Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.3s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 97.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ t_2 := \sinh^{-1} t\_1\\ \mathbf{if}\;eh \leq -1.2 \cdot 10^{-80} \lor \neg \left(eh \leq 7.8 \cdot 10^{-108}\right):\\ \;\;\;\;\left|\left(\mathsf{fma}\left(\tanh t\_2, \frac{\cos t}{ew}, \frac{\sin t}{eh} \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right) \cdot ew\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew + eh \cdot \left(\cos t \cdot t\_1\right)}{\cosh t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)) (t_2 (asinh t_1)))
   (if (or (<= eh -1.2e-80) (not (<= eh 7.8e-108)))
     (fabs
      (*
       (*
        (fma
         (tanh t_2)
         (/ (cos t) ew)
         (* (/ (sin t) eh) (cos (atan (/ (/ eh ew) t)))))
        ew)
       eh))
     (fabs (/ (+ (* (sin t) ew) (* eh (* (cos t) t_1))) (cosh t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double t_2 = asinh(t_1);
	double tmp;
	if ((eh <= -1.2e-80) || !(eh <= 7.8e-108)) {
		tmp = fabs(((fma(tanh(t_2), (cos(t) / ew), ((sin(t) / eh) * cos(atan(((eh / ew) / t))))) * ew) * eh));
	} else {
		tmp = fabs((((sin(t) * ew) + (eh * (cos(t) * t_1))) / cosh(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	t_2 = asinh(t_1)
	tmp = 0.0
	if ((eh <= -1.2e-80) || !(eh <= 7.8e-108))
		tmp = abs(Float64(Float64(fma(tanh(t_2), Float64(cos(t) / ew), Float64(Float64(sin(t) / eh) * cos(atan(Float64(Float64(eh / ew) / t))))) * ew) * eh));
	else
		tmp = abs(Float64(Float64(Float64(sin(t) * ew) + Float64(eh * Float64(cos(t) * t_1))) / cosh(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[ArcSinh[t$95$1], $MachinePrecision]}, If[Or[LessEqual[eh, -1.2e-80], N[Not[LessEqual[eh, 7.8e-108]], $MachinePrecision]], N[Abs[N[(N[(N[(N[Tanh[t$95$2], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] / eh), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.2e-80 or 7.79999999999999989e-108 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in ew around inf

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

   8. Applied rewrites 99.0%

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

   10. Applied rewrites 99.6%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 99.4%

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

   if -1.2e-80 < eh < 7.79999999999999989e-108

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. associate-*l/ N/A

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

      10. lift-sin.f64 N/A

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

   4. Applied rewrites 97.5%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.2 \cdot 10^{-80} \lor \neg \left(eh \leq 7.8 \cdot 10^{-108}\right):\\ \;\;\;\;\left|\left(\mathsf{fma}\left(\tanh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \frac{\cos t}{ew}, \frac{\sin t}{eh} \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)\right) \cdot ew\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew + eh \cdot \left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.05 \cdot 10^{-57} \lor \neg \left(ew \leq 1.5 \cdot 10^{-199}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5.05e-57) (not (<= ew 1.5e-199)))
   (fabs
    (+
     (* (* eh (cos t)) (sin (atan (/ (/ eh ew) t))))
     (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t)))))))
   (fabs (* (sin (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.05e-57) || !(ew <= 1.5e-199)) {
		tmp = fabs((((eh * cos(t)) * sin(atan(((eh / ew) / t)))) + ((ew * sin(t)) * cos(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = fabs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-5.05d-57)) .or. (.not. (ew <= 1.5d-199))) then
        tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / t)))) + ((ew * sin(t)) * cos(atan(((eh / ew) / tan(t)))))))
    else
        tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.05e-57) || !(ew <= 1.5e-199)) {
		tmp = Math.abs((((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / t)))) + ((ew * Math.sin(t)) * Math.cos(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((Math.cos(t) / ew) * (eh / Math.sin(t))))) * (Math.cos(t) * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5.05e-57) or not (ew <= 1.5e-199):
		tmp = math.fabs((((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / t)))) + ((ew * math.sin(t)) * math.cos(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = math.fabs((math.sin(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.cos(t) * eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5.05e-57) || !(ew <= 1.5e-199))
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / t)))) + Float64(Float64(ew * sin(t)) * cos(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5.05e-57) || ~((ew <= 1.5e-199)))
		tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / t)))) + ((ew * sin(t)) * cos(atan(((eh / ew) / tan(t)))))));
	else
		tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5.05e-57], N[Not[LessEqual[ew, 1.5e-199]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.05000000000000008e-57 or 1.49999999999999992e-199 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

