Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ eh (* (tan t) ew)))))
   (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 / (tan(t) * ew)));
	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 / (tan(t) * ew)))
    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 / (Math.tan(t) * ew)));
	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 / (math.tan(t) * ew)))
	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(eh / Float64(tan(t) * ew)))
	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 / (tan(t) * ew)));
	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[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

3. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

Alternative 2: 74.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -8.8 \cdot 10^{+27}:\\ \;\;\;\;eh \cdot \left(-\cos t\right)\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{-48}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew} \cdot eh}{\tan t}, \cos t, t\_1\right)}{\sqrt{1 + {t\_2}^{2}}}\right|\\ \mathbf{elif}\;eh \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{eh \cdot eh}{\tan t \cdot ew}, \cos t, t\_1\right)}{\cosh \sinh^{-1} t\_2}\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)) (t_2 (/ (/ eh (tan t)) ew)))
   (if (<= eh -8.8e+27)
     (* eh (- (cos t)))
     (if (<= eh 5.5e-48)
       (fabs
        (/
         (fma (/ (* (/ eh ew) eh) (tan t)) (cos t) t_1)
         (sqrt (+ 1.0 (pow t_2 2.0)))))
       (if (<= eh 1.7e+106)
         (fabs
          (/
           (fma (/ (* eh eh) (* (tan t) ew)) (cos t) t_1)
           (cosh (asinh t_2))))
         (* eh (cos t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = (eh / tan(t)) / ew;
	double tmp;
	if (eh <= -8.8e+27) {
		tmp = eh * -cos(t);
	} else if (eh <= 5.5e-48) {
		tmp = fabs((fma((((eh / ew) * eh) / tan(t)), cos(t), t_1) / sqrt((1.0 + pow(t_2, 2.0)))));
	} else if (eh <= 1.7e+106) {
		tmp = fabs((fma(((eh * eh) / (tan(t) * ew)), cos(t), t_1) / cosh(asinh(t_2))));
	} else {
		tmp = eh * cos(t);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if (eh <= -8.8e+27)
		tmp = Float64(eh * Float64(-cos(t)));
	elseif (eh <= 5.5e-48)
		tmp = abs(Float64(fma(Float64(Float64(Float64(eh / ew) * eh) / tan(t)), cos(t), t_1) / sqrt(Float64(1.0 + (t_2 ^ 2.0)))));
	elseif (eh <= 1.7e+106)
		tmp = abs(Float64(fma(Float64(Float64(eh * eh) / Float64(tan(t) * ew)), cos(t), t_1) / cosh(asinh(t_2))));
	else
		tmp = Float64(eh * cos(t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -8.8e+27], N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 5.5e-48], N[Abs[N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision] + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.7e+106], N[Abs[N[(N[(N[(N[(eh * eh), $MachinePrecision] / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision] + t$95$1), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if eh < -8.7999999999999995e27

   1. Initial program 99.8%

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

   3. Applied rewrites 17.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 15.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 14.6

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

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{eh \cdot \cos \left(t + \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\cos \left(t + \mathsf{PI}\left(\right)\right)} \]

      2. cos-+PI N/A

         \[\leadsto eh \cdot \left(\mathsf{neg}\left(\cos t\right)\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto eh \cdot \left(-\cos t\right) \]

      4. lift-cos.f64 64.7

         \[\leadsto eh \cdot \left(-\cos t\right) \]

   10. Applied rewrites 64.7%

       \[\leadsto \color{blue}{eh \cdot \left(-\cos t\right)} \]

   if -8.7999999999999995e27 < eh < 5.50000000000000047e-48

   1. Initial program 99.8%

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

   3. Applied rewrites 88.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 80.6%

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

   if 5.50000000000000047e-48 < eh < 1.69999999999999997e106

   1. Initial program 99.8%

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

   3. Applied rewrites 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. frac-times N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if 1.69999999999999997e106 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 52.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 13.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 13.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 68.7

