Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. 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. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 80.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := eh \cdot t\_1\\ t_3 := \sinh^{-1} t\_2\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := \left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4\\ t_6 := \cos t \cdot ew\\ t_7 := \frac{\mathsf{fma}\left(t\_2, \sin t \cdot eh, t\_6\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot eh, \sin t, t\_6\right)}{-\cosh t\_3}\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh t\_3 \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* eh t_1))
        (t_3 (asinh t_2))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4))))
        (t_6 (* (cos t) ew))
        (t_7 (/ (fma t_2 (* (sin t) eh) t_6) (cosh (asinh (* t_1 eh))))))
   (if (<= t_5 -2e-269)
     (/ (fma (* t_2 eh) (sin t) t_6) (- (cosh t_3)))
     (if (<= t_5 5e+82)
       t_7
       (if (<= t_5 5e+202)
         (pow (sqrt (fma (* (tanh t_3) (sin t)) eh ew)) 2.0)
         t_7)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = eh * t_1;
	double t_3 = asinh(t_2);
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = ((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4));
	double t_6 = cos(t) * ew;
	double t_7 = fma(t_2, (sin(t) * eh), t_6) / cosh(asinh((t_1 * eh)));
	double tmp;
	if (t_5 <= -2e-269) {
		tmp = fma((t_2 * eh), sin(t), t_6) / -cosh(t_3);
	} else if (t_5 <= 5e+82) {
		tmp = t_7;
	} else if (t_5 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(t_3) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(eh * t_1)
	t_3 = asinh(t_2)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4)))
	t_6 = Float64(cos(t) * ew)
	t_7 = Float64(fma(t_2, Float64(sin(t) * eh), t_6) / cosh(asinh(Float64(t_1 * eh))))
	tmp = 0.0
	if (t_5 <= -2e-269)
		tmp = Float64(fma(Float64(t_2 * eh), sin(t), t_6) / Float64(-cosh(t_3)));
	elseif (t_5 <= 5e+82)
		tmp = t_7;
	elseif (t_5 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(t_3) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_7;
	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[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[ArcSinh[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$2 * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + t$95$6), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e-269], N[(N[(N[(t$95$2 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$6), $MachinePrecision] / (-N[Cosh[t$95$3], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$5, 5e+82], t$95$7, If[LessEqual[t$95$5, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[t$95$3], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$7]]]]]]]]]]

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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. rem-square-sqrt 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. sqrt-unprod N/A

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

      3. sqr-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

   5. 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 -1.9999999999999999e-269 < (-.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))))) < 5.00000000000000015e82 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if 5.00000000000000015e82 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 63.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.0%

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

Alternative 3: 73.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := eh \cdot t\_1\\ t_3 := eh \cdot \sin t\\ t_4 := ew \cdot \cos t\\ t_5 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_6 := t\_4 \cdot \cos t\_5 - t\_3 \cdot \sin t\_5\\ t_7 := \frac{\mathsf{fma}\left(t\_2, \sin t \cdot eh, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\left|\frac{t\_4}{\cos \tan^{-1} \left(\frac{t\_3}{t\_4}\right)}\right|\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} t\_2 \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* eh t_1))
        (t_3 (* eh (sin t)))
        (t_4 (* ew (cos t)))
        (t_5 (atan (/ (* (- eh) (tan t)) ew)))
        (t_6 (- (* t_4 (cos t_5)) (* t_3 (sin t_5))))
        (t_7
         (/
          (fma t_2 (* (sin t) eh) (* (cos t) ew))
          (cosh (asinh (* t_1 eh))))))
   (if (<= t_6 -2e-269)
     (fabs (/ t_4 (cos (atan (/ t_3 t_4)))))
     (if (<= t_6 5e+82)
       t_7
       (if (<= t_6 5e+202)
         (pow (sqrt (fma (* (tanh (asinh t_2)) (sin t)) eh ew)) 2.0)
         t_7)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = eh * t_1;
	double t_3 = eh * sin(t);
	double t_4 = ew * cos(t);
	double t_5 = atan(((-eh * tan(t)) / ew));
	double t_6 = (t_4 * cos(t_5)) - (t_3 * sin(t_5));
	double t_7 = fma(t_2, (sin(t) * eh), (cos(t) * ew)) / cosh(asinh((t_1 * eh)));
	double tmp;
	if (t_6 <= -2e-269) {
		tmp = fabs((t_4 / cos(atan((t_3 / t_4)))));
	} else if (t_6 <= 5e+82) {
		tmp = t_7;
	} else if (t_6 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh(t_2)) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(eh * t_1)
	t_3 = Float64(eh * sin(t))
	t_4 = Float64(ew * cos(t))
	t_5 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_6 = Float64(Float64(t_4 * cos(t_5)) - Float64(t_3 * sin(t_5)))
	t_7 = Float64(fma(t_2, Float64(sin(t) * eh), Float64(cos(t) * ew)) / cosh(asinh(Float64(t_1 * eh))))
	tmp = 0.0
	if (t_6 <= -2e-269)
		tmp = abs(Float64(t_4 / cos(atan(Float64(t_3 / t_4)))));
	elseif (t_6 <= 5e+82)
		tmp = t_7;
	elseif (t_6 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(t_2)) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_7;
	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[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$4 * N[Cos[t$95$5], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[(t$95$2 * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e-269], N[Abs[N[(t$95$4 / N[Cos[N[ArcTan[N[(t$95$3 / t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$6, 5e+82], t$95$7, If[LessEqual[t$95$6, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$7]]]]]]]]]]

