Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 29.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left(ew \cdot \sin t\right) \cdot \cos t\_1 + \left(eh \cdot \cos t\right) \cdot \sin t\_1 \leq -5 \cdot 10^{-257}:\\ \;\;\;\;\left|t \cdot \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, ew \cdot \left(t \cdot t\right), 0.008333333333333333 \cdot ew\right), -0.16666666666666666 \cdot ew\right), ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sin t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (/ eh ew) (tan t)))))
   (if (<=
        (+ (* (* ew (sin t)) (cos t_1)) (* (* eh (cos t)) (sin t_1)))
        -5e-257)
     (fabs
      (*
       t
       (fma
        (* t t)
        (fma
         (* t t)
         (fma
          -0.0001984126984126984
          (* ew (* t t))
          (* 0.008333333333333333 ew))
         (* -0.16666666666666666 ew))
        ew)))
     (* (sin t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	double tmp;
	if ((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))) <= -5e-257) {
		tmp = fabs((t * fma((t * t), fma((t * t), fma(-0.0001984126984126984, (ew * (t * t)), (0.008333333333333333 * ew)), (-0.16666666666666666 * ew)), ew)));
	} else {
		tmp = sin(t) * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * sin(t)) * cos(t_1)) + Float64(Float64(eh * cos(t)) * sin(t_1))) <= -5e-257)
		tmp = abs(Float64(t * fma(Float64(t * t), fma(Float64(t * t), fma(-0.0001984126984126984, Float64(ew * Float64(t * t)), Float64(0.008333333333333333 * ew)), Float64(-0.16666666666666666 * ew)), ew)));
	else
		tmp = Float64(sin(t) * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-257], N[Abs[N[(t * N[(N[(t * t), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * N[(-0.0001984126984126984 * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision] + N[(0.008333333333333333 * ew), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * ew), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))) < -4.99999999999999989e-257

   1. Initial program 99.8%

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 43.1

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

   7. Applied rewrites 43.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 19.6%

       \[\leadsto \left|t \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, ew \cdot \left(t \cdot t\right), 0.008333333333333333 \cdot ew\right), -0.16666666666666666 \cdot ew\right), ew\right)}\right| \]

   if -4.99999999999999989e-257 < (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))

   1. Initial program 99.8%

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

   4. Applied rewrites 74.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 49.4

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

   7. Applied rewrites 49.4%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 49.4

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

   9. Applied rewrites 49.4%

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

Alternative 3: 87.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -5.5 \cdot 10^{-24} \lor \neg \left(eh \leq 4.5 \cdot 10^{+64}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -5.5e-24) (not (<= eh 4.5e+64)))
     (fabs (* eh (cos t)))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -5.5e-24) || !(eh <= 4.5e+64)) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -5.5e-24) || !(eh <= 4.5e+64))
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -5.5e-24], N[Not[LessEqual[eh, 4.5e+64]], $MachinePrecision]], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.4999999999999999e-24 or 4.49999999999999973e64 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 37.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 30.8%

