Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.4s

Alternatives: 18

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

Alternative 2: 41.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\\ t_3 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|t\_1 \cdot \cos t\_3 + \left(eh \cdot \cos t\right) \cdot \sin t\_3\right| \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\left|\frac{t\_2}{\sqrt{1 + t\_2 \cdot t\_2}} \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2
         (*
          (/ 1.0 ew)
          (/
           eh
           (*
            t
            (+
             1.0
             (*
              (* t t)
              (- (* 0.008333333333333333 (* t t)) 0.16666666666666666)))))))
        (t_3 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (fabs (+ (* t_1 (cos t_3)) (* (* eh (cos t)) (sin t_3)))) 2e-199)
     (fabs (* (/ t_2 (sqrt (+ 1.0 (* t_2 t_2)))) eh))
     (fabs t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = (1.0 / ew) * (eh / (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))));
	double t_3 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs(((t_1 * cos(t_3)) + ((eh * cos(t)) * sin(t_3)))) <= 2e-199) {
		tmp = fabs(((t_2 / sqrt((1.0 + (t_2 * t_2)))) * eh));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = (1.0d0 / ew) * (eh / (t * (1.0d0 + ((t * t) * ((0.008333333333333333d0 * (t * t)) - 0.16666666666666666d0)))))
    t_3 = atan(((eh / ew) / tan(t)))
    if (abs(((t_1 * cos(t_3)) + ((eh * cos(t)) * sin(t_3)))) <= 2d-199) then
        tmp = abs(((t_2 / sqrt((1.0d0 + (t_2 * t_2)))) * eh))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = (1.0 / ew) * (eh / (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))));
	double t_3 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if (Math.abs(((t_1 * Math.cos(t_3)) + ((eh * Math.cos(t)) * Math.sin(t_3)))) <= 2e-199) {
		tmp = Math.abs(((t_2 / Math.sqrt((1.0 + (t_2 * t_2)))) * eh));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = (1.0 / ew) * (eh / (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))))
	t_3 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if math.fabs(((t_1 * math.cos(t_3)) + ((eh * math.cos(t)) * math.sin(t_3)))) <= 2e-199:
		tmp = math.fabs(((t_2 / math.sqrt((1.0 + (t_2 * t_2)))) * eh))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(Float64(1.0 / ew) * Float64(eh / Float64(t * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.008333333333333333 * Float64(t * t)) - 0.16666666666666666))))))
	t_3 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(t_1 * cos(t_3)) + Float64(Float64(eh * cos(t)) * sin(t_3)))) <= 2e-199)
		tmp = abs(Float64(Float64(t_2 / sqrt(Float64(1.0 + Float64(t_2 * t_2)))) * eh));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = (1.0 / ew) * (eh / (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))));
	t_3 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if (abs(((t_1 * cos(t_3)) + ((eh * cos(t)) * sin(t_3)))) <= 2e-199)
		tmp = abs(((t_2 / sqrt((1.0 + (t_2 * t_2)))) * eh));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / ew), $MachinePrecision] * N[(eh / N[(t * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$1 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-199], N[Abs[N[(N[(t$95$2 / N[Sqrt[N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.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))))))) < 1.99999999999999996e-199

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 60.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      4. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      8. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      9. lower-*.f64 58.0

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   7. Applied rewrites 58.0%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 58.0%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]
   11. Step-by-step derivation

       1. lift-tanh.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

       2. lift-asinh.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

       3. tanh-asinh N/A

          \[\leadsto \left|\frac{\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}}{\sqrt{1 + \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right)}} \cdot eh\right| \]

       4. lower-/.f64 N/A

          \[\leadsto \left|\frac{\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}}{\sqrt{1 + \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right)}} \cdot eh\right| \]

   12. Applied rewrites 39.9%

       \[\leadsto \left|\frac{\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}}{\sqrt{1 + \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right)}} \cdot eh\right| \]

   if 1.99999999999999996e-199 < (fabs.f64 (+.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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 41.7