      3. lower-/.f64 90.9

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

   5. Applied rewrites 90.9%

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

   if -5.05000000000000008e-57 < ew < 1.49999999999999992e-199

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.05 \cdot 10^{-57} \lor \neg \left(ew \leq 1.5 \cdot 10^{-199}\right):\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -7.8 \cdot 10^{-29} \lor \neg \left(eh \leq 0.000108\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew + eh \cdot \left(\cos t \cdot t\_1\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -7.8e-29) (not (<= eh 0.000108)))
     (fabs (* (sin (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (cos t) eh)))
     (fabs (/ (+ (* (sin t) ew) (* eh (* (cos t) t_1))) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -7.8e-29) || !(eh <= 0.000108)) {
		tmp = fabs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((((sin(t) * ew) + (eh * (cos(t) * t_1))) / cosh(asinh(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / math.tan(t)) / ew
	tmp = 0
	if (eh <= -7.8e-29) or not (eh <= 0.000108):
		tmp = math.fabs((math.sin(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.cos(t) * eh)))
	else:
		tmp = math.fabs((((math.sin(t) * ew) + (eh * (math.cos(t) * t_1))) / math.cosh(math.asinh(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -7.8e-29) || !(eh <= 0.000108))
		tmp = abs(Float64(sin(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(Float64(Float64(sin(t) * ew) + Float64(eh * Float64(cos(t) * t_1))) / cosh(asinh(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / tan(t)) / ew;
	tmp = 0.0;
	if ((eh <= -7.8e-29) || ~((eh <= 0.000108)))
		tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	else
		tmp = abs((((sin(t) * ew) + (eh * (cos(t) * t_1))) / cosh(asinh(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -7.8e-29], N[Not[LessEqual[eh, 0.000108]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] + N[(eh * N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.7999999999999995e-29 or 1.08e-4 < eh

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -7.7999999999999995e-29 < eh < 1.08e-4

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. associate-*l/ N/A

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

      10. lift-sin.f64 N/A

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

   4. Applied rewrites 95.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.8 \cdot 10^{-29} \lor \neg \left(eh \leq 0.000108\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew + eh \cdot \left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -7.8 \cdot 10^{-29} \lor \neg \left(eh \leq 0.000108\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot t\_1, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -7.8e-29) (not (<= eh 0.000108)))
     (fabs (* (sin (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (cos t) eh)))
     (fabs (/ (fma (* (cos t) t_1) eh (* (sin t) ew)) (cosh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -7.8e-29) || !(eh <= 0.000108)) {
		tmp = fabs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((fma((cos(t) * t_1), eh, (sin(t) * ew)) / cosh(asinh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -7.8e-29) || !(eh <= 0.000108))
		tmp = abs(Float64(sin(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(fma(Float64(cos(t) * t_1), eh, Float64(sin(t) * ew)) / cosh(asinh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -7.8e-29], N[Not[LessEqual[eh, 0.000108]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.7999999999999995e-29 or 1.08e-4 < eh

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -7.7999999999999995e-29 < eh < 1.08e-4

