Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.3s

Alternatives: 16

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\cos t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \left|\left(ew \cdot \sin t\right) \cdot \cos t\_2 + \left(eh \cdot \frac{t\_1 \cdot \cos t}{t\_1}\right) \cdot \sin t\_2\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (cos t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (fabs
    (+
     (* (* ew (sin t)) (cos t_2))
     (* (* eh (/ (* t_1 (cos t)) t_1)) (sin t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -cos(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_2)) + ((eh * ((t_1 * cos(t)) / t_1)) * sin(t_2))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    t_1 = -cos(t)
    t_2 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_2)) + ((eh * ((t_1 * cos(t)) / t_1)) * sin(t_2))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = -Math.cos(t);
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	return Math.abs((((ew * Math.sin(t)) * Math.cos(t_2)) + ((eh * ((t_1 * Math.cos(t)) / t_1)) * Math.sin(t_2))));
}

Python

def code(eh, ew, t):
	t_1 = -math.cos(t)
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	return math.fabs((((ew * math.sin(t)) * math.cos(t_2)) + ((eh * ((t_1 * math.cos(t)) / t_1)) * math.sin(t_2))))

Julia

function code(eh, ew, t)
	t_1 = Float64(-cos(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_2)) + Float64(Float64(eh * Float64(Float64(t_1 * cos(t)) / t_1)) * sin(t_2))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = -cos(t);
	t_2 = atan(((eh / ew) / tan(t)));
	tmp = abs((((ew * sin(t)) * cos(t_2)) + ((eh * ((t_1 * cos(t)) / t_1)) * sin(t_2))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. sin-sum N/A

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

   4. flip-+ N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 50.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ t_2 := ew \cdot \sin t\\ t_3 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ t_4 := t\_2 \cdot \cos t\_3 + t\_1 \cdot \sin t\_3\\ \mathbf{if}\;t\_4 \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sin \tan^{-1} \left(\frac{t\_1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t)))
        (t_2 (* ew (sin t)))
        (t_3 (atan (/ (/ eh ew) (tan t))))
        (t_4 (+ (* t_2 (cos t_3)) (* t_1 (sin t_3)))))
   (if (<= t_4 -5e+254)
     (fabs
      (*
       (sin
        (atan
         (*
          (fma
           (* t t)
           (- (* 0.041666666666666664 (/ (* t t) ew)) (/ 0.5 ew))
           (pow ew -1.0))
          (/ (+ eh (* 0.16666666666666666 (* eh (* t t)))) t))))
       eh))
     (if (<= t_4 -5e-287)
       (* (- (sin t)) ew)
       (* t_1 (sin (atan (/ t_1 t_2))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double t_2 = ew * sin(t);
	double t_3 = atan(((eh / ew) / tan(t)));
	double t_4 = (t_2 * cos(t_3)) + (t_1 * sin(t_3));
	double tmp;
	if (t_4 <= -5e+254) {
		tmp = fabs((sin(atan((fma((t * t), ((0.041666666666666664 * ((t * t) / ew)) - (0.5 / ew)), pow(ew, -1.0)) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	} else if (t_4 <= -5e-287) {
		tmp = -sin(t) * ew;
	} else {
		tmp = t_1 * sin(atan((t_1 / t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	t_2 = Float64(ew * sin(t))
	t_3 = atan(Float64(Float64(eh / ew) / tan(t)))
	t_4 = Float64(Float64(t_2 * cos(t_3)) + Float64(t_1 * sin(t_3)))
	tmp = 0.0
	if (t_4 <= -5e+254)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), Float64(Float64(0.041666666666666664 * Float64(Float64(t * t) / ew)) - Float64(0.5 / ew)), (ew ^ -1.0)) * Float64(Float64(eh + Float64(0.16666666666666666 * Float64(eh * Float64(t * t)))) / t)))) * eh));
	elseif (t_4 <= -5e-287)
		tmp = Float64(Float64(-sin(t)) * ew);
	else
		tmp = Float64(t_1 * sin(atan(Float64(t_1 / t_2))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -5e+254], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] - N[(0.5 / ew), $MachinePrecision]), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(eh + N[(0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, -5e-287], N[((-N[Sin[t], $MachinePrecision]) * ew), $MachinePrecision], N[(t$95$1 * N[Sin[N[ArcTan[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -4.99999999999999994e254