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 68.7%

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

Alternative 3: 79.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -1.2 \cdot 10^{+31}:\\ \;\;\;\;eh \cdot \left(-\cos t\right)\\ \mathbf{elif}\;eh \leq 1.7 \cdot 10^{+106}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(t\_1 \cdot eh, \cos t, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (<= eh -1.2e+31)
     (* eh (- (cos t)))
     (if (<= eh 1.7e+106)
       (fabs (/ (fma (* t_1 eh) (cos t) (* (sin t) ew)) (cosh (asinh t_1))))
       (* eh (cos t))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if (eh <= -1.2e+31) {
		tmp = eh * -cos(t);
	} else if (eh <= 1.7e+106) {
		tmp = fabs((fma((t_1 * eh), cos(t), (sin(t) * ew)) / cosh(asinh(t_1))));
	} else {
		tmp = eh * cos(t);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if (eh <= -1.2e+31)
		tmp = Float64(eh * Float64(-cos(t)));
	elseif (eh <= 1.7e+106)
		tmp = abs(Float64(fma(Float64(t_1 * eh), cos(t), Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	else
		tmp = Float64(eh * cos(t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -1.2e+31], N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 1.7e+106], N[Abs[N[(N[(N[(t$95$1 * eh), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.19999999999999991e31

   1. Initial program 99.8%

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

   3. Applied rewrites 16.9%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 15.1%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 14.2

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

   7. Applied rewrites 14.2%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{eh \cdot \cos \left(t + \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\cos \left(t + \mathsf{PI}\left(\right)\right)} \]

      2. cos-+PI N/A

         \[\leadsto eh \cdot \left(\mathsf{neg}\left(\cos t\right)\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto eh \cdot \left(-\cos t\right) \]

      4. lift-cos.f64 65.0

         \[\leadsto eh \cdot \left(-\cos t\right) \]

   10. Applied rewrites 65.0%

       \[\leadsto \color{blue}{eh \cdot \left(-\cos t\right)} \]

   if -1.19999999999999991e31 < eh < 1.69999999999999997e106

   1. Initial program 99.8%

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. lift-/.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if 1.69999999999999997e106 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 52.7

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 13.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 13.1%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 5.0

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

   7. Applied rewrites 5.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 68.7

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 68.7%

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

Alternative 4: 72.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8.8 \cdot 10^{+27}:\\ \;\;\;\;eh \cdot \left(-\cos t\right)\\ \mathbf{elif}\;eh \leq 1.1 \cdot 10^{+95}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\frac{eh}{ew} \cdot eh}{\tan t}, \cos t, \sin t \cdot ew\right)}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -8.8e+27)
   (* eh (- (cos t)))
   (if (<= eh 1.1e+95)
     (fabs
      (/
       (fma (/ (* (/ eh ew) eh) (tan t)) (cos t) (* (sin t) ew))
       (sqrt (+ 1.0 (pow (/ (/ eh (tan t)) ew) 2.0)))))
     (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -8.8e+27) {
		tmp = eh * -cos(t);
	} else if (eh <= 1.1e+95) {
		tmp = fabs((fma((((eh / ew) * eh) / tan(t)), cos(t), (sin(t) * ew)) / sqrt((1.0 + pow(((eh / tan(t)) / ew), 2.0)))));
	} else {
		tmp = eh * cos(t);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -8.8e+27)
		tmp = Float64(eh * Float64(-cos(t)));
	elseif (eh <= 1.1e+95)
		tmp = abs(Float64(fma(Float64(Float64(Float64(eh / ew) * eh) / tan(t)), cos(t), Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (Float64(Float64(eh / tan(t)) / ew) ^ 2.0)))));
	else
		tmp = Float64(eh * cos(t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -8.8e+27], N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 1.1e+95], N[Abs[N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * eh), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.7999999999999995e27

   1. Initial program 99.8%

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

   3. Applied rewrites 17.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 15.4%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 14.6

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

   7. Applied rewrites 14.6%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{eh \cdot \cos \left(t + \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\cos \left(t + \mathsf{PI}\left(\right)\right)} \]

      2. cos-+PI N/A

         \[\leadsto eh \cdot \left(\mathsf{neg}\left(\cos t\right)\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto eh \cdot \left(-\cos t\right) \]

      4. lift-cos.f64 64.7

         \[\leadsto eh \cdot \left(-\cos t\right) \]

   10. Applied rewrites 64.7%

       \[\leadsto \color{blue}{eh \cdot \left(-\cos t\right)} \]

   if -8.7999999999999995e27 < eh < 1.0999999999999999e95

   1. Initial program 99.8%

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

   3. Applied rewrites 86.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 76.3%

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

   if 1.0999999999999999e95 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 52.4

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 14.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 13.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 6.0

