Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.1s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 88.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \tan^{-1} \left(t\_1 \cdot eh\right)\\ t_3 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(t\_1 \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_2 \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (atan (* t_1 eh)))
        (t_3 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_3)) (* (* eh (sin t)) (sin t_3)))
        -1e-283)
     (fabs
      (*
       (fma
        (/ eh ew)
        (* (- (sin t)) (tanh (asinh (* t_1 (- eh)))))
        (* (cos (atan (* (- eh) (/ t ew)))) (cos t)))
       ew))
     (fma (* (sin t_2) (sin t)) eh (* (* (cos t) ew) (cos t_2))))))

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 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_3)) - ((eh * sin(t)) * sin(t_3))) <= -1e-283) {
		tmp = fabs((fma((eh / ew), (-sin(t) * tanh(asinh((t_1 * -eh)))), (cos(atan((-eh * (t / ew)))) * cos(t))) * ew));
	} else {
		tmp = fma((sin(t_2) * sin(t)), eh, ((cos(t) * ew) * cos(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = atan(Float64(t_1 * eh))
	t_3 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3))) <= -1e-283)
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(Float64(-sin(t)) * tanh(asinh(Float64(t_1 * Float64(-eh))))), Float64(cos(atan(Float64(Float64(-eh) * Float64(t / ew)))) * cos(t))) * ew));
	else
		tmp = fma(Float64(sin(t_2) * sin(t)), eh, Float64(Float64(cos(t) * ew) * cos(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[ArcTan[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-283], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * N[Tanh[N[ArcSinh[N[(t$95$1 * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t$95$2], $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))))) < -9.99999999999999947e-284

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 88.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.8%

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

   10. Applied rewrites 77.8%

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

   if -9.99999999999999947e-284 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 99.8

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -1e-283)
     (fabs (* (cos (atan (* (- eh) (/ t ew)))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -1e-283) {
		tmp = fabs((cos(atan((-eh * (t / ew)))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-1d-283)) then
        tmp = abs((cos(atan((-eh * (t / ew)))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -1e-283) {
		tmp = Math.abs((Math.cos(Math.atan((-eh * (t / ew)))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -1e-283:
		tmp = math.fabs((math.cos(math.atan((-eh * (t / ew)))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -1e-283)
		tmp = abs(Float64(cos(atan(Float64(Float64(-eh) * Float64(t / ew)))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -1e-283)
		tmp = abs((cos(atan((-eh * (t / ew)))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-283], N[Abs[N[(N[Cos[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999947e-284

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{t \cdot \left(1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)}{ew}\right) \cdot \frac{eh}{\cos t}\right) \cdot ew\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \left|\cos \tan^{-1} \left(\left(-\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, t \cdot t, 0.008333333333333333\right) \cdot \left(t \cdot t\right) - 0.16666666666666666, t \cdot t, 1\right) \cdot t}{ew}\right) \cdot \frac{eh}{\cos t}\right) \cdot ew\right| \]

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 43.7%

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

   if -9.99999999999999947e-284 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.0

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

   7. Applied rewrites 63.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -1e-283)
     (fabs (* (cos (atan (/ (* (- eh) t) ew))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -1e-283) {
		tmp = fabs((cos(atan(((-eh * t) / ew))) * ew));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-1d-283)) then
        tmp = abs((cos(atan(((-eh * t) / ew))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -1e-283) {
		tmp = Math.abs((Math.cos(Math.atan(((-eh * t) / ew))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -1e-283:
		tmp = math.fabs((math.cos(math.atan(((-eh * t) / ew))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -1e-283)
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(-eh) * t) / ew))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -1e-283)
		tmp = abs((cos(atan(((-eh * t) / ew))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-283], N[Abs[N[(N[Cos[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -9.99999999999999947e-284

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 43.7%

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

   if -9.99999999999999947e-284 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.0