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

   7. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 85.7

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

   9. Applied rewrites 85.7%

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

   if -5.4999999999999999e-24 < eh < 4.49999999999999973e64

   1. Initial program 99.8%

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

   4. Applied rewrites 94.2%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.5 \cdot 10^{-24} \lor \neg \left(eh \leq 4.5 \cdot 10^{+64}\right):\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \frac{eh}{\tan t}\\ t_3 := \sqrt{1 + {\left(\frac{t\_2}{ew}\right)}^{2}}\\ t_4 := eh \cdot \cos t\\ \mathbf{if}\;ew \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{t\_4}{\tan t \cdot ew}, eh, t\_1\right)}{t\_3}\right|\\ \mathbf{elif}\;ew \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, t\_4\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{t\_2 \cdot \cos t}{ew}, eh, t\_1\right)}{t\_3}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew))
        (t_2 (/ eh (tan t)))
        (t_3 (sqrt (+ 1.0 (pow (/ t_2 ew) 2.0))))
        (t_4 (* eh (cos t))))
   (if (<= ew -4.5e+65)
     (fabs (/ (fma (/ t_4 (* (tan t) ew)) eh t_1) t_3))
     (if (<= ew 4.4e-51)
       (fabs (fma (* ew ew) (* 0.5 (/ (/ (pow (sin t) 2.0) eh) (cos t))) t_4))
       (fabs (/ (fma (/ (* t_2 (cos t)) ew) eh t_1) t_3))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = eh / tan(t);
	double t_3 = sqrt((1.0 + pow((t_2 / ew), 2.0)));
	double t_4 = eh * cos(t);
	double tmp;
	if (ew <= -4.5e+65) {
		tmp = fabs((fma((t_4 / (tan(t) * ew)), eh, t_1) / t_3));
	} else if (ew <= 4.4e-51) {
		tmp = fabs(fma((ew * ew), (0.5 * ((pow(sin(t), 2.0) / eh) / cos(t))), t_4));
	} else {
		tmp = fabs((fma(((t_2 * cos(t)) / ew), eh, t_1) / t_3));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(eh / tan(t))
	t_3 = sqrt(Float64(1.0 + (Float64(t_2 / ew) ^ 2.0)))
	t_4 = Float64(eh * cos(t))
	tmp = 0.0
	if (ew <= -4.5e+65)
		tmp = abs(Float64(fma(Float64(t_4 / Float64(tan(t) * ew)), eh, t_1) / t_3));
	elseif (ew <= 4.4e-51)
		tmp = abs(fma(Float64(ew * ew), Float64(0.5 * Float64(Float64((sin(t) ^ 2.0) / eh) / cos(t))), t_4));
	else
		tmp = abs(Float64(fma(Float64(Float64(t_2 * cos(t)) / ew), eh, t_1) / t_3));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 + N[Power[N[(t$95$2 / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -4.5e+65], N[Abs[N[(N[(N[(t$95$4 / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4.4e-51], N[Abs[N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(t$95$2 * N[Cos[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.5e65

   1. Initial program 99.6%

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

   4. Applied rewrites 95.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-tan.f64 96.2

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

   8. Applied rewrites 96.2%

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

   if -4.5e65 < ew < 4.4e-51

   1. Initial program 99.8%

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

   4. Applied rewrites 54.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 40.6%

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

   7. Taylor expanded in ew around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 83.2

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \color{blue}{\cos t}\right)\right| \]

   9. Applied rewrites 83.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)}\right| \]

   if 4.4e-51 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 86.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 85.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   8. Applied rewrites 85.1%

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

Alternative 5: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.5 \cdot 10^{+65} \lor \neg \left(ew \leq 4.7 \cdot 10^{-51}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{\tan t \cdot ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -4.5e+65) (not (<= ew 4.7e-51)))
   (fabs
    (/
     (fma (* (cos t) (/ (/ eh (tan t)) ew)) eh (* (sin t) ew))
     (sqrt (+ 1.0 (pow (/ eh (* (tan t) ew)) 2.0)))))
   (fabs
    (fma
     (* ew ew)
     (* 0.5 (/ (/ (pow (sin t) 2.0) eh) (cos t)))
     (* eh (cos t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -4.5e+65) || !(ew <= 4.7e-51)) {
		tmp = fabs((fma((cos(t) * ((eh / tan(t)) / ew)), eh, (sin(t) * ew)) / sqrt((1.0 + pow((eh / (tan(t) * ew)), 2.0)))));
	} else {
		tmp = fabs(fma((ew * ew), (0.5 * ((pow(sin(t), 2.0) / eh) / cos(t))), (eh * cos(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -4.5e+65) || !(ew <= 4.7e-51))
		tmp = abs(Float64(fma(Float64(cos(t) * Float64(Float64(eh / tan(t)) / ew)), eh, Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(eh / Float64(tan(t) * ew)) ^ 2.0)))));
	else
		tmp = abs(fma(Float64(ew * ew), Float64(0.5 * Float64(Float64((sin(t) ^ 2.0) / eh) / cos(t))), Float64(eh * cos(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -4.5e+65], N[Not[LessEqual[ew, 4.7e-51]], $MachinePrecision]], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.5e65 or 4.6999999999999997e-51 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 90.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 90.3

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

   8. Applied rewrites 90.3%

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

   if -4.5e65 < ew < 4.6999999999999997e-51

   1. Initial program 99.8%

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

   4. Applied rewrites 54.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 40.6%

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

   7. Taylor expanded in ew around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 83.2

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \color{blue}{\cos t}\right)\right| \]