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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if -1.9999999999999999e-269 < (-.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))))) < 5.00000000000000015e82 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.2

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

   5. Applied rewrites 83.2%

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

   if 5.00000000000000015e82 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 63.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.0%

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

Alternative 4: 72.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := eh \cdot \sin t\\ t_5 := t\_2 \cdot \cos t\_3 - t\_4 \cdot \sin t\_3\\ t_6 := eh \cdot t\_1\\ t_7 := \frac{\mathsf{fma}\left(\cos t, ew, \left(t\_6 \cdot eh\right) \cdot \sin t\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\left|\frac{t\_2}{\cos \tan^{-1} \left(\frac{t\_4}{t\_2}\right)}\right|\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+82}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} t\_6 \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (* eh (sin t)))
        (t_5 (- (* t_2 (cos t_3)) (* t_4 (sin t_3))))
        (t_6 (* eh t_1))
        (t_7
         (/
          (fma (cos t) ew (* (* t_6 eh) (sin t)))
          (cosh (asinh (* t_1 eh))))))
   (if (<= t_5 -2e-269)
     (fabs (/ t_2 (cos (atan (/ t_4 t_2)))))
     (if (<= t_5 5e+82)
       t_7
       (if (<= t_5 5e+202)
         (pow (sqrt (fma (* (tanh (asinh t_6)) (sin t)) eh ew)) 2.0)
         t_7)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = eh * sin(t);
	double t_5 = (t_2 * cos(t_3)) - (t_4 * sin(t_3));
	double t_6 = eh * t_1;
	double t_7 = fma(cos(t), ew, ((t_6 * eh) * sin(t))) / cosh(asinh((t_1 * eh)));
	double tmp;
	if (t_5 <= -2e-269) {
		tmp = fabs((t_2 / cos(atan((t_4 / t_2)))));
	} else if (t_5 <= 5e+82) {
		tmp = t_7;
	} else if (t_5 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh(t_6)) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = Float64(eh * sin(t))
	t_5 = Float64(Float64(t_2 * cos(t_3)) - Float64(t_4 * sin(t_3)))
	t_6 = Float64(eh * t_1)
	t_7 = Float64(fma(cos(t), ew, Float64(Float64(t_6 * eh) * sin(t))) / cosh(asinh(Float64(t_1 * eh))))
	tmp = 0.0
	if (t_5 <= -2e-269)
		tmp = abs(Float64(t_2 / cos(atan(Float64(t_4 / t_2)))));
	elseif (t_5 <= 5e+82)
		tmp = t_7;
	elseif (t_5 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(t_6)) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_7;
	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[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$7 = N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(t$95$6 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -2e-269], N[Abs[N[(t$95$2 / N[Cos[N[ArcTan[N[(t$95$4 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 5e+82], t$95$7, If[LessEqual[t$95$5, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[t$95$6], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$7]]]]]]]]]]

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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 65.2

