Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.5s

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

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = abs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[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

3. Final simplification 99.8%

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

Alternative 2: 56.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(-\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-299)
     (fabs (* t_1 (cos (atan (- (/ (* eh t) ew))))))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299) {
		tmp = fabs((t_1 * cos(atan(-((eh * 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) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-1d-299)) then
        tmp = abs((t_1 * cos(atan(-((eh * 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 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -1e-299) {
		tmp = Math.abs((t_1 * Math.cos(Math.atan(-((eh * t) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -1e-299:
		tmp = math.fabs((t_1 * math.cos(math.atan(-((eh * t) / ew)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = abs(Float64(t_1 * cos(atan(Float64(-Float64(Float64(eh * t) / ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = abs((t_1 * cos(atan(-((eh * t) / ew)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-299], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[(-N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999992e-300

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 90.4%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 54.2%

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

   if -9.99999999999999992e-300 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 63.2%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-299)
     (fabs (* (cos (atan (/ (* (- eh) t) ew))) ew))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299) {
		tmp = fabs((cos(atan(((-eh * t) / ew))) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-1d-299)) then
        tmp = abs((cos(atan(((-eh * t) / ew))) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -1e-299) {
		tmp = Math.abs((Math.cos(Math.atan(((-eh * t) / ew))) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -1e-299:
		tmp = math.fabs((math.cos(math.atan(((-eh * t) / ew))) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-eh) * t) / ew))) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = abs((cos(atan(((-eh * t) / ew))) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-299], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999992e-300

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

   5. Applied rewrites 43.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 42.4%

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

   if -9.99999999999999992e-300 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 63.2%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 42.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;e^{\log \left(ew \cdot ew\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-299)
     (exp (* (log (* ew ew)) 0.5))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299) {
		tmp = exp((log((ew * ew)) * 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-1d-299)) then
        tmp = exp((log((ew * ew)) * 0.5d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -1e-299) {
		tmp = Math.exp((Math.log((ew * ew)) * 0.5));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -1e-299:
		tmp = math.exp((math.log((ew * ew)) * 0.5))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = exp(Float64(log(Float64(ew * ew)) * 0.5));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = exp((log((ew * ew)) * 0.5));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-299], N[Exp[N[(N[Log[N[(ew * ew), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999992e-300

   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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 23.1%

      \[\leadsto e^{\log \color{blue}{\left(ew \cdot ew\right)} \cdot 0.5} \]

   if -9.99999999999999992e-300 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 63.2%

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

4. Final simplification 41.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-299}:\\ \;\;\;\;e^{\log \left(ew \cdot ew\right) \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -1e-299)
     (sqrt (pow (fma (* -0.5 ew) (* t t) ew) 2.0))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -1e-299) {
		tmp = sqrt(pow(fma((-0.5 * ew), (t * t), ew), 2.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -1e-299)
		tmp = sqrt((fma(Float64(-0.5 * ew), Float64(t * t), ew) ^ 2.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-299], N[Sqrt[N[Power[N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999992e-300

   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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 19.6%

      \[\leadsto e^{\log \left({\color{blue}{\left(ew + \left(t \cdot t\right) \cdot \left(\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right) - 0.5 \cdot \frac{eh \cdot eh}{ew}\right)\right)}}^{2}\right) \cdot 0.5} \]

   8. Taylor expanded in eh around 0

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

   10. Applied rewrites 21.1%

       \[\leadsto e^{\log \left({\left(ew + \left(t \cdot t\right) \cdot \left(-0.5 \cdot \color{blue}{ew}\right)\right)}^{2}\right) \cdot 0.5} \]
   11. Step-by-step derivation

       1. lift-exp.f64 N/A

          \[\leadsto \color{blue}{e^{\log \left({\left(ew + \left(t \cdot t\right) \cdot \left(\frac{-1}{2} \cdot ew\right)\right)}^{2}\right) \cdot \frac{1}{2}}} \]

       2. lift-*.f64 N/A

          \[\leadsto e^{\color{blue}{\log \left({\left(ew + \left(t \cdot t\right) \cdot \left(\frac{-1}{2} \cdot ew\right)\right)}^{2}\right) \cdot \frac{1}{2}}} \]

       3. lift-log.f64 N/A

          \[\leadsto e^{\color{blue}{\log \left({\left(ew + \left(t \cdot t\right) \cdot \left(\frac{-1}{2} \cdot ew\right)\right)}^{2}\right)} \cdot \frac{1}{2}} \]

   12. Applied rewrites 22.2%

       \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right)\right)}^{2}}} \]

   if -9.99999999999999992e-300 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 69.9%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 63.2%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-299}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 32.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-196}:\\ \;\;\;\;ew\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -2e-196)
     ew
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-196) {
		tmp = ew;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-2d-196)) then
        tmp = ew
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -2e-196) {
		tmp = ew;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -2e-196:
		tmp = ew
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-196)
		tmp = ew;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-196)
		tmp = ew;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-196], ew, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2.0000000000000001e-196

   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

   4. Applied rewrites 1.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 5.2%

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

   if -2.0000000000000001e-196 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.7%