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 46.1

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

   5. Applied rewrites 46.1%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 46.2%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \frac{1}{24} \cdot \frac{t \cdot t}{ew} - \frac{\frac{1}{2}}{ew}, \frac{1}{ew}\right) \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 46.2%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

   if -4.99999999999999994e254 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -5.00000000000000025e-287

   1. Initial program 99.8%

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

   4. Applied rewrites 1.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 2.0

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

   7. Applied rewrites 2.0%

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

   9. Applied rewrites 20.2%

      \[\leadsto ew \cdot \sin \left(\sqrt{-t} \cdot \sqrt{-t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 46.6%

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

   if -5.00000000000000025e-287 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 37.6

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

   5. Applied rewrites 37.6%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. pow2 N/A

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

   7. Applied rewrites 35.2%

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

   8. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 58.2

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

   10. Applied rewrites 58.2%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \leq -5 \cdot 10^{+254}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{elif}\;\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \leq -5 \cdot 10^{-287}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 + \left(eh \cdot \cos t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (+ (* t_1 (cos t_2)) (* (* eh (cos t)) (sin t_2))) -2e-216)
     (* (- (sin t)) ew)
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -2e-216) {
		tmp = -sin(t) * ew;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = atan(((eh / ew) / tan(t)))
    if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= (-2d-216)) then
        tmp = -sin(t) * ew
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if (((t_1 * Math.cos(t_2)) + ((eh * Math.cos(t)) * Math.sin(t_2))) <= -2e-216) {
		tmp = -Math.sin(t) * ew;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if ((t_1 * math.cos(t_2)) + ((eh * math.cos(t)) * math.sin(t_2))) <= -2e-216:
		tmp = -math.sin(t) * ew
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) + Float64(Float64(eh * cos(t)) * sin(t_2))) <= -2e-216)
		tmp = Float64(Float64(-sin(t)) * ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -2e-216)
		tmp = -sin(t) * ew;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-216], N[((-N[Sin[t], $MachinePrecision]) * ew), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -2.0000000000000001e-216

   1. Initial program 99.8%

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

   4. Applied rewrites 1.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 1.7

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

   7. Applied rewrites 1.7%

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

   9. Applied rewrites 18.9%

      \[\leadsto ew \cdot \sin \left(\sqrt{-t} \cdot \sqrt{-t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 42.9%

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

   if -2.0000000000000001e-216 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 44.3

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

   7. Applied rewrites 44.3%

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

Alternative 4: 32.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 + \left(eh \cdot \cos t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-216}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (+ (* t_1 (cos t_2)) (* (* eh (cos t)) (sin t_2))) -2e-216)
     (* ew (sin (fabs t)))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -2e-216) {
		tmp = ew * sin(fabs(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = atan(((eh / ew) / tan(t)))
    if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= (-2d-216)) then
        tmp = ew * sin(abs(t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if (((t_1 * Math.cos(t_2)) + ((eh * Math.cos(t)) * Math.sin(t_2))) <= -2e-216) {
		tmp = ew * Math.sin(Math.abs(t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if ((t_1 * math.cos(t_2)) + ((eh * math.cos(t)) * math.sin(t_2))) <= -2e-216:
		tmp = ew * math.sin(math.fabs(t))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) + Float64(Float64(eh * cos(t)) * sin(t_2))) <= -2e-216)
		tmp = Float64(ew * sin(abs(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -2e-216)
		tmp = ew * sin(abs(t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-216], N[(ew * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -2.0000000000000001e-216