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

   8. Applied rewrites 65.2%

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

   if -1.9999999999999999e-269 < (-.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))))) < 5.00000000000000015e82 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 70.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 70.1

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 80.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 80.4

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

   5. Applied rewrites 80.4%

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

   if 5.00000000000000015e82 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 63.5%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.0%

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

Alternative 5: 71.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := eh \cdot \sin t\\ t_3 := eh \cdot t\_1\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := ew \cdot \cos t\\ t_6 := t\_5 \cdot \cos t\_4 - t\_2 \cdot \sin t\_4\\ t_7 := \left|\frac{t\_5}{\cos \tan^{-1} \left(\frac{t\_2}{t\_5}\right)}\right|\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1 \cdot eh, \sin t \cdot eh, \cos t \cdot ew\right)}{\sqrt{1 + {t\_3}^{2}}}\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} t\_3 \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* eh (sin t)))
        (t_3 (* eh t_1))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (* ew (cos t)))
        (t_6 (- (* t_5 (cos t_4)) (* t_2 (sin t_4))))
        (t_7 (fabs (/ t_5 (cos (atan (/ t_2 t_5)))))))
   (if (<= t_6 -2e-269)
     t_7
     (if (<= t_6 2e+83)
       (/
        (fma (* t_1 eh) (* (sin t) eh) (* (cos t) ew))
        (sqrt (+ 1.0 (pow t_3 2.0))))
       (if (<= t_6 5e+202)
         (pow (sqrt (fma (* (tanh (asinh t_3)) (sin t)) eh ew)) 2.0)
         t_7)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = eh * sin(t);
	double t_3 = eh * t_1;
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = ew * cos(t);
	double t_6 = (t_5 * cos(t_4)) - (t_2 * sin(t_4));
	double t_7 = fabs((t_5 / cos(atan((t_2 / t_5)))));
	double tmp;
	if (t_6 <= -2e-269) {
		tmp = t_7;
	} else if (t_6 <= 2e+83) {
		tmp = fma((t_1 * eh), (sin(t) * eh), (cos(t) * ew)) / sqrt((1.0 + pow(t_3, 2.0)));
	} else if (t_6 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh(t_3)) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(eh * sin(t))
	t_3 = Float64(eh * t_1)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(ew * cos(t))
	t_6 = Float64(Float64(t_5 * cos(t_4)) - Float64(t_2 * sin(t_4)))
	t_7 = abs(Float64(t_5 / cos(atan(Float64(t_2 / t_5)))))
	tmp = 0.0
	if (t_6 <= -2e-269)
		tmp = t_7;
	elseif (t_6 <= 2e+83)
		tmp = Float64(fma(Float64(t_1 * eh), Float64(sin(t) * eh), Float64(cos(t) * ew)) / sqrt(Float64(1.0 + (t_3 ^ 2.0))));
	elseif (t_6 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(t_3)) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_7;
	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[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Abs[N[(t$95$5 / N[Cos[N[ArcTan[N[(t$95$2 / t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, -2e-269], t$95$7, If[LessEqual[t$95$6, 2e+83], N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$7]]]]]]]]]]

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))))) < -1.9999999999999999e-269 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if -1.9999999999999999e-269 < (-.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))))) < 2.00000000000000006e83

   1. Initial program 99.8%

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

   3. Applied rewrites 79.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 73.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.6

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 82.6

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if 2.00000000000000006e83 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.1%