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

   6. Applied rewrites 41.7%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \left|\mathsf{fma}\left(ew, \sin t \cdot \frac{1}{\sqrt{1 + {t\_1}^{2}}}, \left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} t\_1\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (fma
     ew
     (* (sin t) (/ 1.0 (sqrt (+ 1.0 (pow t_1 2.0)))))
     (* (* (cos t) eh) (tanh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs(fma(ew, (sin(t) * (1.0 / sqrt((1.0 + pow(t_1, 2.0))))), ((cos(t) * eh) * tanh(asinh(t_1)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(fma(ew, Float64(sin(t) * Float64(1.0 / sqrt(Float64(1.0 + (t_1 ^ 2.0))))), Float64(Float64(cos(t) * eh) * tanh(asinh(t_1)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. unpow2 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(ew, \sin t \cdot \frac{1}{\sqrt{1 + {\left(\frac{\frac{eh}{ew}}{\tan t}\right)}^{2}}}, \left(\cos t \cdot eh\right) \cdot \tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   ew
   (* (sin t) (/ 1.0 (sqrt (+ 1.0 (pow (/ (/ eh ew) (tan t)) 2.0)))))
   (* (* (cos t) eh) (tanh (/ (* eh (cos t)) (* ew (sin t))))))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, (sin(t) * (1.0 / sqrt((1.0 + pow(((eh / ew) / tan(t)), 2.0))))), ((cos(t) * eh) * tanh(((eh * cos(t)) / (ew * sin(t)))))));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, Float64(sin(t) * Float64(1.0 / sqrt(Float64(1.0 + (Float64(Float64(eh / ew) / tan(t)) ^ 2.0))))), Float64(Float64(cos(t) * eh) * tanh(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t)))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. unpow2 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. lower-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 99.2

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

8. Applied rewrites 99.2%

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

Alternative 5: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t)))))
   (*
    (* eh (cos t))
    (sin (atan (/ (/ (fma -0.3333333333333333 (* (* t t) eh) eh) ew) t)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * sin(t)) * cos(atan(((eh / ew) / tan(t))))) + ((eh * cos(t)) * sin(atan(((fma(-0.3333333333333333, ((t * t) * eh), eh) / ew) / t))))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(t * t) * eh), eh) / ew) / t))))))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.1

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)\right| \]

4. Applied rewrites 99.1%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)}\right| \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)))
   (fabs
    (+
     (* (* ew (sin t)) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
     (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return fabs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = (eh / ew) / t
    code = abs((((ew * sin(t)) * (1.0d0 / sqrt((1.0d0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return Math.abs((((ew * Math.sin(t)) * (1.0 / Math.sqrt((1.0 + (t_1 * t_1))))) + ((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	return math.fabs((((ew * math.sin(t)) * (1.0 / math.sqrt((1.0 + (t_1 * t_1))))) + ((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t)))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	return abs(Float64(Float64(Float64(ew * sin(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	tmp = abs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 99.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

Alternative 7: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 98.4

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

6. Applied rewrites 98.4%

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

Alternative 8: 98.4% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma ew (sin t) (* (* (cos t) eh) (tanh (asinh (/ (/ eh ew) (tan t))))))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, sin(t), ((cos(t) * eh) * tanh(asinh(((eh / ew) / tan(t)))))));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, sin(t), Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(eh / ew) / tan(t)))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. unpow2 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. pow2 N/A

