Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.0s

Alternatives: 13

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_3 := eh \cdot t\_1\\ t_4 := \cos t \cdot ew\\ t_5 := \left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ t_6 := \tan^{-1} \left(t\_1 \cdot eh\right)\\ \mathbf{if}\;t\_5 \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(t\_1 \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(t\_3 \cdot eh, \sin t, t\_4\right)}{\cosh \sinh^{-1} t\_3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_6 \cdot \sin t, eh, t\_4 \cdot \cos t\_6\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (atan (/ (* eh (tan t)) (- ew))))
        (t_3 (* eh t_1))
        (t_4 (* (cos t) ew))
        (t_5 (- (* (* ew (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2))))
        (t_6 (atan (* t_1 eh))))
   (if (<= t_5 -1e+72)
     (fabs
      (*
       (fma
        (/ eh ew)
        (* (- (sin t)) (tanh (asinh (* t_1 (- eh)))))
        (* (cos (atan (/ (* eh t) ew))) (cos t)))
       ew))
     (if (<= t_5 -2e-300)
       (/ (- (fma (* t_3 eh) (sin t) t_4)) (cosh (asinh t_3)))
       (fma (* (sin t_6) (sin t)) eh (* t_4 (cos t_6)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = atan(((eh * tan(t)) / -ew));
	double t_3 = eh * t_1;
	double t_4 = cos(t) * ew;
	double t_5 = ((ew * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double t_6 = atan((t_1 * eh));
	double tmp;
	if (t_5 <= -1e+72) {
		tmp = fabs((fma((eh / ew), (-sin(t) * tanh(asinh((t_1 * -eh)))), (cos(atan(((eh * t) / ew))) * cos(t))) * ew));
	} else if (t_5 <= -2e-300) {
		tmp = -fma((t_3 * eh), sin(t), t_4) / cosh(asinh(t_3));
	} else {
		tmp = fma((sin(t_6) * sin(t)), eh, (t_4 * cos(t_6)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_3 = Float64(eh * t_1)
	t_4 = Float64(cos(t) * ew)
	t_5 = Float64(Float64(Float64(ew * cos(t)) * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	t_6 = atan(Float64(t_1 * eh))
	tmp = 0.0
	if (t_5 <= -1e+72)
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(Float64(-sin(t)) * tanh(asinh(Float64(t_1 * Float64(-eh))))), Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t))) * ew));
	elseif (t_5 <= -2e-300)
		tmp = Float64(Float64(-fma(Float64(t_3 * eh), sin(t), t_4)) / cosh(asinh(t_3)));
	else
		tmp = fma(Float64(sin(t_6) * sin(t)), eh, Float64(t_4 * cos(t_6)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[ArcTan[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -1e+72], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * N[Tanh[N[ArcSinh[N[(t$95$1 * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, -2e-300], N[((-N[(N[(t$95$3 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$4), $MachinePrecision]) / N[Cosh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[t$95$6], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(t$95$4 * N[Cos[t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999944e71

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   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 87.4%

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

   7. Applied rewrites 87.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.2%

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

   if -9.99999999999999944e71 < (-.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))))) < -2.00000000000000005e-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 80.0%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 80.6

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   6. Applied rewrites 93.6%

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

   if -2.00000000000000005e-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 99.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 99.8

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{+72}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_3 := \frac{\tan t}{ew}\\ t_4 := eh \cdot t\_3\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(t\_4 \cdot eh, \sin t, t\_1\right)}{\cosh \sinh^{-1} t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{\cosh \sinh^{-1} \left(t\_3 \cdot eh\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew))
        (t_2 (atan (/ (* eh (tan t)) (- ew))))
        (t_3 (/ (tan t) ew))
        (t_4 (* eh t_3)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2)))
        -2e-300)
     (/ (- (fma (* t_4 eh) (sin t) t_1)) (cosh (asinh t_4)))
     (/
      (+ t_1 (* (* (sin t) eh) (* (/ eh ew) (tan t))))
      (cosh (asinh (* t_3 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = atan(((eh * tan(t)) / -ew));
	double t_3 = tan(t) / ew;
	double t_4 = eh * t_3;
	double tmp;
	if ((((ew * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-300) {
		tmp = -fma((t_4 * eh), sin(t), t_1) / cosh(asinh(t_4));
	} else {
		tmp = (t_1 + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / cosh(asinh((t_3 * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_3 = Float64(tan(t) / ew)
	t_4 = Float64(eh * t_3)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-300)
		tmp = Float64(Float64(-fma(Float64(t_4 * eh), sin(t), t_1)) / cosh(asinh(t_4)));
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / cosh(asinh(Float64(t_3 * eh))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$4 = N[(eh * t$95$3), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-300], N[((-N[(N[(t$95$4 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$1), $MachinePrecision]) / N[Cosh[N[ArcSinh[t$95$4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$3 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 65.8