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

Alternative 6: 70.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \frac{\tan t}{ew}\\ t_3 := eh \cdot t\_2\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := ew \cdot \cos t\\ t_6 := t\_5 \cdot \cos t\_4 - t\_1 \cdot \sin t\_4\\ t_7 := \left|\frac{t\_5}{\cos \tan^{-1} \left(\frac{t\_1}{t\_5}\right)}\right|\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;t\_7\\ \mathbf{elif}\;t\_6 \leq 2 \cdot 10^{+83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(t\_2 \cdot eh\right) \cdot eh\right) \cdot \sin t\right)}{\sqrt{1 + {t\_3}^{2}}}\\ \mathbf{elif}\;t\_6 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} t\_3 \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_7\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (/ (tan t) ew))
        (t_3 (* eh t_2))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (* ew (cos t)))
        (t_6 (- (* t_5 (cos t_4)) (* t_1 (sin t_4))))
        (t_7 (fabs (/ t_5 (cos (atan (/ t_1 t_5)))))))
   (if (<= t_6 -2e-269)
     t_7
     (if (<= t_6 2e+83)
       (/
        (fma (cos t) ew (* (* (* t_2 eh) eh) (sin t)))
        (sqrt (+ 1.0 (pow t_3 2.0))))
       (if (<= t_6 5e+202)
         (pow (sqrt (fma (* (tanh (asinh t_3)) (sin t)) eh ew)) 2.0)
         t_7)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = tan(t) / ew;
	double t_3 = eh * t_2;
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = ew * cos(t);
	double t_6 = (t_5 * cos(t_4)) - (t_1 * sin(t_4));
	double t_7 = fabs((t_5 / cos(atan((t_1 / t_5)))));
	double tmp;
	if (t_6 <= -2e-269) {
		tmp = t_7;
	} else if (t_6 <= 2e+83) {
		tmp = fma(cos(t), ew, (((t_2 * eh) * eh) * sin(t))) / sqrt((1.0 + pow(t_3, 2.0)));
	} else if (t_6 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh(t_3)) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_7;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(tan(t) / ew)
	t_3 = Float64(eh * t_2)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(ew * cos(t))
	t_6 = Float64(Float64(t_5 * cos(t_4)) - Float64(t_1 * sin(t_4)))
	t_7 = abs(Float64(t_5 / cos(atan(Float64(t_1 / t_5)))))
	tmp = 0.0
	if (t_6 <= -2e-269)
		tmp = t_7;
	elseif (t_6 <= 2e+83)
		tmp = Float64(fma(cos(t), ew, Float64(Float64(Float64(t_2 * eh) * eh) * sin(t))) / sqrt(Float64(1.0 + (t_3 ^ 2.0))));
	elseif (t_6 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(t_3)) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_7;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$3 = N[(eh * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(t$95$5 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Abs[N[(t$95$5 / N[Cos[N[ArcTan[N[(t$95$1 / t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$6, -2e-269], t$95$7, If[LessEqual[t$95$6, 2e+83], N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(t$95$2 * eh), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$6, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$7]]]]]]]]]]

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))))) < -1.9999999999999999e-269 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 65.1

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

   8. Applied rewrites 65.1%

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

   if -1.9999999999999999e-269 < (-.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))))) < 2.00000000000000006e83

   1. Initial program 99.8%

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

   3. Applied rewrites 79.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 73.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 73.5

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

   5. Applied rewrites 73.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 81.0

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 81.0

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

   7. Applied rewrites 81.0%

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

   if 2.00000000000000006e83 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 63.3%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.1%

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

Alternative 7: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := eh \cdot t\_1\\ t_3 := \cosh \sinh^{-1} t\_2\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := \left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4\\ t_6 := \cos t \cdot ew\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2 \cdot eh, \sin t, t\_6\right)}{-t\_3}\\ \mathbf{elif}\;t\_5 \leq 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(\left(-eh\right) \cdot \sin t\right) \cdot t\_1, -eh, t\_6\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \left(\frac{eh \cdot t}{ew}\right) \cdot \sin t, eh, \frac{t\_6}{t\_3}\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* eh t_1))
        (t_3 (cosh (asinh t_2)))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4))))
        (t_6 (* (cos t) ew)))
   (if (<= t_5 -2e-269)
     (/ (fma (* t_2 eh) (sin t) t_6) (- t_3))
     (if (<= t_5 1e+25)
       (/
        (fma (* (* (- eh) (sin t)) t_1) (- eh) t_6)
        (cosh (asinh (* t_1 eh))))
       (pow
        (sqrt (fma (* (tanh (/ (* eh t) ew)) (sin t)) eh (/ t_6 t_3)))
        2.0)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = eh * t_1;
	double t_3 = cosh(asinh(t_2));
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = ((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4));
	double t_6 = cos(t) * ew;
	double tmp;
	if (t_5 <= -2e-269) {
		tmp = fma((t_2 * eh), sin(t), t_6) / -t_3;
	} else if (t_5 <= 1e+25) {
		tmp = fma(((-eh * sin(t)) * t_1), -eh, t_6) / cosh(asinh((t_1 * eh)));
	} else {
		tmp = pow(sqrt(fma((tanh(((eh * t) / ew)) * sin(t)), eh, (t_6 / t_3))), 2.0);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(eh * t_1)
	t_3 = cosh(asinh(t_2))
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4)))
	t_6 = Float64(cos(t) * ew)
	tmp = 0.0
	if (t_5 <= -2e-269)
		tmp = Float64(fma(Float64(t_2 * eh), sin(t), t_6) / Float64(-t_3));
	elseif (t_5 <= 1e+25)
		tmp = Float64(fma(Float64(Float64(Float64(-eh) * sin(t)) * t_1), Float64(-eh), t_6) / cosh(asinh(Float64(t_1 * eh))));
	else
		tmp = sqrt(fma(Float64(tanh(Float64(Float64(eh * t) / ew)) * sin(t)), eh, Float64(t_6 / t_3))) ^ 2.0;
	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[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[t$95$5, -2e-269], N[(N[(N[(t$95$2 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$6), $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$5, 1e+25], N[(N[(N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * (-eh) + t$95$6), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[N[(N[(N[Tanh[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(t$95$6 / t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. rem-square-sqrt 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. sqrt-unprod N/A