   9. Applied rewrites 83.2%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.5 \cdot 10^{+65} \lor \neg \left(ew \leq 4.7 \cdot 10^{-51}\right):\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{eh}{\tan t \cdot ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \frac{\frac{eh}{\tan t}}{ew}\\ t_3 := eh \cdot \cos t\\ t_4 := \tan t \cdot ew\\ \mathbf{if}\;ew \leq -4.5 \cdot 10^{+65}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{t\_3}{t\_4}, eh, t\_1\right)}{\sqrt{1 + {t\_2}^{2}}}\right|\\ \mathbf{elif}\;ew \leq 4.7 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, t\_3\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_2, eh, t\_1\right)}{\sqrt{1 + {\left(\frac{eh}{t\_4}\right)}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew))
        (t_2 (/ (/ eh (tan t)) ew))
        (t_3 (* eh (cos t)))
        (t_4 (* (tan t) ew)))
   (if (<= ew -4.5e+65)
     (fabs (/ (fma (/ t_3 t_4) eh t_1) (sqrt (+ 1.0 (pow t_2 2.0)))))
     (if (<= ew 4.7e-51)
       (fabs (fma (* ew ew) (* 0.5 (/ (/ (pow (sin t) 2.0) eh) (cos t))) t_3))
       (fabs
        (/
         (fma (* (cos t) t_2) eh t_1)
         (sqrt (+ 1.0 (pow (/ eh t_4) 2.0)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = (eh / tan(t)) / ew;
	double t_3 = eh * cos(t);
	double t_4 = tan(t) * ew;
	double tmp;
	if (ew <= -4.5e+65) {
		tmp = fabs((fma((t_3 / t_4), eh, t_1) / sqrt((1.0 + pow(t_2, 2.0)))));
	} else if (ew <= 4.7e-51) {
		tmp = fabs(fma((ew * ew), (0.5 * ((pow(sin(t), 2.0) / eh) / cos(t))), t_3));
	} else {
		tmp = fabs((fma((cos(t) * t_2), eh, t_1) / sqrt((1.0 + pow((eh / t_4), 2.0)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(Float64(eh / tan(t)) / ew)
	t_3 = Float64(eh * cos(t))
	t_4 = Float64(tan(t) * ew)
	tmp = 0.0
	if (ew <= -4.5e+65)
		tmp = abs(Float64(fma(Float64(t_3 / t_4), eh, t_1) / sqrt(Float64(1.0 + (t_2 ^ 2.0)))));
	elseif (ew <= 4.7e-51)
		tmp = abs(fma(Float64(ew * ew), Float64(0.5 * Float64(Float64((sin(t) ^ 2.0) / eh) / cos(t))), t_3));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_2), eh, t_1) / sqrt(Float64(1.0 + (Float64(eh / t_4) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[ew, -4.5e+65], N[Abs[N[(N[(N[(t$95$3 / t$95$4), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 4.7e-51], N[Abs[N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$2), $MachinePrecision] * eh + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(eh / t$95$4), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.5e65

   1. Initial program 99.6%

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

   4. Applied rewrites 95.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 96.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-/l/ N/A

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

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lift-tan.f64 96.2

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

   8. Applied rewrites 96.2%

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

   if -4.5e65 < ew < 4.6999999999999997e-51

   1. Initial program 99.8%

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

   4. Applied rewrites 54.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 40.6%

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

   7. Taylor expanded in ew around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 83.2

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \color{blue}{\cos t}\right)\right| \]

   9. Applied rewrites 83.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)}\right| \]

   if 4.6999999999999997e-51 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 86.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 85.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-/l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 85.1