   1. Initial program 99.8%

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

   4. Applied rewrites 1.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 1.7

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

   7. Applied rewrites 1.7%

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

   9. Applied rewrites 21.4%

      \[\leadsto ew \cdot \sin \left(\left|t\right|\right) \]

   if -2.0000000000000001e-216 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   4. Applied rewrites 66.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 44.3

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

   7. Applied rewrites 44.3%

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

Alternative 5: 27.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 + \left(eh \cdot \cos t\right) \cdot \sin t\_2 \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\left|ew\right| \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (+ (* t_1 (cos t_2)) (* (* eh (cos t)) (sin t_2))) -5e-211)
     (* (fabs ew) t)
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -5e-211) {
		tmp = fabs(ew) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = atan(((eh / ew) / tan(t)))
    if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= (-5d-211)) then
        tmp = abs(ew) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if (((t_1 * Math.cos(t_2)) + ((eh * Math.cos(t)) * Math.sin(t_2))) <= -5e-211) {
		tmp = Math.abs(ew) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if ((t_1 * math.cos(t_2)) + ((eh * math.cos(t)) * math.sin(t_2))) <= -5e-211:
		tmp = math.fabs(ew) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) + Float64(Float64(eh * cos(t)) * sin(t_2))) <= -5e-211)
		tmp = Float64(abs(ew) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) + ((eh * cos(t)) * sin(t_2))) <= -5e-211)
		tmp = abs(ew) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-211], N[(N[Abs[ew], $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -5.0000000000000002e-211

   1. Initial program 99.8%

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

   4. Applied rewrites 1.0%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 1.7

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 2.4%

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

   12. Applied rewrites 8.2%

       \[\leadsto \left|ew\right| \cdot t \]

   if -5.0000000000000002e-211 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   4. Applied rewrites 65.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 44.0

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

   7. Applied rewrites 44.0%

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

Alternative 6: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 7: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 99.3

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

5. Applied rewrites 99.3%

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

Alternative 8: 88.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{+23} \lor \neg \left(eh \leq 3.8 \cdot 10^{+57}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -4.3e+23) (not (<= eh 3.8e+57)))
     (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -4.3e+23) || !(eh <= 3.8e+57)) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -4.3e+23) || !(eh <= 3.8e+57))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -4.3e+23], N[Not[LessEqual[eh, 3.8e+57]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.2999999999999999e23 or 3.7999999999999999e57 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 52.0

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

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 52.1%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-atan.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 85.1

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

   11. Applied rewrites 85.1%

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

   if -4.2999999999999999e23 < eh < 3.7999999999999999e57

   1. Initial program 99.9%

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

   4. Applied rewrites 92.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.3 \cdot 10^{+23} \lor \neg \left(eh \leq 3.8 \cdot 10^{+57}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -2.3 \cdot 10^{-219} \lor \neg \left(eh \leq 1.2 \cdot 10^{-10}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\sin t}\right)\right) \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (or (<= eh -2.3e-219) (not (<= eh 1.2e-10)))
     (fabs (* eh (* (cos t) (sin (atan (/ t_1 (* ew (sin t))))))))
     (fabs (* (* ew (cos (atan (/ (/ t_1 ew) (sin t))))) (sin t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10)) {
		tmp = fabs((eh * (cos(t) * sin(atan((t_1 / (ew * sin(t))))))));
	} else {
		tmp = fabs(((ew * cos(atan(((t_1 / ew) / sin(t))))) * sin(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 = eh * cos(t)
    if ((eh <= (-2.3d-219)) .or. (.not. (eh <= 1.2d-10))) then
        tmp = abs((eh * (cos(t) * sin(atan((t_1 / (ew * sin(t))))))))
    else
        tmp = abs(((ew * cos(atan(((t_1 / ew) / sin(t))))) * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10)) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((t_1 / (ew * Math.sin(t))))))));
	} else {
		tmp = Math.abs(((ew * Math.cos(Math.atan(((t_1 / ew) / Math.sin(t))))) * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if (eh <= -2.3e-219) or not (eh <= 1.2e-10):
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan((t_1 / (ew * math.sin(t))))))))
	else:
		tmp = math.fabs(((ew * math.cos(math.atan(((t_1 / ew) / math.sin(t))))) * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(t_1 / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(Float64(ew * cos(atan(Float64(Float64(t_1 / ew) / sin(t))))) * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if ((eh <= -2.3e-219) || ~((eh <= 1.2e-10)))
		tmp = abs((eh * (cos(t) * sin(atan((t_1 / (ew * sin(t))))))));
	else
		tmp = abs(((ew * cos(atan(((t_1 / ew) / sin(t))))) * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -2.3e-219], N[Not[LessEqual[eh, 1.2e-10]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(t$95$1 / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Cos[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.29999999999999988e-219 or 1.2e-10 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-atan.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 77.5

