Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 0 \cdot \sin t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \left|\left(eh \cdot \sin t\right) \cdot \sin t\_2 - \left(ew \cdot \frac{t\_1 \cdot t\_1 - \cos t \cdot \cos t}{t\_1 - \cos t}\right) \cdot \cos t\_2\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* 0.0 (sin t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (fabs
    (-
     (* (* eh (sin t)) (sin t_2))
     (*
      (* ew (/ (- (* t_1 t_1) (* (cos t) (cos t))) (- t_1 (cos t))))
      (cos t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = 0.0 * sin(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	return fabs((((eh * sin(t)) * sin(t_2)) - ((ew * (((t_1 * t_1) - (cos(t) * cos(t))) / (t_1 - cos(t)))) * cos(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 = 0.0d0 * sin(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    code = abs((((eh * sin(t)) * sin(t_2)) - ((ew * (((t_1 * t_1) - (cos(t) * cos(t))) / (t_1 - cos(t)))) * cos(t_2))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = 0.0 * Math.sin(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	return Math.abs((((eh * Math.sin(t)) * Math.sin(t_2)) - ((ew * (((t_1 * t_1) - (Math.cos(t) * Math.cos(t))) / (t_1 - Math.cos(t)))) * Math.cos(t_2))));
}

Python

def code(eh, ew, t):
	t_1 = 0.0 * math.sin(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	return math.fabs((((eh * math.sin(t)) * math.sin(t_2)) - ((ew * (((t_1 * t_1) - (math.cos(t) * math.cos(t))) / (t_1 - math.cos(t)))) * math.cos(t_2))))

Julia

function code(eh, ew, t)
	t_1 = Float64(0.0 * sin(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	return abs(Float64(Float64(Float64(eh * sin(t)) * sin(t_2)) - Float64(Float64(ew * Float64(Float64(Float64(t_1 * t_1) - Float64(cos(t) * cos(t))) / Float64(t_1 - cos(t)))) * cos(t_2))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = 0.0 * sin(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = abs((((eh * sin(t)) * sin(t_2)) - ((ew * (((t_1 * t_1) - (cos(t) * cos(t))) / (t_1 - cos(t)))) * cos(t_2))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(0.0 * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - N[(N[Cos[t], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 - N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. sin-sum N/A

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

   4. flip-+ N/A

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

   5. lower-/.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(ew \cdot \frac{\left(0 \cdot \sin t\right) \cdot \left(0 \cdot \sin t\right) - \cos t \cdot \cos t}{0 \cdot \sin t - \cos t}\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right| \]
6. Add Preprocessing

Alternative 2: 88.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \tan^{-1} t\_1\\ t_3 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_4 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3 \leq -4 \cdot 10^{-232}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot eh, \sin t, t\_4\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_2 \cdot \sin t, eh, t\_4 \cdot \cos t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2 (atan t_1))
        (t_3 (atan (/ (* eh (tan t)) (- ew))))
        (t_4 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_3)) (* (* eh (sin t)) (sin t_3)))
        -4e-232)
     (fabs (/ (fma (* t_1 eh) (sin t) t_4) (cosh (asinh t_1))))
     (fma (* (sin t_2) (sin t)) eh (* t_4 (cos t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = atan(t_1);
	double t_3 = atan(((eh * tan(t)) / -ew));
	double t_4 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_3)) - ((eh * sin(t)) * sin(t_3))) <= -4e-232) {
		tmp = fabs((fma((t_1 * eh), sin(t), t_4) / cosh(asinh(t_1))));
	} else {
		tmp = fma((sin(t_2) * sin(t)), eh, (t_4 * cos(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = atan(t_1)
	t_3 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_4 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3))) <= -4e-232)
		tmp = abs(Float64(fma(Float64(t_1 * eh), sin(t), t_4) / cosh(asinh(t_1))));
	else
		tmp = fma(Float64(sin(t_2) * sin(t)), eh, Float64(t_4 * cos(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-232], N[Abs[N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + t$95$4), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(t$95$4 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.0000000000000001e-232

   1. Initial program 99.8%

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

   4. Applied rewrites 71.7%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 71.7

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 82.0

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

   6. Applied rewrites 82.0%

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

   if -4.0000000000000001e-232 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.9%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 99.9

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-232}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \sin t, eh, \left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) 2e-219)
     (fabs ew)
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 2e-219) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((eh * tan(t)) / -ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 2d-219) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= 2e-219) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= 2e-219:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= 2e-219)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 2e-219)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e-219], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 2.0000000000000001e-219