   1. Initial program 99.8%

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

   4. Applied rewrites 95.2%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.8 \cdot 10^{-29} \lor \neg \left(eh \leq 0.000108\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -4.7 \cdot 10^{-55} \lor \neg \left(eh \leq 4.2 \cdot 10^{+22}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{eh \cdot \left(\cos t \cdot t\_1\right) + \sin t \cdot ew}{\sqrt{1 + {t\_1}^{2}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -4.7e-55) (not (<= eh 4.2e+22)))
     (fabs (* (sin (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (cos t) eh)))
     (fabs
      (/
       (+ (* eh (* (cos t) t_1)) (* (sin t) ew))
       (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -4.7e-55) || !(eh <= 4.2e+22)) {
		tmp = fabs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((((eh * (cos(t) * t_1)) + (sin(t) * ew)) / sqrt((1.0 + pow(t_1, 2.0)))));
	}
	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 / tan(t)) / ew
    if ((eh <= (-4.7d-55)) .or. (.not. (eh <= 4.2d+22))) then
        tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)))
    else
        tmp = abs((((eh * (cos(t) * t_1)) + (sin(t) * ew)) / sqrt((1.0d0 + (t_1 ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / Math.tan(t)) / ew;
	double tmp;
	if ((eh <= -4.7e-55) || !(eh <= 4.2e+22)) {
		tmp = Math.abs((Math.sin(Math.atan(((Math.cos(t) / ew) * (eh / Math.sin(t))))) * (Math.cos(t) * eh)));
	} else {
		tmp = Math.abs((((eh * (Math.cos(t) * t_1)) + (Math.sin(t) * ew)) / Math.sqrt((1.0 + Math.pow(t_1, 2.0)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / math.tan(t)) / ew
	tmp = 0
	if (eh <= -4.7e-55) or not (eh <= 4.2e+22):
		tmp = math.fabs((math.sin(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.cos(t) * eh)))
	else:
		tmp = math.fabs((((eh * (math.cos(t) * t_1)) + (math.sin(t) * ew)) / math.sqrt((1.0 + math.pow(t_1, 2.0)))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -4.7e-55) || !(eh <= 4.2e+22))
		tmp = abs(Float64(sin(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(Float64(Float64(eh * Float64(cos(t) * t_1)) + Float64(sin(t) * ew)) / sqrt(Float64(1.0 + (t_1 ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / tan(t)) / ew;
	tmp = 0.0;
	if ((eh <= -4.7e-55) || ~((eh <= 4.2e+22)))
		tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	else
		tmp = abs((((eh * (cos(t) * t_1)) + (sin(t) * ew)) / sqrt((1.0 + (t_1 ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -4.7e-55], N[Not[LessEqual[eh, 4.2e+22]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh * N[(N[Cos[t], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.7e-55 or 4.1999999999999996e22 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 87.5

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

   5. Applied rewrites 87.5%

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

   if -4.7e-55 < eh < 4.1999999999999996e22

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. associate-*l/ N/A

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

      10. lift-sin.f64 N/A

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

   4. Applied rewrites 94.2%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-prod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lift-cosh.f64 N/A

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

      5. lift-asinh.f64 N/A

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

      6. cosh-asinh N/A

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

      7. lift-cosh.f64 N/A

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

      8. lift-asinh.f64 N/A

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

      9. cosh-asinh N/A

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

      10. rem-square-sqrt N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. pow2 N/A

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

      14. lower-pow.f64 89.7

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

   6. Applied rewrites 89.7%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.7 \cdot 10^{-55} \lor \neg \left(eh \leq 4.2 \cdot 10^{+22}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{eh \cdot \left(\cos t \cdot \frac{\frac{eh}{\tan t}}{ew}\right) + \sin t \cdot ew}{\sqrt{1 + {\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{-55} \lor \neg \left(eh \leq 4.5 \cdot 10^{-18}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.45e-55) (not (<= eh 4.5e-18)))
   (fabs (* (sin (atan (* (/ (cos t) ew) (/ eh (sin t))))) (* (cos t) eh)))
   (fabs (* (sin t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.45e-55) || !(eh <= 4.5e-18)) {
		tmp = fabs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	} else {
		tmp = fabs((sin(t) * ew));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-1.45d-55)) .or. (.not. (eh <= 4.5d-18))) then
        tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)))
    else
        tmp = abs((sin(t) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.45e-55) || !(eh <= 4.5e-18)) {
		tmp = Math.abs((Math.sin(Math.atan(((Math.cos(t) / ew) * (eh / Math.sin(t))))) * (Math.cos(t) * eh)));
	} else {
		tmp = Math.abs((Math.sin(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.45e-55) or not (eh <= 4.5e-18):
		tmp = math.fabs((math.sin(math.atan(((math.cos(t) / ew) * (eh / math.sin(t))))) * (math.cos(t) * eh)))
	else:
		tmp = math.fabs((math.sin(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.45e-55) || !(eh <= 4.5e-18))
		tmp = abs(Float64(sin(atan(Float64(Float64(cos(t) / ew) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(sin(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.45e-55) || ~((eh <= 4.5e-18)))
		tmp = abs((sin(atan(((cos(t) / ew) * (eh / sin(t))))) * (cos(t) * eh)));
	else
		tmp = abs((sin(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.45e-55], N[Not[LessEqual[eh, 4.5e-18]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.45e-55 or 4.49999999999999994e-18 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 87.2