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

   11. Applied rewrites 77.5%

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

   if -2.29999999999999988e-219 < eh < 1.2e-10

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. sin-sum N/A

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

      4. flip-+ N/A

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

      5. lower-/.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. associate-/r* N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 78.7

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

   7. Applied rewrites 78.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-219} \lor \neg \left(eh \leq 1.2 \cdot 10^{-10}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos \tan^{-1} \left(\frac{\frac{eh \cdot \cos t}{ew}}{\sin t}\right)\right) \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-219} \lor \neg \left(eh \leq 1.2 \cdot 10^{-10}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.3e-219) (not (<= eh 1.2e-10)))
   (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
   (fabs (* (cos (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (sin t) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10)) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else {
		tmp = fabs((cos(atan(((cos(t) / ew) * (eh / sin(t))))) * (sin(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) :: tmp
    if ((eh <= (-2.3d-219)) .or. (.not. (eh <= 1.2d-10))) then
        tmp = abs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))))
    else
        tmp = abs((cos(atan(((cos(t) / ew) * (eh / sin(t))))) * (sin(t) * ew)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10)) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan(((eh * Math.cos(t)) / (ew * Math.sin(t))))))));
	} else {
		tmp = Math.abs((Math.cos(Math.atan(((Math.cos(t) / ew) * (eh / Math.sin(t))))) * (Math.sin(t) * ew)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -2.3e-219) or not (eh <= 1.2e-10):
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan(((eh * math.cos(t)) / (ew * math.sin(t))))))))
	else:
		tmp = math.fabs((math.cos(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.sin(t) * ew)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.3e-219) || !(eh <= 1.2e-10))
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(sin(t) * ew)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -2.3e-219) || ~((eh <= 1.2e-10)))
		tmp = abs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	else
		tmp = abs((cos(atan(((cos(t) / ew) * (eh / sin(t))))) * (sin(t) * ew)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.3e-219], N[Not[LessEqual[eh, 1.2e-10]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.29999999999999988e-219 or 1.2e-10 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-atan.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 77.5

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

   11. Applied rewrites 77.5%

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

   if -2.29999999999999988e-219 < eh < 1.2e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lower-sin.f64 78.7