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

      3. sqr-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

   5. 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 -1.9999999999999999e-269 < (-.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))))) < 1.00000000000000009e25

   1. Initial program 99.8%

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

   3. Applied rewrites 82.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. lower-fma.f64 93.1

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

   5. Applied rewrites 93.1%

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

   if 1.00000000000000009e25 < (-.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%

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

   3. Applied rewrites 59.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 84.7

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

   8. Applied rewrites 84.7%

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

Alternative 8: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := ew \cdot \cos t\\ t_4 := t\_3 \cdot \cos t\_2 - t\_1 \cdot \sin t\_2\\ t_5 := \left|\frac{t\_3}{\cos \tan^{-1} \left(\frac{t\_1}{t\_3}\right)}\right|\\ \mathbf{if}\;t\_4 \leq 10^{-175}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (* ew (cos t)))
        (t_4 (- (* t_3 (cos t_2)) (* t_1 (sin t_2))))
        (t_5 (fabs (/ t_3 (cos (atan (/ t_1 t_3)))))))
   (if (<= t_4 1e-175)
     t_5
     (if (<= t_4 5e+202)
       (pow
        (sqrt (fma (* (tanh (asinh (* eh (/ (tan t) ew)))) (sin t)) eh ew))
        2.0)
       t_5))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = ew * cos(t);
	double t_4 = (t_3 * cos(t_2)) - (t_1 * sin(t_2));
	double t_5 = fabs((t_3 / cos(atan((t_1 / t_3)))));
	double tmp;
	if (t_4 <= 1e-175) {
		tmp = t_5;
	} else if (t_4 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh((eh * (tan(t) / ew)))) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(ew * cos(t))
	t_4 = Float64(Float64(t_3 * cos(t_2)) - Float64(t_1 * sin(t_2)))
	t_5 = abs(Float64(t_3 / cos(atan(Float64(t_1 / t_3)))))
	tmp = 0.0
	if (t_4 <= 1e-175)
		tmp = t_5;
	elseif (t_4 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(Float64(eh * Float64(tan(t) / ew)))) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(t$95$3 / N[Cos[N[ArcTan[N[(t$95$1 / t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, 1e-175], t$95$5, If[LessEqual[t$95$4, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$5]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1e-175 or 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lift-cos.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if 1e-175 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 74.7%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 78.0%

