrites 70.2%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right| \]
   6. 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 70.4

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

   7. Applied rewrites 70.4%

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

   if -9.4999999999999999e51 < eh < 3.5999999999999999e152

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 75.5

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

   4. Applied rewrites 75.5%

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

Alternative 10: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;t \leq -1.65 \cdot 10^{-96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-60}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (- eh) (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t))))))
   (if (<= t -1.65e-96)
     t_1
     (if (<= t 8.5e-60) (fabs (* (cos (atan (- (* (/ eh ew) t)))) ew)) 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 <= -1.65e-96) {
		tmp = t_1;
	} else if (t <= 8.5e-60) {
		tmp = fabs((cos(atan(-((eh / ew) * t))) * 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) :: tmp
    t_1 = abs((-eh * (tanh(((-1.0d0) * ((eh * t) / ew))) * sin(t))))
    if (t <= (-1.65d-96)) then
        tmp = t_1
    else if (t <= 8.5d-60) then
        tmp = abs((cos(atan(-((eh / ew) * t))) * 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 = Math.abs((-eh * (Math.tanh((-1.0 * ((eh * t) / ew))) * Math.sin(t))));
	double tmp;
	if (t <= -1.65e-96) {
		tmp = t_1;
	} else if (t <= 8.5e-60) {
		tmp = Math.abs((Math.cos(Math.atan(-((eh / ew) * t))) * ew));
	} 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 <= -1.65e-96:
		tmp = t_1
	elif t <= 8.5e-60:
		tmp = math.fabs((math.cos(math.atan(-((eh / ew) * t))) * ew))
	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 <= -1.65e-96)
		tmp = t_1;
	elseif (t <= 8.5e-60)
		tmp = abs(Float64(cos(atan(Float64(-Float64(Float64(eh / ew) * t)))) * ew));
	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 <= -1.65e-96)
		tmp = t_1;
	elseif (t <= 8.5e-60)
		tmp = abs((cos(atan(-((eh / ew) * t))) * ew));
	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, -1.65e-96], t$95$1, If[LessEqual[t, 8.5e-60], N[Abs[N[(N[Cos[N[ArcTan[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.64999999999999995e-96 or 8.50000000000000044e-60 < t

   1. Initial program 99.7%

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

   2. Taylor expanded in eh around inf

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

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\left(-eh\right) \cdot \left(\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t\right)\right| \]
   6. 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 50.8

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

   7. Applied rewrites 50.8%

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

   if -1.64999999999999995e-96 < t < 8.50000000000000044e-60

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 76.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 76.7%

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

Alternative 11: 41.5% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 41.5%

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

Alternative 12: 40.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 41.5%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 40.3%

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

Alternative 13: 4.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{{\left(eh \cdot t\right)}^{2}}{ew}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ (pow (* eh t) 2.0) ew)))

C

double code(double eh, double ew, double t) {
	return fabs((pow((eh * t), 2.0) / ew));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((eh * t) ** 2.0d0) / ew))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.pow((eh * t), 2.0) / ew));
}

Python

def code(eh, ew, t):
	return math.fabs((math.pow((eh * t), 2.0) / ew))

Julia

function code(eh, ew, t)
	return abs(Float64((Float64(eh * t) ^ 2.0) / ew))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * t) ^ 2.0) / ew));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Power[N[(eh * t), $MachinePrecision], 2.0], $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around inf

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

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

4. Applied rewrites 41.7%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2} \cdot {t}^{2}}{ew}\right| \]

   2. pow-prod-down N/A

      \[\leadsto \left|\frac{{\left(eh \cdot t\right)}^{2}}{ew}\right| \]

   3. lower-pow.f64 N/A

      \[\leadsto \left|\frac{{\left(eh \cdot t\right)}^{2}}{ew}\right| \]

   4. lower-*.f64 4.6

      \[\leadsto \left|\frac{{\left(eh \cdot t\right)}^{2}}{ew}\right| \]

7. Applied rewrites 4.6%

   \[\leadsto \left|\frac{{\left(eh \cdot t\right)}^{2}}{\color{blue}{ew}}\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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)))))))