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

   7. Applied rewrites 63.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{-283}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ \mathbf{if}\;eh \leq -3.2 \cdot 10^{+255} \lor \neg \left(eh \leq 1.05 \cdot 10^{+222}\right):\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))))
   (if (or (<= eh -3.2e+255) (not (<= eh 1.05e+222)))
     (fabs (* (* t_1 eh) (sin (atan (* (/ t_1 ew) (/ eh (cos t)))))))
     (fabs
      (*
       (fma
        (/ eh ew)
        (* t_1 (tanh (asinh (* (/ (tan t) ew) (- eh)))))
        (* (cos (atan (* (- eh) (/ t ew)))) (cos t)))
       ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double tmp;
	if ((eh <= -3.2e+255) || !(eh <= 1.05e+222)) {
		tmp = fabs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))));
	} else {
		tmp = fabs((fma((eh / ew), (t_1 * tanh(asinh(((tan(t) / ew) * -eh)))), (cos(atan((-eh * (t / ew)))) * cos(t))) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	tmp = 0.0
	if ((eh <= -3.2e+255) || !(eh <= 1.05e+222))
		tmp = abs(Float64(Float64(t_1 * eh) * sin(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t)))))));
	else
		tmp = abs(Float64(fma(Float64(eh / ew), Float64(t_1 * tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh))))), Float64(cos(atan(Float64(Float64(-eh) * Float64(t / ew)))) * cos(t))) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, If[Or[LessEqual[eh, -3.2e+255], N[Not[LessEqual[eh, 1.05e+222]], $MachinePrecision]], N[Abs[N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh / ew), $MachinePrecision] * N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.1999999999999998e255 or 1.05000000000000005e222 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. distribute-lft-neg-in N/A

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

   5. Applied rewrites 85.7%

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

   if -3.1999999999999998e255 < eh < 1.05000000000000005e222

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 96.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 84.1%

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

   10. Applied rewrites 84.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.2 \cdot 10^{+255} \lor \neg \left(eh \leq 1.05 \cdot 10^{+222}\right):\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{eh}{ew}, \left(-\sin t\right) \cdot \tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right), \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ \mathbf{if}\;ew \leq -1.35 \cdot 10^{-49} \lor \neg \left(ew \leq 5.2 \cdot 10^{-82}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))))
   (if (or (<= ew -1.35e-49) (not (<= ew 5.2e-82)))
     (fabs (* (* (cos (atan (* (/ eh ew) (tan t)))) (cos t)) ew))
     (fabs (* (* t_1 eh) (sin (atan (* (/ t_1 ew) (/ eh (cos t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double tmp;
	if ((ew <= -1.35e-49) || !(ew <= 5.2e-82)) {
		tmp = fabs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	} else {
		tmp = fabs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(t)
    if ((ew <= (-1.35d-49)) .or. (.not. (ew <= 5.2d-82))) then
        tmp = abs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew))
    else
        tmp = abs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = -Math.sin(t);
	double tmp;
	if ((ew <= -1.35e-49) || !(ew <= 5.2e-82)) {
		tmp = Math.abs(((Math.cos(Math.atan(((eh / ew) * Math.tan(t)))) * Math.cos(t)) * ew));
	} else {
		tmp = Math.abs(((t_1 * eh) * Math.sin(Math.atan(((t_1 / ew) * (eh / Math.cos(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -math.sin(t)
	tmp = 0
	if (ew <= -1.35e-49) or not (ew <= 5.2e-82):
		tmp = math.fabs(((math.cos(math.atan(((eh / ew) * math.tan(t)))) * math.cos(t)) * ew))
	else:
		tmp = math.fabs(((t_1 * eh) * math.sin(math.atan(((t_1 / ew) * (eh / math.cos(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	tmp = 0.0
	if ((ew <= -1.35e-49) || !(ew <= 5.2e-82))
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(Float64(Float64(t_1 * eh) * sin(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -sin(t);
	tmp = 0.0;
	if ((ew <= -1.35e-49) || ~((ew <= 5.2e-82)))
		tmp = abs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, If[Or[LessEqual[ew, -1.35e-49], N[Not[LessEqual[ew, 5.2e-82]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.35e-49 or 5.2e-82 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-cos.f64 86.6