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

   5. Applied rewrites 78.7%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.3 \cdot 10^{-219} \lor \neg \left(eh \leq 1.2 \cdot 10^{-10}\right):\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ \mathbf{if}\;ew \leq -2.2 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{+163}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{t\_1}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))))
   (if (<= ew -2.2e+140)
     t_1
     (if (<= ew 3.1e+163)
       (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) t_1))))))
       (* ew (sin (fabs t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double tmp;
	if (ew <= -2.2e+140) {
		tmp = t_1;
	} else if (ew <= 3.1e+163) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / t_1))))));
	} else {
		tmp = ew * sin(fabs(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 = ew * sin(t)
    if (ew <= (-2.2d+140)) then
        tmp = t_1
    else if (ew <= 3.1d+163) then
        tmp = abs((eh * (cos(t) * sin(atan(((eh * cos(t)) / t_1))))))
    else
        tmp = ew * sin(abs(t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double tmp;
	if (ew <= -2.2e+140) {
		tmp = t_1;
	} else if (ew <= 3.1e+163) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan(((eh * Math.cos(t)) / t_1))))));
	} else {
		tmp = ew * Math.sin(Math.abs(t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	tmp = 0
	if ew <= -2.2e+140:
		tmp = t_1
	elif ew <= 3.1e+163:
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan(((eh * math.cos(t)) / t_1))))))
	else:
		tmp = ew * math.sin(math.fabs(t))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	tmp = 0.0
	if (ew <= -2.2e+140)
		tmp = t_1;
	elseif (ew <= 3.1e+163)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / t_1))))));
	else
		tmp = Float64(ew * sin(abs(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	tmp = 0.0;
	if (ew <= -2.2e+140)
		tmp = t_1;
	elseif (ew <= 3.1e+163)
		tmp = abs((eh * (cos(t) * sin(atan(((eh * cos(t)) / t_1))))));
	else
		tmp = ew * sin(abs(t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.2e+140], t$95$1, If[LessEqual[ew, 3.1e+163], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(ew * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.1999999999999998e140

   1. Initial program 99.9%

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 53.2

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

   7. Applied rewrites 53.2%

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

   if -2.1999999999999998e140 < ew < 3.10000000000000029e163

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 44.3

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 44.5%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-atan.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 72.6

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

   11. Applied rewrites 72.6%

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

   if 3.10000000000000029e163 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 41.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 36.4

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

   7. Applied rewrites 36.4%

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

   9. Applied rewrites 62.9%

      \[\leadsto ew \cdot \sin \left(\left|t\right|\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+140}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;ew \leq 3.1 \cdot 10^{+163}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -4.3e+229)
   (* ew (sin (fabs t)))
   (if (<= t -4.2e-25)
     (* ew (sin t))
     (if (<= t 1e-12)
       (fabs
        (*
         (sin
          (atan
           (*
            (fma
             (* t t)
             (-
              (*
               (* t t)
               (fma
                -0.001388888888888889
                (/ (* t t) ew)
                (/ 0.041666666666666664 ew)))
              (/ 0.5 ew))
             (pow ew -1.0))
            (/ eh (sin t)))))
         eh))
       (* (- (sin t)) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -4.3e+229) {
		tmp = ew * sin(fabs(t));
	} else if (t <= -4.2e-25) {
		tmp = ew * sin(t);
	} else if (t <= 1e-12) {
		tmp = fabs((sin(atan((fma((t * t), (((t * t) * fma(-0.001388888888888889, ((t * t) / ew), (0.041666666666666664 / ew))) - (0.5 / ew)), pow(ew, -1.0)) * (eh / sin(t))))) * eh));
	} else {
		tmp = -sin(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -4.3e+229)
		tmp = Float64(ew * sin(abs(t)));
	elseif (t <= -4.2e-25)
		tmp = Float64(ew * sin(t));
	elseif (t <= 1e-12)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), Float64(Float64(Float64(t * t) * fma(-0.001388888888888889, Float64(Float64(t * t) / ew), Float64(0.041666666666666664 / ew))) - Float64(0.5 / ew)), (ew ^ -1.0)) * Float64(eh / sin(t))))) * eh));
	else
		tmp = Float64(Float64(-sin(t)) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -4.3e+229], N[(ew * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.2e-25], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-12], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * N[(-0.001388888888888889 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[(0.041666666666666664 / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / ew), $MachinePrecision]), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[((-N[Sin[t], $MachinePrecision]) * ew), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.29999999999999991e229

   1. Initial program 99.6%

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 4.8

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

   7. Applied rewrites 4.8%

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

   9. Applied rewrites 42.9%

      \[\leadsto ew \cdot \sin \left(\left|t\right|\right) \]

   if -4.29999999999999991e229 < t < -4.20000000000000005e-25

   1. Initial program 99.6%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if -4.20000000000000005e-25 < t < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-1}{720} \cdot \frac{{t}^{2}}{ew} + \frac{1}{24} \cdot \frac{1}{ew}\right) - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   if 9.9999999999999998e-13 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 23.2