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

   8. Applied rewrites 85.1%

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

Alternative 7: 74.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)\right|\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{+140}:\\ \;\;\;\;\left|\frac{\frac{\mathsf{fma}\left(ew \cdot ew, \sin t, \frac{eh \cdot eh}{t}\right)}{ew}}{\cosh \sinh^{-1} t\_1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (<= ew -1.9e+78)
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) 1.0))
     (if (<= ew 8e-51)
       (fabs
        (fma
         (* ew ew)
         (* 0.5 (/ (/ (pow (sin t) 2.0) eh) (cos t)))
         (* eh (cos t))))
       (if (<= ew 1.35e+140)
         (fabs
          (/
           (/ (fma (* ew ew) (sin t) (/ (* eh eh) t)) ew)
           (cosh (asinh t_1))))
         (fabs (* ew (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if (ew <= -1.9e+78) {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / 1.0));
	} else if (ew <= 8e-51) {
		tmp = fabs(fma((ew * ew), (0.5 * ((pow(sin(t), 2.0) / eh) / cos(t))), (eh * cos(t))));
	} else if (ew <= 1.35e+140) {
		tmp = fabs(((fma((ew * ew), sin(t), ((eh * eh) / t)) / ew) / cosh(asinh(t_1))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if (ew <= -1.9e+78)
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / 1.0));
	elseif (ew <= 8e-51)
		tmp = abs(fma(Float64(ew * ew), Float64(0.5 * Float64(Float64((sin(t) ^ 2.0) / eh) / cos(t))), Float64(eh * cos(t))));
	elseif (ew <= 1.35e+140)
		tmp = abs(Float64(Float64(fma(Float64(ew * ew), sin(t), Float64(Float64(eh * eh) / t)) / ew) / cosh(asinh(t_1))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[ew, -1.9e+78], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8e-51], N[Abs[N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.35e+140], N[Abs[N[(N[(N[(N[(ew * ew), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[(eh * eh), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if ew < -1.9e78

   1. Initial program 99.6%

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 87.8%

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

   if -1.9e78 < ew < 8.0000000000000001e-51

   1. Initial program 99.8%

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

   4. Applied rewrites 55.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 41.5%

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

   7. Taylor expanded in ew around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 82.8

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \color{blue}{\cos t}\right)\right| \]

   9. Applied rewrites 82.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)}\right| \]

   if 8.0000000000000001e-51 < ew < 1.35000000000000009e140

   1. Initial program 99.9%

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

   4. Applied rewrites 84.7%

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

   5. Taylor expanded in ew around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-sin.f64 77.9

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

   7. Applied rewrites 77.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 73.4%

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

   if 1.35000000000000009e140 < ew

   1. Initial program 99.6%

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

   4. Applied rewrites 90.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 84.6

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

   7. Applied rewrites 84.6%

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

Alternative 8: 74.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{elif}\;ew \leq 6.4 \cdot 10^{+34}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.9e+78)
   (fabs (/ (fma (* (cos t) (/ (/ eh (tan t)) ew)) eh (* (sin t) ew)) 1.0))
   (if (<= ew 6.4e+34)
     (fabs
      (fma
       (* ew ew)
       (* 0.5 (/ (/ (pow (sin t) 2.0) eh) (cos t)))
       (* eh (cos t))))
     (fabs (* ew (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.9e+78) {
		tmp = fabs((fma((cos(t) * ((eh / tan(t)) / ew)), eh, (sin(t) * ew)) / 1.0));
	} else if (ew <= 6.4e+34) {
		tmp = fabs(fma((ew * ew), (0.5 * ((pow(sin(t), 2.0) / eh) / cos(t))), (eh * cos(t))));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.9e+78)
		tmp = abs(Float64(fma(Float64(cos(t) * Float64(Float64(eh / tan(t)) / ew)), eh, Float64(sin(t) * ew)) / 1.0));
	elseif (ew <= 6.4e+34)
		tmp = abs(fma(Float64(ew * ew), Float64(0.5 * Float64(Float64((sin(t) ^ 2.0) / eh) / cos(t))), Float64(eh * cos(t))));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.9e+78], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 6.4e+34], N[Abs[N[(N[(ew * ew), $MachinePrecision] * N[(0.5 * N[(N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.9e78

   1. Initial program 99.6%

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 87.8%

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

   if -1.9e78 < ew < 6.3999999999999997e34

   1. Initial program 99.8%

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

   4. Applied rewrites 58.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in ew around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. distribute-lft1-in N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 79.2

          \[\leadsto \left|\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \color{blue}{\cos t}\right)\right| \]