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

   2. cos-atan-rev N/A

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

   3. lift-sin.f64 98.4

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

8. Applied rewrites 98.4%

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

Alternative 9: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \frac{\frac{eh}{ew}}{t}\\ t_3 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -2.2 \cdot 10^{+185}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{t\_3}{t\_1}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \frac{1}{\sqrt{1 + t\_2 \cdot t\_2}} + t\_3 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t))) (t_2 (/ (/ eh ew) t)) (t_3 (* eh (cos t))))
   (if (<= eh -2.2e+185)
     (fabs (* eh (* (cos t) (sin (atan (/ t_3 t_1))))))
     (fabs
      (+
       (* t_1 (/ 1.0 (sqrt (+ 1.0 (* t_2 t_2)))))
       (* t_3 (sin (atan (/ eh (* ew t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = (eh / ew) / t;
	double t_3 = eh * cos(t);
	double tmp;
	if (eh <= -2.2e+185) {
		tmp = fabs((eh * (cos(t) * sin(atan((t_3 / t_1))))));
	} else {
		tmp = fabs(((t_1 * (1.0 / sqrt((1.0 + (t_2 * t_2))))) + (t_3 * sin(atan((eh / (ew * t)))))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = (eh / ew) / t
    t_3 = eh * cos(t)
    if (eh <= (-2.2d+185)) then
        tmp = abs((eh * (cos(t) * sin(atan((t_3 / t_1))))))
    else
        tmp = abs(((t_1 * (1.0d0 / sqrt((1.0d0 + (t_2 * t_2))))) + (t_3 * sin(atan((eh / (ew * t)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = (eh / ew) / t;
	double t_3 = eh * Math.cos(t);
	double tmp;
	if (eh <= -2.2e+185) {
		tmp = Math.abs((eh * (Math.cos(t) * Math.sin(Math.atan((t_3 / t_1))))));
	} else {
		tmp = Math.abs(((t_1 * (1.0 / Math.sqrt((1.0 + (t_2 * t_2))))) + (t_3 * Math.sin(Math.atan((eh / (ew * t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = (eh / ew) / t
	t_3 = eh * math.cos(t)
	tmp = 0
	if eh <= -2.2e+185:
		tmp = math.fabs((eh * (math.cos(t) * math.sin(math.atan((t_3 / t_1))))))
	else:
		tmp = math.fabs(((t_1 * (1.0 / math.sqrt((1.0 + (t_2 * t_2))))) + (t_3 * math.sin(math.atan((eh / (ew * t)))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(Float64(eh / ew) / t)
	t_3 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -2.2e+185)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(t_3 / t_1))))));
	else
		tmp = abs(Float64(Float64(t_1 * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_2 * t_2))))) + Float64(t_3 * sin(atan(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = (eh / ew) / t;
	t_3 = eh * cos(t);
	tmp = 0.0;
	if (eh <= -2.2e+185)
		tmp = abs((eh * (cos(t) * sin(atan((t_3 / t_1))))));
	else
		tmp = abs(((t_1 * (1.0 / sqrt((1.0 + (t_2 * t_2))))) + (t_3 * sin(atan((eh / (ew * t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.2e+185], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(t$95$3 / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$1 * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.2000000000000001e185

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 57.2%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 94.2

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

   7. Applied rewrites 94.2%

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

   if -2.2000000000000001e185 < 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. Taylor expanded in t around 0

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

   4. Applied rewrites 99.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 99.1

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 89.2

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

   9. Applied rewrites 89.2%

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

Alternative 10: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(ew, \sin t, \left(\cos t \cdot eh\right) \cdot \tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (fma
   ew
   (sin t)
   (* (* (cos t) eh) (tanh (/ (* eh (cos t)) (* ew (sin t))))))))

C

double code(double eh, double ew, double t) {
	return fabs(fma(ew, sin(t), ((cos(t) * eh) * tanh(((eh * cos(t)) / (ew * sin(t)))))));
}

Julia

function code(eh, ew, t)
	return abs(fma(ew, sin(t), Float64(Float64(cos(t) * eh) * tanh(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t)))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. unpow2 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. lower-/.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-*.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Taylor expanded in eh around 0

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

    1. pow2 N/A

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

    2. lift-sin.f64 98.4

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

11. Applied rewrites 98.4%

    \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\sin t}, \left(\cos t \cdot eh\right) \cdot \tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right| \]
12. Add Preprocessing

Alternative 11: 89.1% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)))
   (fabs
    (+
     (* (* ew (sin t)) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
     (* (* eh (cos t)) (sin (atan (/ eh (* ew t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return fabs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan((eh / (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
    real(8) :: t_1
    t_1 = (eh / ew) / t
    code = abs((((ew * sin(t)) * (1.0d0 / sqrt((1.0d0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan((eh / (ew * t)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return Math.abs((((ew * Math.sin(t)) * (1.0 / Math.sqrt((1.0 + (t_1 * t_1))))) + ((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * t)))))));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	return math.fabs((((ew * math.sin(t)) * (1.0 / math.sqrt((1.0 + (t_1 * t_1))))) + ((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * t)))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	return abs(Float64(Float64(Float64(ew * sin(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * t)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	tmp = abs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan((eh / (ew * t)))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in t around 0

