Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 34.6s

Alternatives: 6

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \frac{-1}{\cos t} \cdot \frac{t\_1}{ew}\\ t_3 := \tanh \sinh^{-1} t\_2\\ t_4 := \cos \tan^{-1} t\_2\\ \mathbf{if}\;ew \leq -4 \cdot 10^{+24} \lor \neg \left(ew \leq 5 \cdot 10^{-30}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_1 \cdot t\_3}{ew}, -1, t\_4 \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\left(\cos t \cdot ew\right) \cdot t\_4}{eh} - t\_3 \cdot \sin t\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* (/ -1.0 (cos t)) (/ t_1 ew)))
        (t_3 (tanh (asinh t_2)))
        (t_4 (cos (atan t_2))))
   (if (or (<= ew -4e+24) (not (<= ew 5e-30)))
     (fabs (* (fma (/ (* t_1 t_3) ew) -1.0 (* t_4 (cos t))) ew))
     (fabs (* (- (/ (* (* (cos t) ew) t_4) eh) (* t_3 (sin t))) eh)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = (-1.0 / cos(t)) * (t_1 / ew);
	double t_3 = tanh(asinh(t_2));
	double t_4 = cos(atan(t_2));
	double tmp;
	if ((ew <= -4e+24) || !(ew <= 5e-30)) {
		tmp = fabs((fma(((t_1 * t_3) / ew), -1.0, (t_4 * cos(t))) * ew));
	} else {
		tmp = fabs((((((cos(t) * ew) * t_4) / eh) - (t_3 * sin(t))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(Float64(-1.0 / cos(t)) * Float64(t_1 / ew))
	t_3 = tanh(asinh(t_2))
	t_4 = cos(atan(t_2))
	tmp = 0.0
	if ((ew <= -4e+24) || !(ew <= 5e-30))
		tmp = abs(Float64(fma(Float64(Float64(t_1 * t_3) / ew), -1.0, Float64(t_4 * cos(t))) * ew));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(cos(t) * ew) * t_4) / eh) - Float64(t_3 * sin(t))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-1.0 / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -4e+24], N[Not[LessEqual[ew, 5e-30]], $MachinePrecision]], N[Abs[N[(N[(N[(N[(t$95$1 * t$95$3), $MachinePrecision] / ew), $MachinePrecision] * -1.0 + N[(t$95$4 * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * t$95$4), $MachinePrecision] / eh), $MachinePrecision] - N[(t$95$3 * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.9999999999999999e24 or 4.99999999999999972e-30 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

   if -3.9999999999999999e24 < ew < 4.99999999999999972e-30

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4 \cdot 10^{+24} \lor \neg \left(ew \leq 5 \cdot 10^{-30}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{-1}{\cos t} \cdot \frac{\sin t \cdot eh}{ew}\right)}{ew}, -1, \cos \tan^{-1} \left(\frac{-1}{\cos t} \cdot \frac{\sin t \cdot eh}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\frac{\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-1}{\cos t} \cdot \frac{\sin t \cdot eh}{ew}\right)}{eh} - \tanh \sinh^{-1} \left(\frac{-1}{\cos t} \cdot \frac{\sin t \cdot eh}{ew}\right) \cdot \sin t\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\\ \left|\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin t\_1 + \left(\left(--1 \cdot ew\right) \cdot \mathsf{fma}\left(\cos t, \sin \left(0.5 \cdot \pi\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin t\right)\right) \cdot \cos t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* -1.0 (/ (* eh (tan t)) ew)))))
   (fabs
    (+
     (* (* -1.0 (* eh (sin t))) (sin t_1))
     (*
      (*
       (- (* -1.0 ew))
       (fma
        (cos t)
        (sin (* 0.5 PI))
        (* (sin (fma 0.5 PI (/ PI 2.0))) (sin t))))
      (cos t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((-1.0 * ((eh * tan(t)) / ew)));
	return fabs((((-1.0 * (eh * sin(t))) * sin(t_1)) + ((-(-1.0 * ew) * fma(cos(t), sin((0.5 * ((double) M_PI))), (sin(fma(0.5, ((double) M_PI), (((double) M_PI) / 2.0))) * sin(t)))) * cos(t_1))));
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(-1.0 * Float64(Float64(eh * tan(t)) / ew)))
	return abs(Float64(Float64(Float64(-1.0 * Float64(eh * sin(t))) * sin(t_1)) + Float64(Float64(Float64(-Float64(-1.0 * ew)) * fma(cos(t), sin(Float64(0.5 * pi)), Float64(sin(fma(0.5, pi, Float64(pi / 2.0))) * sin(t)))) * cos(t_1))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(-1.0 * N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(-1.0 * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] + N[(N[((-N[(-1.0 * ew), $MachinePrecision]) * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(0.5 * Pi + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