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

   8. Applied rewrites 86.6%

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

   10. Applied rewrites 86.6%

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

   if -1.35e-49 < ew < 5.2e-82

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

      15. distribute-lft-neg-in N/A

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

   5. Applied rewrites 72.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-49} \lor \neg \left(ew \leq 5.2 \cdot 10^{-82}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.7 \cdot 10^{-182} \lor \neg \left(ew \leq 1.2 \cdot 10^{-149}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) \cdot eh}{-ew} \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5.7e-182) (not (<= ew 1.2e-149)))
   (fabs (* (* (cos (atan (* (/ eh ew) (tan t)))) (cos t)) ew))
   (fabs
    (*
     (/ (* (* (sin t) (tanh (asinh (* (- eh) (/ (tan t) ew))))) eh) (- ew))
     ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.7e-182) || !(ew <= 1.2e-149)) {
		tmp = fabs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	} else {
		tmp = fabs(((((sin(t) * tanh(asinh((-eh * (tan(t) / ew))))) * eh) / -ew) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5.7e-182) or not (ew <= 1.2e-149):
		tmp = math.fabs(((math.cos(math.atan(((eh / ew) * math.tan(t)))) * math.cos(t)) * ew))
	else:
		tmp = math.fabs(((((math.sin(t) * math.tanh(math.asinh((-eh * (math.tan(t) / ew))))) * eh) / -ew) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5.7e-182) || !(ew <= 1.2e-149))
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * tanh(asinh(Float64(Float64(-eh) * Float64(tan(t) / ew))))) * eh) / Float64(-ew)) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5.7e-182) || ~((ew <= 1.2e-149)))
		tmp = abs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(((((sin(t) * tanh(asinh((-eh * (tan(t) / ew))))) * eh) / -ew) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5.7e-182], N[Not[LessEqual[ew, 1.2e-149]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * N[Tanh[N[ArcSinh[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.6999999999999998e-182 or 1.2000000000000001e-149 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-cos.f64 79.1

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 79.1%

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

   if -5.6999999999999998e-182 < ew < 1.2000000000000001e-149

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.7%

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

   9. Taylor expanded in eh around inf

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

   11. Applied rewrites 58.2%

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

   13. Applied rewrites 58.2%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.7 \cdot 10^{-182} \lor \neg \left(ew \leq 1.2 \cdot 10^{-149}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right) \cdot eh}{-ew} \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{-182} \lor \neg \left(ew \leq 7.5 \cdot 10^{-150}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{-eh}{ew} \cdot \left(\sin t \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5.3e-182) (not (<= ew 7.5e-150)))
   (fabs (* (* (cos (atan (* (/ eh ew) (tan t)))) (cos t)) ew))
   (fabs
    (*
     (* (/ (- eh) ew) (* (sin t) (tanh (asinh (* (- eh) (/ (tan t) ew))))))
     ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.3e-182) || !(ew <= 7.5e-150)) {
		tmp = fabs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	} else {
		tmp = fabs((((-eh / ew) * (sin(t) * tanh(asinh((-eh * (tan(t) / ew)))))) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5.3e-182) or not (ew <= 7.5e-150):
		tmp = math.fabs(((math.cos(math.atan(((eh / ew) * math.tan(t)))) * math.cos(t)) * ew))
	else:
		tmp = math.fabs((((-eh / ew) * (math.sin(t) * math.tanh(math.asinh((-eh * (math.tan(t) / ew)))))) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5.3e-182) || !(ew <= 7.5e-150))
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(Float64(Float64(Float64(Float64(-eh) / ew) * Float64(sin(t) * tanh(asinh(Float64(Float64(-eh) * Float64(tan(t) / ew)))))) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5.3e-182) || ~((ew <= 7.5e-150)))
		tmp = abs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs((((-eh / ew) * (sin(t) * tanh(asinh((-eh * (tan(t) / ew)))))) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5.3e-182], N[Not[LessEqual[ew, 7.5e-150]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[((-eh) / ew), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * N[Tanh[N[ArcSinh[N[((-eh) * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000005e-182 or 7.5000000000000004e-150 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-cos.f64 79.1