   1. Initial program 99.8%

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

   4. Applied rewrites 71.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 44.0%

      \[\leadsto \left|\color{blue}{ew}\right| \]

   if 2.0000000000000001e-219 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.9%

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

   4. Applied rewrites 49.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 65.5%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 2 \cdot 10^{-219}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;ew \cdot \cos t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 5: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 6: 87.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ \mathbf{if}\;eh \leq -1 \cdot 10^{+139} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh)))
   (if (or (<= eh -1e+139) (not (<= eh 1.9e+227)))
     (fabs (* (sin t) eh))
     (fabs (/ (fma (* t_1 eh) (sin t) (* (cos t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double tmp;
	if ((eh <= -1e+139) || !(eh <= 1.9e+227)) {
		tmp = fabs((sin(t) * eh));
	} else {
		tmp = fabs((fma((t_1 * eh), sin(t), (cos(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	tmp = 0.0
	if ((eh <= -1e+139) || !(eh <= 1.9e+227))
		tmp = abs(Float64(sin(t) * eh));
	else
		tmp = abs(Float64(fma(Float64(t_1 * eh), sin(t), Float64(cos(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[eh, -1e+139], N[Not[LessEqual[eh, 1.9e+227]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[Cos[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 < -1.00000000000000003e139 or 1.90000000000000018e227 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 5.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 4.2

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

   6. Applied rewrites 4.2%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 79.2%

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

   if -1.00000000000000003e139 < eh < 1.90000000000000018e227

   1. Initial program 99.8%

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

   4. Applied rewrites 88.2%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 88.2

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-*.f64 94.2

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

   6. Applied rewrites 94.2%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1 \cdot 10^{+139} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh, \sin t, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ \mathbf{if}\;eh \leq -8.8 \cdot 10^{+138} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\sin t \cdot t\_1\right) \cdot eh, eh, \cos t \cdot ew\right)}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* t_1 eh)))
   (if (or (<= eh -8.8e+138) (not (<= eh 1.9e+227)))
     (fabs (* (sin t) eh))
     (fabs
      (/
       (fma (* (* (sin t) t_1) eh) eh (* (cos t) ew))
       (sqrt (fma t_2 t_2 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double tmp;
	if ((eh <= -8.8e+138) || !(eh <= 1.9e+227)) {
		tmp = fabs((sin(t) * eh));
	} else {
		tmp = fabs((fma(((sin(t) * t_1) * eh), eh, (cos(t) * ew)) / sqrt(fma(t_2, t_2, 1.0))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	tmp = 0.0
	if ((eh <= -8.8e+138) || !(eh <= 1.9e+227))
		tmp = abs(Float64(sin(t) * eh));
	else
		tmp = abs(Float64(fma(Float64(Float64(sin(t) * t_1) * eh), eh, Float64(cos(t) * ew)) / sqrt(fma(t_2, t_2, 1.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * eh), $MachinePrecision]}, If[Or[LessEqual[eh, -8.8e+138], N[Not[LessEqual[eh, 1.9e+227]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh), $MachinePrecision] * eh + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.8000000000000003e138 or 1.90000000000000018e227 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 5.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 4.2

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

   6. Applied rewrites 4.2%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 79.2%

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

   if -8.8000000000000003e138 < eh < 1.90000000000000018e227

   1. Initial program 99.8%

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

   4. Applied rewrites 88.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 85.9

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

   6. Applied rewrites 85.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.5

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

   8. Applied rewrites 92.5%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+138} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\sin t \cdot \frac{\tan t}{ew}\right) \cdot eh, eh, \cos t \cdot ew\right)}{\sqrt{\mathsf{fma}\left(\frac{\tan t}{ew} \cdot eh, \frac{\tan t}{ew} \cdot eh, 1\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 84.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -8.8 \cdot 10^{+138} \lor \neg \left(eh \leq 1.1 \cdot 10^{+154}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\sqrt{{\left(eh \cdot t\_1\right)}^{2} - -1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= eh -8.8e+138) (not (<= eh 1.1e+154)))
     (fabs (* (sin t) eh))
     (fabs
      (/
       (fma (sin t) (* t_1 (* eh eh)) (* (cos t) ew))
       (sqrt (- (pow (* eh t_1) 2.0) -1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((eh <= -8.8e+138) || !(eh <= 1.1e+154)) {
		tmp = fabs((sin(t) * eh));
	} else {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), (cos(t) * ew)) / sqrt((pow((eh * t_1), 2.0) - -1.0))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((eh <= -8.8e+138) || !(eh <= 1.1e+154))
		tmp = abs(Float64(sin(t) * eh));
	else
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), Float64(cos(t) * ew)) / sqrt(Float64((Float64(eh * t_1) ^ 2.0) - -1.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -8.8e+138], N[Not[LessEqual[eh, 1.1e+154]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(eh * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.8000000000000003e138 or 1.1000000000000001e154 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 4.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 3.8