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

4. Applied rewrites 99.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 99.1

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.1

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

9. Applied rewrites 89.1%

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

Alternative 12: 82.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ t_2 := eh \cdot \cos t\\ t_3 := \frac{eh}{ew \cdot t}\\ t_4 := \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}}\\ t_5 := \left|\left(ew \cdot \sin t\right) \cdot t\_4 + t\_2 \cdot \frac{t\_3}{\sqrt{1 + t\_3 \cdot t\_3}}\right|\\ \mathbf{if}\;t \leq -70000000000:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t \leq 2.3 \cdot 10^{-15}:\\ \;\;\;\;\left|\left(ew \cdot t\right) \cdot t\_4 + t\_2 \cdot \sin \tan^{-1} t\_3\right|\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t))
        (t_2 (* eh (cos t)))
        (t_3 (/ eh (* ew t)))
        (t_4 (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
        (t_5
         (fabs
          (+
           (* (* ew (sin t)) t_4)
           (* t_2 (/ t_3 (sqrt (+ 1.0 (* t_3 t_3)))))))))
   (if (<= t -70000000000.0)
     t_5
     (if (<= t 2.3e-15)
       (fabs (+ (* (* ew t) t_4) (* t_2 (sin (atan t_3)))))
       t_5))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = eh * cos(t);
	double t_3 = eh / (ew * t);
	double t_4 = 1.0 / sqrt((1.0 + (t_1 * t_1)));
	double t_5 = fabs((((ew * sin(t)) * t_4) + (t_2 * (t_3 / sqrt((1.0 + (t_3 * t_3)))))));
	double tmp;
	if (t <= -70000000000.0) {
		tmp = t_5;
	} else if (t <= 2.3e-15) {
		tmp = fabs((((ew * t) * t_4) + (t_2 * sin(atan(t_3)))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_1 = (eh / ew) / t
    t_2 = eh * cos(t)
    t_3 = eh / (ew * t)
    t_4 = 1.0d0 / sqrt((1.0d0 + (t_1 * t_1)))
    t_5 = abs((((ew * sin(t)) * t_4) + (t_2 * (t_3 / sqrt((1.0d0 + (t_3 * t_3)))))))
    if (t <= (-70000000000.0d0)) then
        tmp = t_5
    else if (t <= 2.3d-15) then
        tmp = abs((((ew * t) * t_4) + (t_2 * sin(atan(t_3)))))
    else
        tmp = t_5
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = eh * Math.cos(t);
	double t_3 = eh / (ew * t);
	double t_4 = 1.0 / Math.sqrt((1.0 + (t_1 * t_1)));
	double t_5 = Math.abs((((ew * Math.sin(t)) * t_4) + (t_2 * (t_3 / Math.sqrt((1.0 + (t_3 * t_3)))))));
	double tmp;
	if (t <= -70000000000.0) {
		tmp = t_5;
	} else if (t <= 2.3e-15) {
		tmp = Math.abs((((ew * t) * t_4) + (t_2 * Math.sin(Math.atan(t_3)))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	t_2 = eh * math.cos(t)
	t_3 = eh / (ew * t)
	t_4 = 1.0 / math.sqrt((1.0 + (t_1 * t_1)))
	t_5 = math.fabs((((ew * math.sin(t)) * t_4) + (t_2 * (t_3 / math.sqrt((1.0 + (t_3 * t_3)))))))
	tmp = 0
	if t <= -70000000000.0:
		tmp = t_5
	elif t <= 2.3e-15:
		tmp = math.fabs((((ew * t) * t_4) + (t_2 * math.sin(math.atan(t_3)))))
	else:
		tmp = t_5
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	t_2 = Float64(eh * cos(t))
	t_3 = Float64(eh / Float64(ew * t))
	t_4 = Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))
	t_5 = abs(Float64(Float64(Float64(ew * sin(t)) * t_4) + Float64(t_2 * Float64(t_3 / sqrt(Float64(1.0 + Float64(t_3 * t_3)))))))
	tmp = 0.0
	if (t <= -70000000000.0)
		tmp = t_5;
	elseif (t <= 2.3e-15)
		tmp = abs(Float64(Float64(Float64(ew * t) * t_4) + Float64(t_2 * sin(atan(t_3)))));
	else
		tmp = t_5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	t_2 = eh * cos(t);
	t_3 = eh / (ew * t);
	t_4 = 1.0 / sqrt((1.0 + (t_1 * t_1)));
	t_5 = abs((((ew * sin(t)) * t_4) + (t_2 * (t_3 / sqrt((1.0 + (t_3 * t_3)))))));
	tmp = 0.0;
	if (t <= -70000000000.0)
		tmp = t_5;
	elseif (t <= 2.3e-15)
		tmp = abs((((ew * t) * t_4) + (t_2 * sin(atan(t_3)))));
	else
		tmp = t_5;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$2 * N[(t$95$3 / N[Sqrt[N[(1.0 + N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -70000000000.0], t$95$5, If[LessEqual[t, 2.3e-15], N[Abs[N[(N[(N[(ew * t), $MachinePrecision] * t$95$4), $MachinePrecision] + N[(t$95$2 * N[Sin[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$5]]]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -7e10 or 2.2999999999999999e-15 < t