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   6. Applied rewrites 80.1%

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

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 78.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ t_2 := \mathsf{fma}\left(t\_1 \cdot eh, \sin t, \cos t \cdot ew\right)\\ t_3 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_4 := \cosh \sinh^{-1} t\_1\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-t\_2}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_2}{t\_4}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew)))
        (t_2 (fma (* t_1 eh) (sin t) (* (cos t) ew)))
        (t_3 (atan (/ (* eh (tan t)) (- ew))))
        (t_4 (cosh (asinh t_1))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_3)) (* (* eh (sin t)) (sin t_3)))
        -2e-300)
     (/ (- t_2) t_4)
     (/ t_2 t_4))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double t_2 = fma((t_1 * eh), sin(t), (cos(t) * ew));
	double t_3 = atan(((eh * tan(t)) / -ew));
	double t_4 = cosh(asinh(t_1));
	double tmp;
	if ((((ew * cos(t)) * cos(t_3)) - ((eh * sin(t)) * sin(t_3))) <= -2e-300) {
		tmp = -t_2 / t_4;
	} else {
		tmp = t_2 / t_4;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	t_2 = fma(Float64(t_1 * eh), sin(t), Float64(cos(t) * ew))
	t_3 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_4 = cosh(asinh(t_1))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3))) <= -2e-300)
		tmp = Float64(Float64(-t_2) / t_4);
	else
		tmp = Float64(t_2 / t_4);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-300], N[((-t$95$2) / t$95$4), $MachinePrecision], N[(t$95$2 / t$95$4), $MachinePrecision]]]]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

      1. lift-*.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-square-sqrt 65.8

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   6. Applied rewrites 80.1%

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

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. sqr-neg-rev N/A

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

      3. distribute-rgt-neg-out N/A

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

   6. Applied rewrites 78.1%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-ew\right) \cdot \mathsf{fma}\left(\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \sin t, \frac{-0.5}{-ew}, \cos t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-300)
     (*
      (- ew)
      (fma (* (* (* (/ (tan t) ew) eh) eh) (sin t)) (/ -0.5 (- ew)) (cos t)))
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300) {
		tmp = -ew * fma(((((tan(t) / ew) * eh) * eh) * sin(t)), (-0.5 / -ew), cos(t));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = Float64(Float64(-ew) * fma(Float64(Float64(Float64(Float64(tan(t) / ew) * eh) * eh) * sin(t)), Float64(-0.5 / Float64(-ew)), cos(t)));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-300], N[((-ew) * N[(N[(N[(N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(-0.5 / (-ew)), $MachinePrecision] + N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

   7. Applied rewrites 49.6%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \mathsf{fma}\left(-\cos t, -1, -\frac{\frac{\left(eh \cdot eh\right) \cdot {\sin t}^{2}}{\cos t} \cdot -0.5}{ew \cdot ew}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 64.5%

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

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.6%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-ew\right) \cdot \mathsf{fma}\left(\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \sin t, \frac{-0.5}{-ew}, \cos t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 62.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-300)
     (* (- (cos t)) ew)
     (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300) {
		tmp = -cos(t) * ew;
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-2d-300)) then
        tmp = -cos(t) * ew
    else
        tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -2e-300) {
		tmp = -Math.cos(t) * ew;
	} else {
		tmp = ((Math.cos(t) * ew) + ((Math.sin(t) * eh) * ((eh / ew) * Math.tan(t)))) / 1.0;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -2e-300:
		tmp = -math.cos(t) * ew
	else:
		tmp = ((math.cos(t) * ew) + ((math.sin(t) * eh) * ((eh / ew) * math.tan(t)))) / 1.0
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = Float64(Float64(-cos(t)) * ew);
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = -cos(t) * ew;
	else
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-300], N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto ew \cdot \color{blue}{-1} \]

      3. lower-*.f64 47.5

         \[\leadsto \color{blue}{ew \cdot -1} \]

   7. Applied rewrites 47.5%

      \[\leadsto \color{blue}{ew \cdot -1} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos t \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot ew} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos t\right)} \cdot ew \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \cos t\right) \cdot ew \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot \cos t\right) \cdot ew \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \cos t\right) \cdot ew} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos t\right)\right)} \cdot ew \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\cos t\right)} \cdot ew \]

      8. lower-cos.f64 64.0

         \[\leadsto \left(-\color{blue}{\cos t}\right) \cdot ew \]