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

   7. Applied rewrites 6.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 68.4

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 68.4%

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

Alternative 5: 63.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{+27}:\\ \;\;\;\;eh \cdot \left(-\cos t\right)\\ \mathbf{elif}\;eh \leq 3.35 \cdot 10^{-15}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -2e+27)
   (* eh (- (cos t)))
   (if (<= eh 3.35e-15) (fabs (* ew (sin t))) (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e+27) {
		tmp = eh * -cos(t);
	} else if (eh <= 3.35e-15) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = eh * cos(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-2d+27)) then
        tmp = eh * -cos(t)
    else if (eh <= 3.35d-15) then
        tmp = abs((ew * sin(t)))
    else
        tmp = eh * cos(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e+27) {
		tmp = eh * -Math.cos(t);
	} else if (eh <= 3.35e-15) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = eh * Math.cos(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -2e+27:
		tmp = eh * -math.cos(t)
	elif eh <= 3.35e-15:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = eh * math.cos(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -2e+27)
		tmp = Float64(eh * Float64(-cos(t)));
	elseif (eh <= 3.35e-15)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = Float64(eh * cos(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -2e+27)
		tmp = eh * -cos(t);
	elseif (eh <= 3.35e-15)
		tmp = abs((ew * sin(t)));
	else
		tmp = eh * cos(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -2e+27], N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 3.35e-15], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -2e27

   1. Initial program 99.8%

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

   3. Applied rewrites 17.5%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 15.5%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 14.5

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

   7. Applied rewrites 14.5%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{eh \cdot \cos \left(t + \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\cos \left(t + \mathsf{PI}\left(\right)\right)} \]

      2. cos-+PI N/A

         \[\leadsto eh \cdot \left(\mathsf{neg}\left(\cos t\right)\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto eh \cdot \left(-\cos t\right) \]

      4. lift-cos.f64 64.7

         \[\leadsto eh \cdot \left(-\cos t\right) \]

   10. Applied rewrites 64.7%

       \[\leadsto \color{blue}{eh \cdot \left(-\cos t\right)} \]

   if -2e27 < eh < 3.35e-15

   1. Initial program 99.8%

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

   3. Applied rewrites 88.7%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. lower-tan.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f64 N/A

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

      6. lower-PI.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 62.4

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

   8. Applied rewrites 62.4%

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

   if 3.35e-15 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 51.1

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 20.4%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 17.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 12.5

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

   7. Applied rewrites 12.5%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 63.4

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 63.4%

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

Alternative 6: 51.9% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.3 \cdot 10^{-248}:\\ \;\;\;\;eh \cdot \left(-\cos t\right)\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{-169}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;eh \cdot \cos t\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.3e-248)
   (* eh (- (cos t)))
   (if (<= eh 1.65e-169) (* ew (sin t)) (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.3e-248) {
		tmp = eh * -cos(t);
	} else if (eh <= 1.65e-169) {
		tmp = ew * sin(t);
	} else {
		tmp = eh * cos(t);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.3d-248)) then
        tmp = eh * -cos(t)
    else if (eh <= 1.65d-169) then
        tmp = ew * sin(t)
    else
        tmp = eh * cos(t)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.3e-248) {
		tmp = eh * -Math.cos(t);
	} else if (eh <= 1.65e-169) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = eh * Math.cos(t);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.3e-248:
		tmp = eh * -math.cos(t)
	elif eh <= 1.65e-169:
		tmp = ew * math.sin(t)
	else:
		tmp = eh * math.cos(t)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.3e-248)
		tmp = Float64(eh * Float64(-cos(t)));
	elseif (eh <= 1.65e-169)
		tmp = Float64(ew * sin(t));
	else
		tmp = Float64(eh * cos(t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.3e-248)
		tmp = eh * -cos(t);
	elseif (eh <= 1.65e-169)
		tmp = ew * sin(t);
	else
		tmp = eh * cos(t);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.3e-248], N[(eh * (-N[Cos[t], $MachinePrecision])), $MachinePrecision], If[LessEqual[eh, 1.65e-169], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.30000000000000003e-248

   1. Initial program 99.8%

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

   3. Applied rewrites 30.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 27.1%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 26.7

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

   7. Applied rewrites 26.7%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{eh \cdot \cos \left(t + \mathsf{PI}\left(\right)\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto eh \cdot \color{blue}{\cos \left(t + \mathsf{PI}\left(\right)\right)} \]

      2. cos-+PI N/A

         \[\leadsto eh \cdot \left(\mathsf{neg}\left(\cos t\right)\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto eh \cdot \left(-\cos t\right) \]

      4. lift-cos.f64 52.2

         \[\leadsto eh \cdot \left(-\cos t\right) \]