   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. Taylor expanded in t around 0

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

   4. Applied rewrites 98.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 98.2

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 78.4

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

   9. Applied rewrites 78.4%

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

       1. lift-sin.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. sin-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 65.9

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

   11. Applied rewrites 65.9%

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

   if -7e10 < t < 2.2999999999999999e-15

   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. Taylor expanded in t around 0

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

   4. Applied rewrites 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.9

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

   9. Applied rewrites 99.9%

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

   10. Taylor expanded in t around 0

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

       1. lift-*.f64 99.0

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

   12. Applied rewrites 99.0%

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

Alternative 13: 69.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ t_2 := \frac{eh}{ew \cdot t}\\ t_3 := \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}} + \left(eh \cdot \cos t\right) \cdot \frac{t\_2}{\sqrt{1 + t\_2 \cdot t\_2}}\right|\\ \mathbf{if}\;t \leq -4.5 \cdot 10^{-62}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-35}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t))
        (t_2 (/ eh (* ew t)))
        (t_3
         (fabs
          (+
           (* (* ew (sin t)) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
           (* (* eh (cos t)) (/ t_2 (sqrt (+ 1.0 (* t_2 t_2)))))))))
   (if (<= t -4.5e-62)
     t_3
     (if (<= t 1.4e-35)
       (fabs
        (*
         (tanh
          (asinh
           (* (/ 1.0 ew) (/ (+ eh (* 0.16666666666666666 (* eh (* t t)))) t))))
         eh))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = eh / (ew * t);
	double t_3 = fabs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * (t_2 / sqrt((1.0 + (t_2 * t_2)))))));
	double tmp;
	if (t <= -4.5e-62) {
		tmp = t_3;
	} else if (t <= 1.4e-35) {
		tmp = fabs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	t_2 = eh / (ew * t)
	t_3 = math.fabs((((ew * math.sin(t)) * (1.0 / math.sqrt((1.0 + (t_1 * t_1))))) + ((eh * math.cos(t)) * (t_2 / math.sqrt((1.0 + (t_2 * t_2)))))))
	tmp = 0
	if t <= -4.5e-62:
		tmp = t_3
	elif t <= 1.4e-35:
		tmp = math.fabs((math.tanh(math.asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	t_2 = Float64(eh / Float64(ew * t))
	t_3 = abs(Float64(Float64(Float64(ew * sin(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))) + Float64(Float64(eh * cos(t)) * Float64(t_2 / sqrt(Float64(1.0 + Float64(t_2 * t_2)))))))
	tmp = 0.0
	if (t <= -4.5e-62)
		tmp = t_3;
	elseif (t <= 1.4e-35)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(1.0 / ew) * Float64(Float64(eh + Float64(0.16666666666666666 * Float64(eh * Float64(t * t)))) / t)))) * eh));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	t_2 = eh / (ew * t);
	t_3 = abs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * (t_2 / sqrt((1.0 + (t_2 * t_2)))))));
	tmp = 0.0;
	if (t <= -4.5e-62)
		tmp = t_3;
	elseif (t <= 1.4e-35)
		tmp = abs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[N[(1.0 + N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.5e-62], t$95$3, If[LessEqual[t, 1.4e-35], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(N[(eh + N[(0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.50000000000000018e-62 or 1.4e-35 < 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. Taylor expanded in t around 0

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

   4. Applied rewrites 98.4%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 98.4

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.2

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

   9. Applied rewrites 81.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. sin-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 65.6

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

   11. Applied rewrites 65.6%

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

   if -4.50000000000000018e-62 < t < 1.4e-35

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 74.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      4. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      8. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      9. lower-*.f64 74.9

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   7. Applied rewrites 74.9%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 74.9%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       2. lower-+.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       3. lower-*.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       4. lower-*.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       5. pow2 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