   9. Applied rewrites 79.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew \cdot ew, 0.5 \cdot \frac{\frac{{\sin t}^{2}}{eh}}{\cos t}, eh \cdot \cos t\right)}\right| \]

   if 6.3999999999999997e34 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 9: 74.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+78}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{1}\right|\\ \mathbf{elif}\;ew \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.9e+78)
   (fabs (/ (fma (* (cos t) (/ (/ eh (tan t)) ew)) eh (* (sin t) ew)) 1.0))
   (if (<= ew 3.7e+34) (fabs (* eh (cos t))) (fabs (* ew (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.9e+78) {
		tmp = fabs((fma((cos(t) * ((eh / tan(t)) / ew)), eh, (sin(t) * ew)) / 1.0));
	} else if (ew <= 3.7e+34) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((ew * sin(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.9e+78)
		tmp = abs(Float64(fma(Float64(cos(t) * Float64(Float64(eh / tan(t)) / ew)), eh, Float64(sin(t) * ew)) / 1.0));
	elseif (ew <= 3.7e+34)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(ew * sin(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.9e+78], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.7e+34], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.9e78

   1. Initial program 99.6%

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

   4. Applied rewrites 95.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 87.8%

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

   if -1.9e78 < ew < 3.70000000000000009e34

   1. Initial program 99.8%

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

   4. Applied rewrites 58.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 79.0

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

   9. Applied rewrites 79.0%

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

   if 3.70000000000000009e34 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 88.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 77.2

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

   7. Applied rewrites 77.2%

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

Alternative 10: 74.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+78} \lor \neg \left(ew \leq 3.7 \cdot 10^{+34}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.9e+78) (not (<= ew 3.7e+34)))
   (fabs (* ew (sin t)))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e+78) || !(ew <= 3.7e+34)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.9d+78)) .or. (.not. (ew <= 3.7d+34))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.9e+78) || !(ew <= 3.7e+34)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.9e+78) or not (ew <= 3.7e+34):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.9e+78) || !(ew <= 3.7e+34))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.9e+78) || ~((ew <= 3.7e+34)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.9e+78], N[Not[LessEqual[ew, 3.7e+34]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.9e78 or 3.70000000000000009e34 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 92.5%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 83.2

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

   7. Applied rewrites 83.2%

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

   if -1.9e78 < ew < 3.70000000000000009e34

   1. Initial program 99.8%

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

   4. Applied rewrites 58.8%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 45.9%

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

   7. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 79.0

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

   9. Applied rewrites 79.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{+78} \lor \neg \left(ew \leq 3.7 \cdot 10^{+34}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 43.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-223} \lor \neg \left(t \leq 1.7 \cdot 10^{-246}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -1.2e-223) (not (<= t 1.7e-246)))
   (fabs (* ew (sin t)))
   (fabs
    (fma
     (* t t)
     (/
      (fma
       -0.5
       (* (* ew ew) (fma -0.6666666666666666 (/ (* eh eh) (* ew ew)) 1.0))
       (* ew (+ ew (* (/ (* eh eh) ew) -0.8333333333333334))))
      eh)
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -1.2e-223) || !(t <= 1.7e-246)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(fma((t * t), (fma(-0.5, ((ew * ew) * fma(-0.6666666666666666, ((eh * eh) / (ew * ew)), 1.0)), (ew * (ew + (((eh * eh) / ew) * -0.8333333333333334)))) / eh), eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -1.2e-223) || !(t <= 1.7e-246))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(fma(Float64(t * t), Float64(fma(-0.5, Float64(Float64(ew * ew) * fma(-0.6666666666666666, Float64(Float64(eh * eh) / Float64(ew * ew)), 1.0)), Float64(ew * Float64(ew + Float64(Float64(Float64(eh * eh) / ew) * -0.8333333333333334)))) / eh), eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -1.2e-223], N[Not[LessEqual[t, 1.7e-246]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(N[(ew * ew), $MachinePrecision] * N[(-0.6666666666666666 * N[(N[(eh * eh), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(ew * N[(ew + N[(N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.19999999999999993e-223 or 1.7000000000000001e-246 < t

   1. Initial program 99.7%

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 50.9

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

   7. Applied rewrites 50.9%

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

   if -1.19999999999999993e-223 < t < 1.7000000000000001e-246

   1. Initial program 100.0%

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

   4. Applied rewrites 41.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 32.7%

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

   7. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{eh + {t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}\right)}\right| \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}\right) + eh}\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}, eh\right)}\right| \]