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

Alternative 9: 59.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := t\_2 \cdot \cos t\_3 - t\_1 \cdot \sin t\_3\\ \mathbf{if}\;t\_4 \leq 5 \cdot 10^{-255}:\\ \;\;\;\;\left|\frac{ew}{\cos \tan^{-1} \left(\frac{t\_1}{t\_2}\right)}\right|\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+202}:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(\tanh \sinh^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \sin t, eh, ew\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (- (* t_2 (cos t_3)) (* t_1 (sin t_3)))))
   (if (<= t_4 5e-255)
     (fabs (/ ew (cos (atan (/ t_1 t_2)))))
     (if (<= t_4 5e+202)
       (pow
        (sqrt (fma (* (tanh (asinh (* eh (/ (tan t) ew)))) (sin t)) eh ew))
        2.0)
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = (t_2 * cos(t_3)) - (t_1 * sin(t_3));
	double tmp;
	if (t_4 <= 5e-255) {
		tmp = fabs((ew / cos(atan((t_1 / t_2)))));
	} else if (t_4 <= 5e+202) {
		tmp = pow(sqrt(fma((tanh(asinh((eh * (tan(t) / ew)))) * sin(t)), eh, ew)), 2.0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * sin(t_3)))
	tmp = 0.0
	if (t_4 <= 5e-255)
		tmp = abs(Float64(ew / cos(atan(Float64(t_1 / t_2)))));
	elseif (t_4 <= 5e+202)
		tmp = sqrt(fma(Float64(tanh(asinh(Float64(eh * Float64(tan(t) / ew)))) * sin(t)), eh, ew)) ^ 2.0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, 5e-255], N[Abs[N[(ew / N[Cos[N[ArcTan[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 5e+202], N[Power[N[Sqrt[N[(N[(N[Tanh[N[ArcSinh[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + ew), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], t$95$2]]]]]]

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))))) < 4.9999999999999996e-255

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 45.0

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

   8. Applied rewrites 45.0%

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

   if 4.9999999999999996e-255 < (-.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))))) < 4.9999999999999999e202

   1. Initial program 99.8%

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

   3. Applied rewrites 75.1%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. div-add N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 78.1%

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

   if 4.9999999999999999e202 < (-.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%

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

   3. Applied rewrites 50.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 66.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-/.f64 N/A

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

      6. lower-sqrt.f64 66.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 66.8

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

   7. Applied rewrites 66.8%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 63.8

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

   10. Applied rewrites 63.8%

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

Alternative 10: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \tan^{-1} \left(t\_1 \cdot eh\right)\\ t_3 := eh \cdot t\_1\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3 \cdot eh, \sin t, \cos t \cdot ew\right)}{-\cosh \sinh^{-1} t\_3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_2 \cdot \cos t, ew, \sin t\_2 \cdot \left(\sin t \cdot eh\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (atan (* t_1 eh)))
        (t_3 (* eh t_1))
        (t_4 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4)))
        -2e-269)
     (/ (fma (* t_3 eh) (sin t) (* (cos t) ew)) (- (cosh (asinh t_3))))
     (fma (* (cos t_2) (cos t)) ew (* (sin t_2) (* (sin t) eh))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = atan((t_1 * eh));
	double t_3 = eh * t_1;
	double t_4 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4))) <= -2e-269) {
		tmp = fma((t_3 * eh), sin(t), (cos(t) * ew)) / -cosh(asinh(t_3));
	} else {
		tmp = fma((cos(t_2) * cos(t)), ew, (sin(t_2) * (sin(t) * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = atan(Float64(t_1 * eh))
	t_3 = Float64(eh * t_1)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4))) <= -2e-269)
		tmp = Float64(fma(Float64(t_3 * eh), sin(t), Float64(cos(t) * ew)) / Float64(-cosh(asinh(t_3))));
	else
		tmp = fma(Float64(cos(t_2) * cos(t)), ew, Float64(sin(t_2) * Float64(sin(t) * eh)));
	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[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(eh * t$95$1), $MachinePrecision]}, Block[{t$95$4 = 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$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-269], N[(N[(N[(t$95$3 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / (-N[Cosh[N[ArcSinh[t$95$3], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew + N[(N[Sin[t$95$2], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. rem-square-sqrt 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. sqrt-unprod N/A

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

      3. sqr-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-neg.f64 N/A

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

   5. 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 -1.9999999999999999e-269 < (-.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%