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 59.5%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-196}:\\ \;\;\;\;ew\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

3. Taylor expanded in ew around inf

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

5. Applied rewrites 90.0%

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

7. Applied rewrites 94.1%

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

9. Applied rewrites 94.1%

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

10. Taylor expanded in t around 0

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

12. Applied rewrites 93.8%

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

13. Final simplification 93.8%

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

Alternative 8: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.7 \cdot 10^{-105} \lor \neg \left(ew \leq 2.25 \cdot 10^{-61}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.7e-105) (not (<= ew 2.25e-61)))
   (fabs
    (*
     (* ew (cos t))
     (cos (atan (/ (* (- eh) (sin t)) (+ ew (* -0.5 (* ew (* t t)))))))))
   (fabs
    (*
     (* (sin t) (- eh))
     (sin (atan (* (/ (sin t) (- ew)) (/ eh (cos t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61)) {
		tmp = fabs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = fabs(((sin(t) * -eh) * sin(atan(((sin(t) / -ew) * (eh / cos(t)))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-3.7d-105)) .or. (.not. (ew <= 2.25d-61))) then
        tmp = abs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + ((-0.5d0) * (ew * (t * t)))))))))
    else
        tmp = abs(((sin(t) * -eh) * sin(atan(((sin(t) / -ew) * (eh / cos(t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61)) {
		tmp = Math.abs(((ew * Math.cos(t)) * Math.cos(Math.atan(((-eh * Math.sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = Math.abs(((Math.sin(t) * -eh) * Math.sin(Math.atan(((Math.sin(t) / -ew) * (eh / Math.cos(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -3.7e-105) or not (ew <= 2.25e-61):
		tmp = math.fabs(((ew * math.cos(t)) * math.cos(math.atan(((-eh * math.sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))))
	else:
		tmp = math.fabs(((math.sin(t) * -eh) * math.sin(math.atan(((math.sin(t) / -ew) * (eh / math.cos(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61))
		tmp = abs(Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / Float64(ew + Float64(-0.5 * Float64(ew * Float64(t * t)))))))));
	else
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(atan(Float64(Float64(sin(t) / Float64(-ew)) * Float64(eh / cos(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -3.7e-105) || ~((ew <= 2.25e-61)))
		tmp = abs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
	else
		tmp = abs(((sin(t) * -eh) * sin(atan(((sin(t) / -ew) * (eh / cos(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.7e-105], N[Not[LessEqual[ew, 2.25e-61]], $MachinePrecision]], N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[(ew + N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.70000000000000008e-105 or 2.25e-61 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 82.0%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]

   if -3.70000000000000008e-105 < ew < 2.25e-61

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

   5. Applied rewrites 71.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.7 \cdot 10^{-105} \lor \neg \left(ew \leq 2.25 \cdot 10^{-61}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{\sin t}{-ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ t_2 := ew \cdot \cos t\\ \mathbf{if}\;ew \leq -3.7 \cdot 10^{-105} \lor \neg \left(ew \leq 2.25 \cdot 10^{-61}\right):\\ \;\;\;\;\left|t\_2 \cdot \cos \tan^{-1} \left(\frac{t\_1}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{t\_2}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))) (t_2 (* ew (cos t))))
   (if (or (<= ew -3.7e-105) (not (<= ew 2.25e-61)))
     (fabs (* t_2 (cos (atan (/ t_1 (+ ew (* -0.5 (* ew (* t t)))))))))
     (fabs (* t_1 (sin (atan (/ t_1 t_2))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double t_2 = ew * cos(t);
	double tmp;
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61)) {
		tmp = fabs((t_2 * cos(atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = fabs((t_1 * sin(atan((t_1 / 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 = -eh * sin(t)
    t_2 = ew * cos(t)
    if ((ew <= (-3.7d-105)) .or. (.not. (ew <= 2.25d-61))) then
        tmp = abs((t_2 * cos(atan((t_1 / (ew + ((-0.5d0) * (ew * (t * t)))))))))
    else
        tmp = abs((t_1 * sin(atan((t_1 / t_2)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = -eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double tmp;
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61)) {
		tmp = Math.abs((t_2 * Math.cos(Math.atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((t_1 / t_2)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh * math.sin(t)
	t_2 = ew * math.cos(t)
	tmp = 0
	if (ew <= -3.7e-105) or not (ew <= 2.25e-61):
		tmp = math.fabs((t_2 * math.cos(math.atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))))
	else:
		tmp = math.fabs((t_1 * math.sin(math.atan((t_1 / t_2)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	t_2 = Float64(ew * cos(t))
	tmp = 0.0
	if ((ew <= -3.7e-105) || !(ew <= 2.25e-61))
		tmp = abs(Float64(t_2 * cos(atan(Float64(t_1 / Float64(ew + Float64(-0.5 * Float64(ew * Float64(t * t)))))))));
	else
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh * sin(t);
	t_2 = ew * cos(t);
	tmp = 0.0;
	if ((ew <= -3.7e-105) || ~((ew <= 2.25e-61)))
		tmp = abs((t_2 * cos(atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	else
		tmp = abs((t_1 * sin(atan((t_1 / t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -3.7e-105], N[Not[LessEqual[ew, 2.25e-61]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Cos[N[ArcTan[N[(t$95$1 / N[(ew + N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.70000000000000008e-105 or 2.25e-61 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 82.0%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]