   9. Applied rewrites 48.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-223} \lor \neg \left(t \leq 1.7 \cdot 10^{-246}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 21.3% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.3 \cdot 10^{+69} \lor \neg \left(ew \leq 5.2 \cdot 10^{-12}\right):\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.3e+69) (not (<= ew 5.2e-12)))
   (fabs (* ew t))
   (fabs
    (fma
     (* t t)
     (/
      (fma
       -0.5
       (* (* ew ew) (fma -0.6666666666666666 (/ (* eh eh) (* ew ew)) 1.0))
       (* ew (+ ew (* (/ (* eh eh) ew) -0.8333333333333334))))
      eh)
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.3e+69) || !(ew <= 5.2e-12)) {
		tmp = fabs((ew * t));
	} else {
		tmp = fabs(fma((t * t), (fma(-0.5, ((ew * ew) * fma(-0.6666666666666666, ((eh * eh) / (ew * ew)), 1.0)), (ew * (ew + (((eh * eh) / ew) * -0.8333333333333334)))) / eh), eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.3e+69) || !(ew <= 5.2e-12))
		tmp = abs(Float64(ew * t));
	else
		tmp = abs(fma(Float64(t * t), Float64(fma(-0.5, Float64(Float64(ew * ew) * fma(-0.6666666666666666, Float64(Float64(eh * eh) / Float64(ew * ew)), 1.0)), Float64(ew * Float64(ew + Float64(Float64(Float64(eh * eh) / ew) * -0.8333333333333334)))) / eh), eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.3e+69], N[Not[LessEqual[ew, 5.2e-12]], $MachinePrecision]], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(N[(ew * ew), $MachinePrecision] * N[(-0.6666666666666666 * N[(N[(eh * eh), $MachinePrecision] / N[(ew * ew), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(ew * N[(ew + N[(N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.8333333333333334), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.30000000000000017e69 or 5.19999999999999965e-12 < ew

   1. Initial program 99.7%

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

   4. Applied rewrites 90.5%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 78.5

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

   7. Applied rewrites 78.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 35.0%

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

   if -2.30000000000000017e69 < ew < 5.19999999999999965e-12

   1. Initial program 99.8%

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

   4. Applied rewrites 56.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. rem-square-sqrt N/A

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

      6. +-commutative N/A

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

      7. cosh-asinh N/A

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

      8. lift-asinh.f64 N/A

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

      9. lift-cosh.f64 N/A

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

      10. +-commutative N/A

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

      11. cosh-asinh N/A

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

      12. lift-asinh.f64 N/A

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

      13. lift-cosh.f64 N/A

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

   6. Applied rewrites 42.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{eh + {t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}\right)}\right| \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left|\color{blue}{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}\right) + eh}\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{{ew}^{2} \cdot \left(1 - \frac{2}{3} \cdot \frac{{eh}^{2}}{{ew}^{2}}\right)}{eh} + \frac{ew \cdot \left(\left(ew + \frac{-1}{2} \cdot \frac{{eh}^{2}}{ew}\right) - \frac{1}{3} \cdot \frac{{eh}^{2}}{ew}\right)}{eh}, eh\right)}\right| \]

   9. Applied rewrites 20.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.3 \cdot 10^{+69} \lor \neg \left(ew \leq 5.2 \cdot 10^{-12}\right):\\ \;\;\;\;\left|ew \cdot t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t \cdot t, \frac{\mathsf{fma}\left(-0.5, \left(ew \cdot ew\right) \cdot \mathsf{fma}\left(-0.6666666666666666, \frac{eh \cdot eh}{ew \cdot ew}, 1\right), ew \cdot \left(ew + \frac{eh \cdot eh}{ew} \cdot -0.8333333333333334\right)\right)}{eh}, eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 18.6% accurate, 108.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 71.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 46.4

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

7. Applied rewrites 46.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 20.7%

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

Alternative 14: 10.2% accurate, 145.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 71.0%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 46.4

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

7. Applied rewrites 46.4%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 20.7%

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

    1. lift-fabs.f64 N/A

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

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

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

    3. sqrt-prod N/A

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

    4. rem-square-sqrt 12.3

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

12. Applied rewrites 12.3%

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

Reproduce

herbie shell --seed 2025005 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))