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

   5. Applied rewrites 87.2%

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

   if -1.45e-55 < eh < 4.49999999999999994e-18

   1. Initial program 99.8%

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      4. lower-sin.f64 40.8

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

   7. Applied rewrites 40.8%

      \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 70.1

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

   10. Applied rewrites 70.1%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.45 \cdot 10^{-55} \lor \neg \left(eh \leq 4.5 \cdot 10^{-18}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot t\_1\right|\\ \mathbf{if}\;eh \leq -4.7 \cdot 10^{-55}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 4.5 \cdot 10^{-18}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+182}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{eh}{ew} \cdot -0.3333333333333333, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)) (t_2 (fabs (* (sin (atan (/ (/ eh ew) t))) t_1))))
   (if (<= eh -4.7e-55)
     t_2
     (if (<= eh 4.5e-18)
       (fabs (* (sin t) ew))
       (if (<= eh 4.8e+182)
         (fabs
          (*
           (sin
            (atan
             (/ (fma (* (/ eh ew) -0.3333333333333333) (* t t) (/ eh ew)) t)))
           t_1))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((sin(atan(((eh / ew) / t))) * t_1));
	double tmp;
	if (eh <= -4.7e-55) {
		tmp = t_2;
	} else if (eh <= 4.5e-18) {
		tmp = fabs((sin(t) * ew));
	} else if (eh <= 4.8e+182) {
		tmp = fabs((sin(atan((fma(((eh / ew) * -0.3333333333333333), (t * t), (eh / ew)) / t))) * t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(sin(atan(Float64(Float64(eh / ew) / t))) * t_1))
	tmp = 0.0
	if (eh <= -4.7e-55)
		tmp = t_2;
	elseif (eh <= 4.5e-18)
		tmp = abs(Float64(sin(t) * ew));
	elseif (eh <= 4.8e+182)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(Float64(eh / ew) * -0.3333333333333333), Float64(t * t), Float64(eh / ew)) / t))) * t_1));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.7e-55], t$95$2, If[LessEqual[eh, 4.5e-18], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 4.8e+182], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(eh / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -4.7e-55 or 4.80000000000000019e182 < eh

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 91.9

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

   5. Applied rewrites 91.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 80.8%

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

   if -4.7e-55 < eh < 4.49999999999999994e-18

   1. Initial program 99.8%

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      4. lower-sin.f64 40.8

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

   7. Applied rewrites 40.8%

      \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 70.1

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

   10. Applied rewrites 70.1%

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

   if 4.49999999999999994e-18 < eh < 4.80000000000000019e182

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 76.5

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

   5. Applied rewrites 76.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.6%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{eh}{ew} \cdot -0.3333333333333333, t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 68.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.7 \cdot 10^{-55} \lor \neg \left(eh \leq 10^{-14}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -4.7e-55) (not (<= eh 1e-14)))
   (fabs (* (sin (atan (/ (/ eh ew) t))) (* (cos t) eh)))
   (fabs (* (sin t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.7e-55) || !(eh <= 1e-14)) {
		tmp = fabs((sin(atan(((eh / ew) / t))) * (cos(t) * eh)));
	} else {
		tmp = fabs((sin(t) * ew));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-4.7d-55)) .or. (.not. (eh <= 1d-14))) then
        tmp = abs((sin(atan(((eh / ew) / t))) * (cos(t) * eh)))
    else
        tmp = abs((sin(t) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -4.7e-55) || !(eh <= 1e-14)) {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / t))) * (Math.cos(t) * eh)));
	} else {
		tmp = Math.abs((Math.sin(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -4.7e-55) or not (eh <= 1e-14):
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / t))) * (math.cos(t) * eh)))
	else:
		tmp = math.fabs((math.sin(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -4.7e-55) || !(eh <= 1e-14))
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / t))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(sin(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -4.7e-55) || ~((eh <= 1e-14)))
		tmp = abs((sin(atan(((eh / ew) / t))) * (cos(t) * eh)));
	else
		tmp = abs((sin(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -4.7e-55], N[Not[LessEqual[eh, 1e-14]], $MachinePrecision]], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.7e-55 or 9.99999999999999999e-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. Add Preprocessing