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

   3. Applied rewrites 98.7%

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

Alternative 11: 52.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_2 \cdot \cos t\_3 - t\_1 \cdot \sin t\_3 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;\left|\frac{ew}{\cos \tan^{-1} \left(\frac{t\_1}{t\_2}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_2 (cos t_3)) (* t_1 (sin t_3))) -2e-269)
     (fabs (/ ew (cos (atan (/ t_1 t_2)))))
     t_2)))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -2e-269) {
		tmp = fabs((ew / cos(atan((t_1 / t_2)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = eh * sin(t)
    t_2 = ew * cos(t)
    t_3 = atan(((-eh * tan(t)) / ew))
    if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= (-2d-269)) then
        tmp = abs((ew / cos(atan((t_1 / t_2)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_2 * Math.cos(t_3)) - (t_1 * Math.sin(t_3))) <= -2e-269) {
		tmp = Math.abs((ew / Math.cos(Math.atan((t_1 / t_2)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	t_2 = ew * math.cos(t)
	t_3 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_2 * math.cos(t_3)) - (t_1 * math.sin(t_3))) <= -2e-269:
		tmp = math.fabs((ew / math.cos(math.atan((t_1 / t_2)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * sin(t_3))) <= -2e-269)
		tmp = abs(Float64(ew / cos(atan(Float64(t_1 / t_2)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	t_2 = ew * cos(t);
	t_3 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -2e-269)
		tmp = abs((ew / cos(atan((t_1 / t_2)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-269], N[Abs[N[(ew / N[Cos[N[ArcTan[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \color{blue}{\frac{\left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) \cdot \left(0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)\right) - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 \cdot \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Applied rewrites 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{0 - {\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-2}}{0 - \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}, \cos t \cdot ew, \left(-\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 44.2

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

   8. Applied rewrites 44.2%

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

   if -1.9999999999999999e-269 < (-.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%

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

   3. Applied rewrites 68.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-/.f64 N/A

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

      6. lower-sqrt.f64 78.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 60.8

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

   10. Applied rewrites 60.8%

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

Alternative 12: 40.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_2 \cdot \cos t\_3 - t\_1 \cdot \sin t\_3 \leq -1 \cdot 10^{-226}:\\ \;\;\;\;-1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_2 (cos t_3)) (* t_1 (sin t_3))) -1e-226) (* -1.0 t_1) t_2)))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -1e-226) {
		tmp = -1.0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = eh * sin(t)
    t_2 = ew * cos(t)
    t_3 = atan(((-eh * tan(t)) / ew))
    if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= (-1d-226)) then
        tmp = (-1.0d0) * t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_2 * Math.cos(t_3)) - (t_1 * Math.sin(t_3))) <= -1e-226) {
		tmp = -1.0 * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	t_2 = ew * math.cos(t)
	t_3 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_2 * math.cos(t_3)) - (t_1 * math.sin(t_3))) <= -1e-226:
		tmp = -1.0 * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * sin(t_3))) <= -1e-226)
		tmp = Float64(-1.0 * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	t_2 = ew * cos(t);
	t_3 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -1e-226)
		tmp = -1.0 * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-226], N[(-1.0 * t$95$1), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999921e-227

   1. Initial program 99.8%

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

   3. Applied rewrites 1.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 1.1

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 1.1

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

   5. Applied rewrites 1.1%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 1.1

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 1.1

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 20.5

         \[\leadsto -1 \cdot \left(eh \cdot \sin t\right) \]

   10. Applied rewrites 20.5%

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

   if -9.99999999999999921e-227 < (-.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%

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

   3. Applied rewrites 66.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-/.f64 N/A

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

      6. lower-sqrt.f64 75.8

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 59.3

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

   10. Applied rewrites 59.3%

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

Alternative 13: 41.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_2 \cdot \cos t\_3 - t\_1 \cdot \sin t\_3 \leq -2 \cdot 10^{-269}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_2 (cos t_3)) (* t_1 (sin t_3))) -2e-269) t_1 t_2)))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -2e-269) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = eh * sin(t)
    t_2 = ew * cos(t)
    t_3 = atan(((-eh * tan(t)) / ew))
    if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= (-2d-269)) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_2 * Math.cos(t_3)) - (t_1 * Math.sin(t_3))) <= -2e-269) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	t_2 = ew * math.cos(t)
	t_3 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_2 * math.cos(t_3)) - (t_1 * math.sin(t_3))) <= -2e-269:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_2 * cos(t_3)) - Float64(t_1 * sin(t_3))) <= -2e-269)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	t_2 = ew * cos(t);
	t_3 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_2 * cos(t_3)) - (t_1 * sin(t_3))) <= -2e-269)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-269], t$95$1, t$95$2]]]]

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))))) < -1.9999999999999999e-269

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 1.2

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 1.2

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

   5. Applied rewrites 1.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 1.2

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 1.2

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

   7. Applied rewrites 1.2%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \color{blue}{eh \cdot \sin t} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\sin t} \]