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 0.0%

      \[\leadsto ew \cdot \sin \left(\sqrt{-t} \cdot \sqrt{-t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.3%

       \[\leadsto \left(-\sin t\right) \cdot \color{blue}{ew} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \left(t \cdot t\right) \cdot \mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -4.3e+229)
   (* ew (sin (fabs t)))
   (if (<= t -4.2e-25)
     (* ew (sin t))
     (if (<= t 1e-12)
       (fabs
        (*
         (sin
          (atan
           (*
            (fma
             (* t t)
             (- (* 0.041666666666666664 (/ (* t t) ew)) (/ 0.5 ew))
             (pow ew -1.0))
            (/
             eh
             (*
              t
              (+
               1.0
               (*
                (* t t)
                (- (* 0.008333333333333333 (* t t)) 0.16666666666666666))))))))
         eh))
       (* (- (sin t)) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -4.3e+229) {
		tmp = ew * sin(fabs(t));
	} else if (t <= -4.2e-25) {
		tmp = ew * sin(t);
	} else if (t <= 1e-12) {
		tmp = fabs((sin(atan((fma((t * t), ((0.041666666666666664 * ((t * t) / ew)) - (0.5 / ew)), pow(ew, -1.0)) * (eh / (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))))))) * eh));
	} else {
		tmp = -sin(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -4.3e+229)
		tmp = Float64(ew * sin(abs(t)));
	elseif (t <= -4.2e-25)
		tmp = Float64(ew * sin(t));
	elseif (t <= 1e-12)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), Float64(Float64(0.041666666666666664 * Float64(Float64(t * t) / ew)) - Float64(0.5 / ew)), (ew ^ -1.0)) * Float64(eh / Float64(t * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.008333333333333333 * Float64(t * t)) - 0.16666666666666666)))))))) * eh));
	else
		tmp = Float64(Float64(-sin(t)) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -4.3e+229], N[(ew * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.2e-25], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-12], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] - N[(0.5 / ew), $MachinePrecision]), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[(t * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[((-N[Sin[t], $MachinePrecision]) * ew), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.29999999999999991e229

   1. Initial program 99.6%

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 4.8

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

   7. Applied rewrites 4.8%

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

   9. Applied rewrites 42.9%

      \[\leadsto ew \cdot \sin \left(\left|t\right|\right) \]

   if -4.29999999999999991e229 < t < -4.20000000000000005e-25

   1. Initial program 99.6%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if -4.20000000000000005e-25 < t < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \frac{1}{24} \cdot \frac{t \cdot t}{ew} - \frac{\frac{1}{2}}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 70.6%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   if 9.9999999999999998e-13 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 23.2

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 0.0%

      \[\leadsto ew \cdot \sin \left(\sqrt{-t} \cdot \sqrt{-t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.3%

       \[\leadsto \left(-\sin t\right) \cdot \color{blue}{ew} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 48.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -4.3e+229)
   (* ew (sin (fabs t)))
   (if (<= t -4.2e-25)
     (* ew (sin t))
     (if (<= t 1e-12)
       (fabs
        (*
         (sin
          (atan
           (*
            (fma
             (* t t)
             (- (* 0.041666666666666664 (/ (* t t) ew)) (/ 0.5 ew))
             (pow ew -1.0))
            (/ (+ eh (* 0.16666666666666666 (* eh (* t t)))) t))))
         eh))
       (* (- (sin t)) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -4.3e+229) {
		tmp = ew * sin(fabs(t));
	} else if (t <= -4.2e-25) {
		tmp = ew * sin(t);
	} else if (t <= 1e-12) {
		tmp = fabs((sin(atan((fma((t * t), ((0.041666666666666664 * ((t * t) / ew)) - (0.5 / ew)), pow(ew, -1.0)) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	} else {
		tmp = -sin(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -4.3e+229)
		tmp = Float64(ew * sin(abs(t)));
	elseif (t <= -4.2e-25)
		tmp = Float64(ew * sin(t));
	elseif (t <= 1e-12)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(t * t), Float64(Float64(0.041666666666666664 * Float64(Float64(t * t) / ew)) - Float64(0.5 / ew)), (ew ^ -1.0)) * Float64(Float64(eh + Float64(0.16666666666666666 * Float64(eh * Float64(t * t)))) / t)))) * eh));
	else
		tmp = Float64(Float64(-sin(t)) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -4.3e+229], N[(ew * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, -4.2e-25], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1e-12], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] - N[(0.5 / ew), $MachinePrecision]), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(N[(eh + N[(0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[((-N[Sin[t], $MachinePrecision]) * ew), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -4.29999999999999991e229