   10. Applied rewrites 64.0%

       \[\leadsto \color{blue}{\left(-\cos t\right) \cdot ew} \]

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 78.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 60.6%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-300)
     (* (- (cos t)) ew)
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300) {
		tmp = -cos(t) * ew;
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-2d-300)) then
        tmp = -cos(t) * ew
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -2e-300) {
		tmp = -Math.cos(t) * ew;
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -2e-300:
		tmp = -math.cos(t) * ew
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = Float64(Float64(-cos(t)) * ew);
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = -cos(t) * ew;
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-300], N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto ew \cdot \color{blue}{-1} \]

      3. lower-*.f64 47.5

         \[\leadsto \color{blue}{ew \cdot -1} \]

   7. Applied rewrites 47.5%

      \[\leadsto \color{blue}{ew \cdot -1} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos t \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot ew} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos t\right)} \cdot ew \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \cos t\right) \cdot ew \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot \cos t\right) \cdot ew \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \cos t\right) \cdot ew} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos t\right)\right)} \cdot ew \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\cos t\right)} \cdot ew \]

      8. lower-cos.f64 64.0

         \[\leadsto \left(-\color{blue}{\cos t}\right) \cdot ew \]

   10. Applied rewrites 64.0%

       \[\leadsto \color{blue}{\left(-\cos t\right) \cdot ew} \]

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 0.0

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

   6. Applied rewrites 77.7%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.2

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

   9. Applied rewrites 60.2%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -2 \cdot 10^{-300}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -2e-300)
     (- ew)
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300) {
		tmp = -ew;
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-2d-300)) then
        tmp = -ew
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -2e-300) {
		tmp = -ew;
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -2e-300:
		tmp = -ew
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = Float64(-ew);
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -2e-300)
		tmp = -ew;
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -2e-300], (-ew), N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

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))))) < -2.00000000000000005e-300

   1. Initial program 99.9%

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

   4. Applied rewrites 65.4%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto ew \cdot \color{blue}{-1} \]

      3. lower-*.f64 47.5

         \[\leadsto \color{blue}{ew \cdot -1} \]

   7. Applied rewrites 47.5%

      \[\leadsto \color{blue}{ew \cdot -1} \]

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 47.5

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

   10. Applied rewrites 47.5%

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

   if -2.00000000000000005e-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 99.8%

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

   4. Applied rewrites 0.0%

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

      1. lift-*.f64 N/A

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

      2. pow2 N/A

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

      3. lower-pow.f64 0.0

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

   6. Applied rewrites 77.7%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 60.2

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

   9. Applied rewrites 60.2%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -2 \cdot 10^{-300}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \cos t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))))
   (if (<= eh 1.55e+110)
     (fabs
      (*
       (fma
        (/ eh ew)
        (* (- (sin t)) (tanh (asinh (* (/ (tan t) ew) (- eh)))))
        (* (cos (atan (/ (* eh t) ew))) (cos t)))
       ew))
     (fabs (* t_1 (sin (atan (/ t_1 (* ew (cos t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double tmp;
	if (eh <= 1.55e+110) {
		tmp = fabs((fma((eh / ew), (-sin(t) * tanh(asinh(((tan(t) / ew) * -eh)))), (cos(atan(((eh * t) / ew))) * cos(t))) * ew));
	} else {
		tmp = fabs((t_1 * sin(atan((t_1 / (ew * cos(t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if (eh <= 1.55e+110)
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(Float64(-sin(t)) * tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh))))), Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t))) * ew));
	else
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / Float64(ew * cos(t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 1.55e+110], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.55000000000000009e110

   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 95.4%

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

   7. Applied rewrites 95.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 87.0%

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

   if 1.55000000000000009e110 < eh

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   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 59.5%