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

   6. Applied rewrites 3.8%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 72.1%

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

   if -8.8000000000000003e138 < eh < 1.1000000000000001e154

   1. Initial program 99.8%

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

   4. Applied rewrites 94.1%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 91.7

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

   6. Applied rewrites 91.7%

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

      1. lift-fma.f64 N/A

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

      2. metadata-eval N/A

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

      3. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower--.f64 N/A

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

      7. pow2 N/A

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

      8. lower-pow.f64 91.7

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 91.7

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

   8. Applied rewrites 91.7%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+138} \lor \neg \left(eh \leq 1.1 \cdot 10^{+154}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\sqrt{{\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} - -1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.2 \cdot 10^{+56} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -5.2e+56) (not (<= eh 1.9e+227)))
   (fabs (* (sin t) eh))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.2e+56) || !(eh <= 1.9e+227)) {
		tmp = fabs((sin(t) * eh));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-5.2d+56)) .or. (.not. (eh <= 1.9d+227))) then
        tmp = abs((sin(t) * eh))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.2e+56) || !(eh <= 1.9e+227)) {
		tmp = Math.abs((Math.sin(t) * eh));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -5.2e+56) or not (eh <= 1.9e+227):
		tmp = math.fabs((math.sin(t) * eh))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -5.2e+56) || !(eh <= 1.9e+227))
		tmp = abs(Float64(sin(t) * eh));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -5.2e+56) || ~((eh <= 1.9e+227)))
		tmp = abs((sin(t) * eh));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -5.2e+56], N[Not[LessEqual[eh, 1.9e+227]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.20000000000000022e56 or 1.90000000000000018e227 < eh

   1. Initial program 99.9%

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

   4. Applied rewrites 19.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 18.0

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

   6. Applied rewrites 18.0%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 75.5%

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

   if -5.20000000000000022e56 < eh < 1.90000000000000018e227

   1. Initial program 99.8%

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

   4. Applied rewrites 89.2%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 81.2%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.2 \cdot 10^{+56} \lor \neg \left(eh \leq 1.9 \cdot 10^{+227}\right):\\ \;\;\;\;\left|\sin t \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.4% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.7 \cdot 10^{+231}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.7e+231) (fabs (* ew (cos t))) (fabs (* t eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.7e+231) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((t * eh));
	}
	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 <= 1.7d+231) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((t * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.7e+231) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((t * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.7e+231:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((t * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.7e+231)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(t * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.7e+231)
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((t * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.7e+231], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.7e231

   1. Initial program 99.8%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 71.2%

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

   if 1.7e231 < eh

   1. Initial program 100.0%

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

   4. Applied rewrites 3.3%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 2.6

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 89.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|t \cdot eh\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 53.9%

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

Alternative 11: 44.0% accurate, 61.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 1.2 \cdot 10^{+231}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 1.2e+231) (fabs ew) (fabs (* t eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.2e+231) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((t * eh));
	}
	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 <= 1.2d+231) then
        tmp = abs(ew)
    else
        tmp = abs((t * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 1.2e+231) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((t * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 1.2e+231:
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((t * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 1.2e+231)
		tmp = abs(ew);
	else
		tmp = abs(Float64(t * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 1.2e+231)
		tmp = abs(ew);
	else
		tmp = abs((t * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 1.2e+231], N[Abs[ew], $MachinePrecision], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.20000000000000003e231

   1. Initial program 99.8%

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

   4. Applied rewrites 75.8%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 46.0%

      \[\leadsto \left|\color{blue}{ew}\right| \]

   if 1.20000000000000003e231 < eh

   1. Initial program 100.0%

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

   4. Applied rewrites 3.3%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-fma.f64 2.6

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

   6. Applied rewrites 2.6%

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

   7. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 89.8%

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

   10. Taylor expanded in t around 0

       \[\leadsto \left|t \cdot eh\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 53.9%

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

Alternative 12: 42.7% accurate, 287.3× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 70.7%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 43.2%

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

Alternative 13: 22.0% accurate, 862.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 47.9%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 22.2%

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

Reproduce

herbie shell --seed 2025018 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))