   10. Applied rewrites 52.2%

       \[\leadsto \color{blue}{eh \cdot \left(-\cos t\right)} \]

   if -1.30000000000000003e-248 < eh < 1.65000000000000013e-169

   1. Initial program 99.8%

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

   3. Applied rewrites 45.6%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 44.1%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 43.5

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto ew \cdot \sin t \]

      2. lift-*.f64 41.0

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

   10. Applied rewrites 41.0%

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

   if 1.65000000000000013e-169 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 51.0

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 29.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 24.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 19.1

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

   7. Applied rewrites 19.1%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 56.0

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 56.0%

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

Alternative 7: 39.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \cos t\\ \mathbf{if}\;eh \leq -5.6 \cdot 10^{-15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{-169}:\\ \;\;\;\;ew \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (cos t))))
   (if (<= eh -5.6e-15) t_1 (if (<= eh 1.65e-169) (* ew (sin t)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * cos(t);
	double tmp;
	if (eh <= -5.6e-15) {
		tmp = t_1;
	} else if (eh <= 1.65e-169) {
		tmp = ew * sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = eh * cos(t)
    if (eh <= (-5.6d-15)) then
        tmp = t_1
    else if (eh <= 1.65d-169) then
        tmp = ew * sin(t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.cos(t);
	double tmp;
	if (eh <= -5.6e-15) {
		tmp = t_1;
	} else if (eh <= 1.65e-169) {
		tmp = ew * Math.sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.cos(t)
	tmp = 0
	if eh <= -5.6e-15:
		tmp = t_1
	elif eh <= 1.65e-169:
		tmp = ew * math.sin(t)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * cos(t))
	tmp = 0.0
	if (eh <= -5.6e-15)
		tmp = t_1;
	elseif (eh <= 1.65e-169)
		tmp = Float64(ew * sin(t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * cos(t);
	tmp = 0.0;
	if (eh <= -5.6e-15)
		tmp = t_1;
	elseif (eh <= 1.65e-169)
		tmp = ew * sin(t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -5.6e-15], t$95$1, If[LessEqual[eh, 1.65e-169], N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.60000000000000028e-15 or 1.65000000000000013e-169 < 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 51.5

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

   3. Applied rewrites 25.7%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 22.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-/r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r* N/A

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

      10. lift-*.f64 N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 17.0

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 41.4

         \[\leadsto eh \cdot \cos t \]

   10. Applied rewrites 41.4%

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

   if -5.60000000000000028e-15 < eh < 1.65000000000000013e-169

   1. Initial program 99.8%

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

   3. Applied rewrites 45.0%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. cosh-asinh N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-/r* N/A

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

      12. lift-*.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-/.f64 N/A

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

   5. Applied rewrites 41.6%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-quot N/A

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

      4. sin-+PI-rev N/A

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

      5. lift-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lower-PI.f64 41.7

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

   7. Applied rewrites 41.7%

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

   8. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

         \[\leadsto ew \cdot \sin t \]

      2. lift-*.f64 35.6

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

   10. Applied rewrites 35.6%

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

Alternative 8: 32.1% accurate, 8.2× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return eh * cos(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 = eh * cos(t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(eh * N[Cos[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-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 51.4

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

3. Applied rewrites 33.8%

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

   1. lift-cosh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. cosh-asinh N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

5. Applied rewrites 30.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 22.3

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

7. Applied rewrites 22.3%

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

8. Taylor expanded in eh around inf

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

   1. lower-*.f64 N/A

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

   2. lift-cos.f64 32.1

      \[\leadsto eh \cdot \cos t \]

10. Applied rewrites 32.1%

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

Alternative 9: 22.4% accurate, 870.0× speedup

Math

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

FPCore

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

C

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

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 = eh
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := eh

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-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

   4. rem-square-sqrt 51.4

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

   5. lift-+.f64 N/A

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

   6. +-commutative N/A

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

3. Applied rewrites 33.8%

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

   1. lift-cosh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. cosh-asinh N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. associate-/r* N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. associate-/r* N/A

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

   12. lift-*.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-/.f64 N/A

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

5. Applied rewrites 30.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. associate-/r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r* N/A

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

   10. lift-*.f64 N/A

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

   11. frac-times N/A

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

   12. lower-/.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 22.3

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

7. Applied rewrites 22.3%

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

8. Taylor expanded in t around 0

   \[\leadsto \color{blue}{eh} \]
9. Step-by-step derivation

10. Applied rewrites 22.4%

    \[\leadsto \color{blue}{eh} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(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))))))))