       6. lift-*.f64 74.9

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

   13. Applied rewrites 74.9%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 61.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)\right)}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -4.9e-14)
     t_1
     (if (<= t 7.5e-35)
       (fabs
        (*
         (tanh
          (asinh
           (/
            (* (cos t) eh)
            (*
             ew
             (*
              t
              (+
               1.0
               (*
                (* t t)
                (- (* 0.008333333333333333 (* t t)) 0.16666666666666666))))))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -4.9e-14) {
		tmp = t_1;
	} else if (t <= 7.5e-35) {
		tmp = fabs((tanh(asinh(((cos(t) * eh) / (ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))))))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -4.9e-14:
		tmp = t_1
	elif t <= 7.5e-35:
		tmp = math.fabs((math.tanh(math.asinh(((math.cos(t) * eh) / (ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))))))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(cos(t) * eh) / Float64(ew * Float64(t * Float64(1.0 + Float64(Float64(t * t) * Float64(Float64(0.008333333333333333 * Float64(t * t)) - 0.16666666666666666)))))))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs((tanh(asinh(((cos(t) * eh) / (ew * (t * (1.0 + ((t * t) * ((0.008333333333333333 * (t * t)) - 0.16666666666666666)))))))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.9e-14], t$95$1, If[LessEqual[t, 7.5e-35], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / N[(ew * N[(t * N[(1.0 + N[(N[(t * t), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.89999999999999995e-14 or 7.5e-35 < 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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if -4.89999999999999995e-14 < t < 7.5e-35

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      4. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      8. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      9. lower-*.f64 73.7

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   7. Applied rewrites 73.7%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      2. lift-/.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      3. lift-cos.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      4. lift-/.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      5. frac-times N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot eh\right| \]

      6. lower-/.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot eh\right| \]

      7. lift-cos.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot eh\right| \]

      8. lift-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)\right)}\right) \cdot eh\right| \]

      9. lower-*.f64 73.7

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)\right)}\right) \cdot eh\right| \]

   9. Applied rewrites 73.7%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t \cdot eh}{ew \cdot \left(t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)\right)}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 61.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -4.9e-14)
     t_1
     (if (<= t 7.5e-35)
       (fabs
        (*
         (tanh
          (asinh
           (* (/ 1.0 ew) (/ (+ eh (* 0.16666666666666666 (* eh (* t t)))) t))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -4.9e-14) {
		tmp = t_1;
	} else if (t <= 7.5e-35) {
		tmp = fabs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -4.9e-14:
		tmp = t_1
	elif t <= 7.5e-35:
		tmp = math.fabs((math.tanh(math.asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(1.0 / ew) * Float64(Float64(eh + Float64(0.16666666666666666 * Float64(eh * Float64(t * t)))) / t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.9e-14], t$95$1, If[LessEqual[t, 7.5e-35], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(N[(eh + N[(0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.89999999999999995e-14 or 7.5e-35 < 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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if -4.89999999999999995e-14 < t < 7.5e-35

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      4. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      6. lower--.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      7. lower-*.f64 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot {t}^{2} - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      8. unpow2 N/A

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]

      9. lower-*.f64 73.7

         \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   7. Applied rewrites 73.7%

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(\frac{1}{120} \cdot \left(t \cdot t\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 73.7%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t \cdot \left(1 + \left(t \cdot t\right) \cdot \left(0.008333333333333333 \cdot \left(t \cdot t\right) - 0.16666666666666666\right)\right)}\right) \cdot eh\right| \]

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       2. lower-+.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       3. lower-*.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       4. lower-*.f64 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot {t}^{2}\right)}{t}\right) \cdot eh\right| \]

       5. pow2 N/A

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + \frac{1}{6} \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

       6. lift-*.f64 73.7

          \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]

   13. Applied rewrites 73.7%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 61.5% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{-35}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -4.9e-14)
     t_1
     (if (<= t 7.5e-35) (fabs (* (tanh (asinh (/ eh (* ew t)))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -4.9e-14) {
		tmp = t_1;
	} else if (t <= 7.5e-35) {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -4.9e-14:
		tmp = t_1
	elif t <= 7.5e-35:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 7.5e-35)
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.9e-14], t$95$1, If[LessEqual[t, 7.5e-35], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.89999999999999995e-14 or 7.5e-35 < 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. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 51.0

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

   6. Applied rewrites 51.0%

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

   if -4.89999999999999995e-14 < t < 7.5e-35

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 73.7%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 73.7

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

   7. Applied rewrites 73.7%

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

Alternative 17: 41.2% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 41.2

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

6. Applied rewrites 41.2%

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

Alternative 18: 18.5% 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. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 41.2

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

6. Applied rewrites 41.2%

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

7. Taylor expanded in t around 0

   \[\leadsto \left|ew \cdot t\right| \]
8. Step-by-step derivation

9. Applied rewrites 18.5%

   \[\leadsto \left|ew \cdot t\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025105 
(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))))))))