   1. Initial program 99.6%

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

   4. Applied rewrites 24.9%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 4.8

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

   7. Applied rewrites 4.8%

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

   9. Applied rewrites 42.9%

      \[\leadsto ew \cdot \sin \left(\left|t\right|\right) \]

   if -4.29999999999999991e229 < t < -4.20000000000000005e-25

   1. Initial program 99.6%

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

   4. Applied rewrites 57.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 40.3

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

   7. Applied rewrites 40.3%

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

   if -4.20000000000000005e-25 < t < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{{t}^{2}}{ew} - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

   9. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, \frac{1}{24} \cdot \frac{t \cdot t}{ew} - \frac{\frac{1}{2}}{ew}, \frac{1}{ew}\right) \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 70.6%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, \frac{1}{ew}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

   if 9.9999999999999998e-13 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 29.3%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 23.2

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

   7. Applied rewrites 23.2%

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

   9. Applied rewrites 0.0%

      \[\leadsto ew \cdot \sin \left(\sqrt{-t} \cdot \sqrt{-t}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.3%

       \[\leadsto \left(-\sin t\right) \cdot \color{blue}{ew} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+229}:\\ \;\;\;\;ew \cdot \sin \left(\left|t\right|\right)\\ \mathbf{elif}\;t \leq -4.2 \cdot 10^{-25}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{elif}\;t \leq 10^{-12}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(t \cdot t, 0.041666666666666664 \cdot \frac{t \cdot t}{ew} - \frac{0.5}{ew}, {ew}^{-1}\right) \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left(-\sin t\right) \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 14.7% accurate, 62.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.1 \cdot 10^{-282}:\\ \;\;\;\;ew \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right| \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -1.1e-282) (* ew t) (* (fabs ew) t)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.1e-282) {
		tmp = ew * t;
	} else {
		tmp = fabs(ew) * t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.1d-282)) then
        tmp = ew * t
    else
        tmp = abs(ew) * t
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -1.1e-282) {
		tmp = ew * t;
	} else {
		tmp = Math.abs(ew) * t;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -1.1e-282:
		tmp = ew * t
	else:
		tmp = math.fabs(ew) * t
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -1.1e-282)
		tmp = Float64(ew * t);
	else
		tmp = Float64(abs(ew) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -1.1e-282)
		tmp = ew * t;
	else
		tmp = abs(ew) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -1.1e-282], N[(ew * t), $MachinePrecision], N[(N[Abs[ew], $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.09999999999999991e-282

   1. Initial program 99.8%

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

   4. Applied rewrites 44.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 30.3

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

   7. Applied rewrites 30.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 16.3%

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

   if -1.09999999999999991e-282 < t

   1. Initial program 99.8%

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

   4. Applied rewrites 26.2%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 17.9

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

   7. Applied rewrites 17.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 5.9%

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

   12. Applied rewrites 11.9%

       \[\leadsto \left|ew\right| \cdot t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 10.3% accurate, 145.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 34.8%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 23.8

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

7. Applied rewrites 23.8%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 10.9%

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

Reproduce

herbie shell --seed 2024351 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))