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 79.1%

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

   if -5.30000000000000005e-182 < ew < 7.5000000000000004e-150

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.7%

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

   9. Taylor expanded in eh around inf

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

   11. Applied rewrites 58.2%

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

   13. Applied rewrites 52.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{-182} \lor \neg \left(ew \leq 7.5 \cdot 10^{-150}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{-eh}{ew} \cdot \left(\sin t \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{-182} \lor \neg \left(ew \leq 7.5 \cdot 10^{-150}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot \left(eh \cdot \frac{\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}{ew}\right)\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5.3e-182) (not (<= ew 7.5e-150)))
   (fabs (* (* (cos (atan (* (/ eh ew) (tan t)))) (cos t)) ew))
   (fabs
    (* (* (sin t) (* eh (/ (tanh (asinh (* (/ (tan t) ew) eh))) ew))) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.3e-182) || !(ew <= 7.5e-150)) {
		tmp = fabs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	} else {
		tmp = fabs(((sin(t) * (eh * (tanh(asinh(((tan(t) / ew) * eh))) / ew))) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5.3e-182) or not (ew <= 7.5e-150):
		tmp = math.fabs(((math.cos(math.atan(((eh / ew) * math.tan(t)))) * math.cos(t)) * ew))
	else:
		tmp = math.fabs(((math.sin(t) * (eh * (math.tanh(math.asinh(((math.tan(t) / ew) * eh))) / ew))) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5.3e-182) || !(ew <= 7.5e-150))
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(Float64(Float64(sin(t) * Float64(eh * Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * eh))) / ew))) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5.3e-182) || ~((ew <= 7.5e-150)))
		tmp = abs(((cos(atan(((eh / ew) * tan(t)))) * cos(t)) * ew));
	else
		tmp = abs(((sin(t) * (eh * (tanh(asinh(((tan(t) / ew) * eh))) / ew))) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5.3e-182], N[Not[LessEqual[ew, 7.5e-150]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(eh * N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000005e-182 or 7.5000000000000004e-150 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-cos.f64 79.1

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 79.1%

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

   if -5.30000000000000005e-182 < ew < 7.5000000000000004e-150

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.7%

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

   9. Taylor expanded in eh around inf

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

   11. Applied rewrites 58.2%

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

   13. Applied rewrites 52.3%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{-182} \lor \neg \left(ew \leq 7.5 \cdot 10^{-150}\right):\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot \left(eh \cdot \frac{\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}{ew}\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around inf

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

5. Applied rewrites 88.4%

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

6. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-atan.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-cos.f64 66.7

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

8. Applied rewrites 66.7%

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

10. Applied rewrites 66.7%

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

Alternative 11: 31.5% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \cos t \cdot ew \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return Float64(cos(t) * ew)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 38.4%

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

5. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 34.9

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

7. Applied rewrites 34.9%

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

Alternative 12: 20.1% accurate, 50.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right) \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fma (* -0.5 ew) (* t t) ew))

C

double code(double eh, double ew, double t) {
	return fma((-0.5 * ew), (t * t), ew);
}

Julia

function code(eh, ew, t)
	return fma(Float64(-0.5 * ew), Float64(t * t), ew)
end

Wolfram

code[eh_, ew_, t_] := N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 38.4%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 18.0

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

7. Applied rewrites 18.0%

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

8. Taylor expanded in eh around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{2} \cdot ew, \color{blue}{t} \cdot t, ew\right) \]
9. Step-by-step derivation

10. Applied rewrites 21.3%

    \[\leadsto \mathsf{fma}\left(-0.5 \cdot ew, \color{blue}{t} \cdot t, ew\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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)))))))