      2. lift-sin.f64 21.4

         \[\leadsto eh \cdot \sin t \]

   10. Applied rewrites 21.4%

       \[\leadsto \color{blue}{eh \cdot \sin t} \]

   if -1.9999999999999999e-269 < (-.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%

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

   3. Applied rewrites 68.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 78.0%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-/.f64 N/A

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

      6. lower-sqrt.f64 78.0

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 78.0

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 60.8

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

   10. Applied rewrites 60.8%

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

Alternative 14: 77.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := -1 \cdot \left(eh \cdot \sin t\right)\\ \mathbf{if}\;eh \leq -2.8 \cdot 10^{+147}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2.75 \cdot 10^{+144}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* -1.0 (* eh (sin t)))))
   (if (<= eh -2.8e+147)
     t_2
     (if (<= eh 2.75e+144)
       (fabs
        (/
         (fma (sin t) (* t_1 (* eh eh)) (* (cos t) ew))
         (cosh (asinh (* t_1 eh)))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = -1.0 * (eh * sin(t));
	double tmp;
	if (eh <= -2.8e+147) {
		tmp = t_2;
	} else if (eh <= 2.75e+144) {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), (cos(t) * ew)) / cosh(asinh((t_1 * eh)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(-1.0 * Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -2.8e+147)
		tmp = t_2;
	elseif (eh <= 2.75e+144)
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), Float64(cos(t) * ew)) / cosh(asinh(Float64(t_1 * eh)))));
	else
		tmp = t_2;
	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[(-1.0 * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.8e+147], t$95$2, If[LessEqual[eh, 2.75e+144], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.8000000000000001e147 or 2.75000000000000011e144 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 3.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 2.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 2.5

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

   5. Applied rewrites 2.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 19.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 19.8

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

   7. Applied rewrites 19.8%

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

   8. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-sin.f64 37.3

         \[\leadsto -1 \cdot \left(eh \cdot \sin t\right) \]

   10. Applied rewrites 37.3%

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

   if -2.8000000000000001e147 < eh < 2.75000000000000011e144

   1. Initial program 99.8%

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

   3. Applied rewrites 91.2%

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

Alternative 15: 34.5% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 3.7 \cdot 10^{-163}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (if (<= ew 3.7e-163) (* eh (sin t)) ew))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.7e-163) {
		tmp = eh * sin(t);
	} else {
		tmp = 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) :: tmp
    if (ew <= 3.7d-163) then
        tmp = eh * sin(t)
    else
        tmp = ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 3.7e-163) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 3.7e-163:
		tmp = eh * math.sin(t)
	else:
		tmp = ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 3.7e-163)
		tmp = Float64(eh * sin(t));
	else
		tmp = ew;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 3.7e-163)
		tmp = eh * sin(t);
	else
		tmp = ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 3.7e-163], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], ew]

Derivation

1. Split input into 2 regimes

2. if ew < 3.6999999999999999e-163

   1. Initial program 99.8%

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

   3. Applied rewrites 22.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 19.6

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 22.5

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 22.5

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

   7. Applied rewrites 22.5%

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

   8. Taylor expanded in eh around inf

      \[\leadsto \color{blue}{eh \cdot \sin t} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\sin t} \]

      2. lift-sin.f64 25.4

         \[\leadsto eh \cdot \sin t \]

   10. Applied rewrites 25.4%

       \[\leadsto \color{blue}{eh \cdot \sin t} \]

   if 3.6999999999999999e-163 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 56.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 65.9%

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

      1. lift-/.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqrt-undiv N/A

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

      5. lift-/.f64 N/A

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

      6. lower-sqrt.f64 65.9

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 65.9

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 49.6%

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

Alternative 16: 22.1% accurate, 862.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 34.7%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. sqrt-div N/A

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

   4. lower-/.f64 N/A

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

5. Applied rewrites 39.8%

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

   1. lift-/.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. sqrt-undiv N/A

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

   5. lift-/.f64 N/A

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

   6. lower-sqrt.f64 39.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lift-*.f64 39.8

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

7. Applied rewrites 39.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 22.1%

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

Reproduce

herbie shell --seed 2025106 
(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)))))))