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

   7. Applied rewrites 59.5%

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

   8. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 81.7

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

   10. Applied rewrites 81.7%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 1.55 \cdot 10^{+110}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -4.5 \cdot 10^{+94} \lor \neg \left(eh \leq 4 \cdot 10^{+56}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))))
   (if (or (<= eh -4.5e+94) (not (<= eh 4e+56)))
     (fabs (* t_1 (sin (atan (/ t_1 (* ew (cos t)))))))
     (fabs
      (* (cos (atan (* (/ (- (sin t)) ew) (/ eh (cos t))))) (* (cos t) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double tmp;
	if ((eh <= -4.5e+94) || !(eh <= 4e+56)) {
		tmp = fabs((t_1 * sin(atan((t_1 / (ew * cos(t)))))));
	} else {
		tmp = fabs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -eh * sin(t)
    if ((eh <= (-4.5d+94)) .or. (.not. (eh <= 4d+56))) then
        tmp = abs((t_1 * sin(atan((t_1 / (ew * cos(t)))))))
    else
        tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)))
    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 tmp;
	if ((eh <= -4.5e+94) || !(eh <= 4e+56)) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((t_1 / (ew * Math.cos(t)))))));
	} else {
		tmp = Math.abs((Math.cos(Math.atan(((-Math.sin(t) / ew) * (eh / Math.cos(t))))) * (Math.cos(t) * ew)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh * math.sin(t)
	tmp = 0
	if (eh <= -4.5e+94) or not (eh <= 4e+56):
		tmp = math.fabs((t_1 * math.sin(math.atan((t_1 / (ew * math.cos(t)))))))
	else:
		tmp = math.fabs((math.cos(math.atan(((-math.sin(t) / ew) * (eh / math.cos(t))))) * (math.cos(t) * ew)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -4.5e+94) || !(eh <= 4e+56))
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / Float64(ew * cos(t)))))));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t))))) * Float64(cos(t) * ew)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh * sin(t);
	tmp = 0.0;
	if ((eh <= -4.5e+94) || ~((eh <= 4e+56)))
		tmp = abs((t_1 * sin(atan((t_1 / (ew * cos(t)))))));
	else
		tmp = abs((cos(atan(((-sin(t) / ew) * (eh / cos(t))))) * (cos(t) * ew)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -4.5e+94], N[Not[LessEqual[eh, 4e+56]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.49999999999999972e94 or 4.00000000000000037e56 < eh

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   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 73.7%

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

   7. Applied rewrites 73.7%

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

   8. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 75.7

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

   10. Applied rewrites 75.7%

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

   if -4.49999999999999972e94 < eh < 4.00000000000000037e56

   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 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 83.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+94} \lor \neg \left(eh \leq 4 \cdot 10^{+56}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 65.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;ew \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{elif}\;ew \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t))))
   (if (<= ew -3.4e-77)
     (* (- (cos t)) ew)
     (if (<= ew 2.3e-18)
       (fabs (* t_1 (sin (atan (/ t_1 (* ew (cos t)))))))
       (/ (+ (* (cos t) ew) (* (* (sin t) eh) (* (/ eh ew) (tan t)))) 1.0)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double tmp;
	if (ew <= -3.4e-77) {
		tmp = -cos(t) * ew;
	} else if (ew <= 2.3e-18) {
		tmp = fabs((t_1 * sin(atan((t_1 / (ew * cos(t)))))));
	} else {
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -eh * sin(t)
    if (ew <= (-3.4d-77)) then
        tmp = -cos(t) * ew
    else if (ew <= 2.3d-18) then
        tmp = abs((t_1 * sin(atan((t_1 / (ew * cos(t)))))))
    else
        tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0d0
    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 tmp;
	if (ew <= -3.4e-77) {
		tmp = -Math.cos(t) * ew;
	} else if (ew <= 2.3e-18) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((t_1 / (ew * Math.cos(t)))))));
	} else {
		tmp = ((Math.cos(t) * ew) + ((Math.sin(t) * eh) * ((eh / ew) * Math.tan(t)))) / 1.0;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -eh * math.sin(t)
	tmp = 0
	if ew <= -3.4e-77:
		tmp = -math.cos(t) * ew
	elif ew <= 2.3e-18:
		tmp = math.fabs((t_1 * math.sin(math.atan((t_1 / (ew * math.cos(t)))))))
	else:
		tmp = ((math.cos(t) * ew) + ((math.sin(t) * eh) * ((eh / ew) * math.tan(t)))) / 1.0
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if (ew <= -3.4e-77)
		tmp = Float64(Float64(-cos(t)) * ew);
	elseif (ew <= 2.3e-18)
		tmp = abs(Float64(t_1 * sin(atan(Float64(t_1 / Float64(ew * cos(t)))))));
	else
		tmp = Float64(Float64(Float64(cos(t) * ew) + Float64(Float64(sin(t) * eh) * Float64(Float64(eh / ew) * tan(t)))) / 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -eh * sin(t);
	tmp = 0.0;
	if (ew <= -3.4e-77)
		tmp = -cos(t) * ew;
	elseif (ew <= 2.3e-18)
		tmp = abs((t_1 * sin(atan((t_1 / (ew * cos(t)))))));
	else
		tmp = ((cos(t) * ew) + ((sin(t) * eh) * ((eh / ew) * tan(t)))) / 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -3.4e-77], N[((-N[Cos[t], $MachinePrecision]) * ew), $MachinePrecision], If[LessEqual[ew, 2.3e-18], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.39999999999999983e-77