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.0%

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

   if -4.7e-55 < eh < 9.99999999999999999e-15

   1. Initial program 99.8%

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

   4. Applied rewrites 53.8%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      4. lower-sin.f64 40.8

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

   7. Applied rewrites 40.8%

      \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 70.1

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

   10. Applied rewrites 70.1%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.7 \cdot 10^{-55} \lor \neg \left(eh \leq 10^{-14}\right):\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-48} \lor \neg \left(t \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, t \cdot t, 0.008333333333333333\right) \cdot \left(t \cdot t\right) - 0.16666666666666666, t \cdot t, 1\right) \cdot t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -7.5e-48) (not (<= t 8e-6)))
   (fabs (* (sin t) ew))
   (fabs
    (*
     (sin
      (atan
       (*
        (/ (fma -0.5 (* t t) 1.0) ew)
        (/
         eh
         (*
          (fma
           (-
            (*
             (fma -0.0001984126984126984 (* t t) 0.008333333333333333)
             (* t t))
            0.16666666666666666)
           (* t t)
           1.0)
          t)))))
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7.5e-48) || !(t <= 8e-6)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs((sin(atan(((fma(-0.5, (t * t), 1.0) / ew) * (eh / (fma(((fma(-0.0001984126984126984, (t * t), 0.008333333333333333) * (t * t)) - 0.16666666666666666), (t * t), 1.0) * t))))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -7.5e-48) || !(t <= 8e-6))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.5, Float64(t * t), 1.0) / ew) * Float64(eh / Float64(fma(Float64(Float64(fma(-0.0001984126984126984, Float64(t * t), 0.008333333333333333) * Float64(t * t)) - 0.16666666666666666), Float64(t * t), 1.0) * t))))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -7.5e-48], N[Not[LessEqual[t, 8e-6]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[(N[(N[(N[(N[(-0.0001984126984126984 * N[(t * t), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] * N[(t * t), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000042e-48 or 7.99999999999999964e-6 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      4. lower-sin.f64 22.9

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

   7. Applied rewrites 22.9%

      \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if -7.50000000000000042e-48 < t < 7.99999999999999964e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.5%

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

   9. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{2}, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{t \cdot \left(1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {t}^{2}\right) - \frac{1}{6}\right)\right)}\right) \cdot eh\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 77.5%

       \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, t \cdot t, 0.008333333333333333\right) \cdot \left(t \cdot t\right) - 0.16666666666666666, t \cdot t, 1\right) \cdot t}\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-48} \lor \neg \left(t \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\mathsf{fma}\left(\mathsf{fma}\left(-0.0001984126984126984, t \cdot t, 0.008333333333333333\right) \cdot \left(t \cdot t\right) - 0.16666666666666666, t \cdot t, 1\right) \cdot t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-48} \lor \neg \left(t \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -7.5e-48) (not (<= t 8e-6)))
   (fabs (* (sin t) ew))
   (fabs (* (sin (atan (/ (/ eh ew) t))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7.5e-48) || !(t <= 8e-6)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs((sin(atan(((eh / ew) / t))) * eh));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-7.5d-48)) .or. (.not. (t <= 8d-6))) then
        tmp = abs((sin(t) * ew))
    else
        tmp = abs((sin(atan(((eh / ew) / t))) * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -7.5e-48) || !(t <= 8e-6)) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / t))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -7.5e-48) or not (t <= 8e-6):
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / t))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -7.5e-48) || !(t <= 8e-6))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / t))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -7.5e-48) || ~((t <= 8e-6)))
		tmp = abs((sin(t) * ew));
	else
		tmp = abs((sin(atan(((eh / ew) / t))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -7.5e-48], N[Not[LessEqual[t, 8e-6]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.50000000000000042e-48 or 7.99999999999999964e-6 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 31.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
   6. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

      4. lower-sin.f64 22.9

         \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

   7. Applied rewrites 22.9%

      \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

   8. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 45.9

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

   10. Applied rewrites 45.9%

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

   if -7.50000000000000042e-48 < t < 7.99999999999999964e-6

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-sin.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.5%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-48} \lor \neg \left(t \leq 8 \cdot 10^{-6}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 41.5% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 31.0%

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

5. Taylor expanded in eh around 0

   \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

   4. lower-sin.f64 20.4

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

7. Applied rewrites 20.4%

   \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

8. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 37.0

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

10. Applied rewrites 37.0%

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

Alternative 13: 18.8% accurate, 108.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 31.0%

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

5. Taylor expanded in eh around 0

   \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]
6. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \left|{\color{blue}{\left(\sqrt{ew \cdot \sin t}\right)}}^{2}\right| \]

   2. *-commutative N/A

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t \cdot ew}}\right)}^{2}\right| \]

   4. lower-sin.f64 20.4

      \[\leadsto \left|{\left(\sqrt{\color{blue}{\sin t} \cdot ew}\right)}^{2}\right| \]

7. Applied rewrites 20.4%

   \[\leadsto \left|{\color{blue}{\left(\sqrt{\sin t \cdot ew}\right)}}^{2}\right| \]

8. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 37.0

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

10. Applied rewrites 37.0%

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

11. Taylor expanded in t around 0

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

13. Applied rewrites 17.2%

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

Reproduce

herbie shell --seed 2024350 
(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))))))))