   if -3.70000000000000008e-105 < ew < 2.25e-61

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in eh around inf

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

   8. Applied rewrites 71.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.7 \cdot 10^{-105} \lor \neg \left(ew \leq 2.25 \cdot 10^{-61}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ t_2 := ew \cdot \cos t\\ \mathbf{if}\;ew \leq -4.1 \cdot 10^{-232} \lor \neg \left(ew \leq 8.6 \cdot 10^{-193}\right):\\ \;\;\;\;\left|t\_2 \cdot \cos \tan^{-1} \left(\frac{t\_1}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))) (t_2 (* ew (cos t))))
   (if (or (<= ew -4.1e-232) (not (<= ew 8.6e-193)))
     (fabs (* t_2 (cos (atan (/ t_1 (+ ew (* -0.5 (* ew (* t t)))))))))
     (* t_1 (sin (atan (/ t_1 t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double t_2 = ew * cos(t);
	double tmp;
	if ((ew <= -4.1e-232) || !(ew <= 8.6e-193)) {
		tmp = fabs((t_2 * cos(atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = t_1 * sin(atan((t_1 / 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 = -eh * sin(t)
    t_2 = ew * cos(t)
    if ((ew <= (-4.1d-232)) .or. (.not. (ew <= 8.6d-193))) then
        tmp = abs((t_2 * cos(atan((t_1 / (ew + ((-0.5d0) * (ew * (t * t)))))))))
    else
        tmp = t_1 * sin(atan((t_1 / t_2)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = -eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double tmp;
	if ((ew <= -4.1e-232) || !(ew <= 8.6e-193)) {
		tmp = Math.abs((t_2 * Math.cos(Math.atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	} else {
		tmp = t_1 * Math.sin(Math.atan((t_1 / t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh * math.sin(t)
	t_2 = ew * math.cos(t)
	tmp = 0
	if (ew <= -4.1e-232) or not (ew <= 8.6e-193):
		tmp = math.fabs((t_2 * math.cos(math.atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))))
	else:
		tmp = t_1 * math.sin(math.atan((t_1 / t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	t_2 = Float64(ew * cos(t))
	tmp = 0.0
	if ((ew <= -4.1e-232) || !(ew <= 8.6e-193))
		tmp = abs(Float64(t_2 * cos(atan(Float64(t_1 / Float64(ew + Float64(-0.5 * Float64(ew * Float64(t * t)))))))));
	else
		tmp = Float64(t_1 * sin(atan(Float64(t_1 / t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh * sin(t);
	t_2 = ew * cos(t);
	tmp = 0.0;
	if ((ew <= -4.1e-232) || ~((ew <= 8.6e-193)))
		tmp = abs((t_2 * cos(atan((t_1 / (ew + (-0.5 * (ew * (t * t)))))))));
	else
		tmp = t_1 * sin(atan((t_1 / t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[ew, -4.1e-232], N[Not[LessEqual[ew, 8.6e-193]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Cos[N[ArcTan[N[(t$95$1 / N[(ew + N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.09999999999999973e-232 or 8.6000000000000004e-193 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 95.7%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 72.5%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 72.6%

       \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]

   if -4.09999999999999973e-232 < ew < 8.6000000000000004e-193

   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. Add Preprocessing

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 60.1%

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

   7. Applied rewrites 30.0%

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

   8. Taylor expanded in eh around inf

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

   10. Applied rewrites 46.0%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.1 \cdot 10^{-232} \lor \neg \left(ew \leq 8.6 \cdot 10^{-193}\right):\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (* ew (cos t))
   (cos (atan (/ (* (- eh) (sin t)) (+ ew (* -0.5 (* ew (* t t))))))))))

C

double code(double eh, double ew, double t) {
	return fabs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + ((-0.5d0) * (ew * (t * t)))))))))
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.sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
}

Python

def code(eh, ew, t):
	return math.fabs(((ew * math.cos(t)) * math.cos(math.atan(((-eh * math.sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / Float64(ew + Float64(-0.5 * Float64(ew * Float64(t * t)))))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(((ew * cos(t)) * cos(atan(((-eh * sin(t)) / (ew + (-0.5 * (ew * (t * t)))))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / N[(ew + N[(-0.5 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around inf

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in eh around 0

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 63.5%

    \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(-\frac{eh \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]

12. Final simplification 63.5%

    \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew + -0.5 \cdot \left(ew \cdot \left(t \cdot t\right)\right)}\right)\right| \]
13. Add Preprocessing

Alternative 12: 61.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in ew around inf

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

5. Applied rewrites 90.0%

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

6. Taylor expanded in eh around 0

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

8. Applied rewrites 63.4%

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

10. Applied rewrites 63.4%

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

Alternative 13: 21.5% accurate, 862.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 31.9%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 21.1%

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

Reproduce

herbie shell --seed 2025019 
(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)))))))