   1. Initial program 99.9%

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

   4. Applied rewrites 56.4%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto ew \cdot \color{blue}{-1} \]

      3. lower-*.f64 54.9

         \[\leadsto \color{blue}{ew \cdot -1} \]

   7. Applied rewrites 54.9%

      \[\leadsto \color{blue}{ew \cdot -1} \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos t \cdot {\left(\sqrt{-1}\right)}^{2}\right) \cdot ew} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left({\left(\sqrt{-1}\right)}^{2} \cdot \cos t\right)} \cdot ew \]

      3. unpow2 N/A

         \[\leadsto \left(\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot \cos t\right) \cdot ew \]

      4. rem-square-sqrt N/A

         \[\leadsto \left(\color{blue}{-1} \cdot \cos t\right) \cdot ew \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \cos t\right) \cdot ew} \]

      6. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\cos t\right)\right)} \cdot ew \]

      7. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\cos t\right)} \cdot ew \]

      8. lower-cos.f64 65.1

         \[\leadsto \left(-\color{blue}{\cos t}\right) \cdot ew \]

   10. Applied rewrites 65.1%

       \[\leadsto \color{blue}{\left(-\cos t\right) \cdot ew} \]

   if -3.39999999999999983e-77 < ew < 2.3000000000000001e-18

   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 74.7%

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

   7. Applied rewrites 74.7%

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

   8. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 71.4

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

   10. Applied rewrites 71.4%

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

   if 2.3000000000000001e-18 < 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 78.8%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 70.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.4 \cdot 10^{-77}:\\ \;\;\;\;\left(-\cos t\right) \cdot ew\\ \mathbf{elif}\;ew \leq 2.3 \cdot 10^{-18}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{eh}{ew} \cdot \tan t\right)}{1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.7% accurate, 18.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 1.2 \cdot 10^{-295}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, -0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5, ew\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 1.2e-295)
   (- ew)
   (fma (* t t) (- (* -0.5 ew) (* (/ (* eh eh) ew) -0.5)) ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.2e-295) {
		tmp = -ew;
	} else {
		tmp = fma((t * t), ((-0.5 * ew) - (((eh * eh) / ew) * -0.5)), ew);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 1.2e-295)
		tmp = Float64(-ew);
	else
		tmp = fma(Float64(t * t), Float64(Float64(-0.5 * ew) - Float64(Float64(Float64(eh * eh) / ew) * -0.5)), ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 1.2e-295], (-ew), N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * ew), $MachinePrecision] - N[(N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.1999999999999999e-295

   1. Initial program 99.9%

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

   4. Applied rewrites 49.2%

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

   5. Taylor expanded in t around 0

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

      1. unpow2 N/A

         \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

      2. rem-square-sqrt N/A

         \[\leadsto ew \cdot \color{blue}{-1} \]

      3. lower-*.f64 43.6

         \[\leadsto \color{blue}{ew \cdot -1} \]

   7. Applied rewrites 43.6%

      \[\leadsto \color{blue}{ew \cdot -1} \]

   8. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

      3. rem-square-sqrt N/A

         \[\leadsto \color{blue}{-1} \cdot ew \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

      5. lower-neg.f64 43.6

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

   10. Applied rewrites 43.6%

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

   if 1.1999999999999999e-295 < 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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. distribute-rgt-out N/A

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

      8. metadata-eval N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 32.4

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

   7. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot t, -0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5, ew\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 1.2 \cdot 10^{-295}:\\ \;\;\;\;-ew\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot t, -0.5 \cdot ew - \frac{eh \cdot eh}{ew} \cdot -0.5, ew\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 22.0% accurate, 287.3× 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 Float64(-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 32.5%

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

5. Taylor expanded in t around 0

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

   1. unpow2 N/A

      \[\leadsto ew \cdot \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \]

   2. rem-square-sqrt N/A

      \[\leadsto ew \cdot \color{blue}{-1} \]

   3. lower-*.f64 26.2

      \[\leadsto \color{blue}{ew \cdot -1} \]

7. Applied rewrites 26.2%

   \[\leadsto \color{blue}{ew \cdot -1} \]

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

      \[\leadsto \color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot ew} \]

   2. unpow2 N/A

      \[\leadsto \color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right)} \cdot ew \]

   3. rem-square-sqrt N/A

      \[\leadsto \color{blue}{-1} \cdot ew \]

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(ew\right)} \]

   5. lower-neg.f64 26.2

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

10. Applied rewrites 26.2%

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

Reproduce

herbie shell --seed 2024356 
(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)))))))