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

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

   3. sin-sum N/A

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

   4. lower-fma.f64 N/A

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

   5. lift-sin.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-PI.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-PI.f64 99.6

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in t around inf

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

   1. lower-fma.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-*.f64 N/A

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

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

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

   8. lower-sin.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-fma.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-sin.f64 99.5

       \[\leadsto \left|\left(ew \cdot \mathsf{fma}\left(\cos t, \sin \left(0.5 \cdot \pi\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin t\right)\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

   \[\leadsto \left|\left(-1 \cdot \left(eh \cdot \sin t\right)\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) + \left(\left(--1 \cdot ew\right) \cdot \mathsf{fma}\left(\cos t, \sin \left(0.5 \cdot \pi\right), \sin \left(\mathsf{fma}\left(0.5, \pi, \frac{\pi}{2}\right)\right) \cdot \sin t\right)\right) \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right| \]
9. Add Preprocessing

Alternative 4: 87.7% accurate, N/A× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ -1.0 (cos t)) (/ (* (sin t) eh) ew))))
   (fabs
    (*
     (-
      (/ (* (* (cos t) ew) (cos (atan t_1))) eh)
      (* (tanh (asinh t_1)) (sin t)))
     eh))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-1.0 / cos(t)) * ((sin(t) * eh) / ew);
	return fabs((((((cos(t) * ew) * cos(atan(t_1))) / eh) - (tanh(asinh(t_1)) * sin(t))) * eh));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in eh around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 86.2%

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

Alternative 5: 46.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-1 \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-1 \cdot \frac{eh}{ew}\right) \cdot \tan t\right)\\ t_2 := e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)}\\ t_3 := {t\_2}^{-1}\\ \mathbf{if}\;ew \leq -3.5 \cdot 10^{-116} \lor \neg \left(ew \leq 2.05 \cdot 10^{-177}\right):\\ \;\;\;\;\left|\left(ew \cdot \mathsf{fma}\left(\frac{eh}{ew \cdot ew}, \frac{{\sin t}^{2}}{\cos t}, \frac{t\_1}{-1 \cdot eh}\right)\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-1 \cdot eh\right) \cdot \left(\frac{t\_1}{eh} + \frac{\sin t}{ew} \cdot \frac{t\_2 - t\_3}{t\_2 + t\_3}\right)\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (* -1.0 (cos t)) (cos (atan (* (* -1.0 (/ eh ew)) (tan t))))))
        (t_2
         (*
          (exp (+ (log 2.0) (log (/ (sin t) (* ew (cos t))))))
          (exp (log (pow (/ 1.0 (* -1.0 eh)) -1.0)))))
        (t_3 (pow t_2 -1.0)))
   (if (or (<= ew -3.5e-116) (not (<= ew 2.05e-177)))
     (fabs
      (*
       (*
        ew
        (fma
         (/ eh (* ew ew))
         (/ (pow (sin t) 2.0) (cos t))
         (/ t_1 (* -1.0 eh))))
       eh))
     (fabs
      (*
       (*
        (* -1.0 eh)
        (+ (/ t_1 eh) (* (/ (sin t) ew) (/ (- t_2 t_3) (+ t_2 t_3)))))
       ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-1.0 * cos(t)) * cos(atan(((-1.0 * (eh / ew)) * tan(t))));
	double t_2 = exp((log(2.0) + log((sin(t) / (ew * cos(t)))))) * exp(log(pow((1.0 / (-1.0 * eh)), -1.0)));
	double t_3 = pow(t_2, -1.0);
	double tmp;
	if ((ew <= -3.5e-116) || !(ew <= 2.05e-177)) {
		tmp = fabs(((ew * fma((eh / (ew * ew)), (pow(sin(t), 2.0) / cos(t)), (t_1 / (-1.0 * eh)))) * eh));
	} else {
		tmp = fabs((((-1.0 * eh) * ((t_1 / eh) + ((sin(t) / ew) * ((t_2 - t_3) / (t_2 + t_3))))) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-1.0 * cos(t)) * cos(atan(Float64(Float64(-1.0 * Float64(eh / ew)) * tan(t)))))
	t_2 = Float64(exp(Float64(log(2.0) + log(Float64(sin(t) / Float64(ew * cos(t)))))) * exp(log((Float64(1.0 / Float64(-1.0 * eh)) ^ -1.0))))
	t_3 = t_2 ^ -1.0
	tmp = 0.0
	if ((ew <= -3.5e-116) || !(ew <= 2.05e-177))
		tmp = abs(Float64(Float64(ew * fma(Float64(eh / Float64(ew * ew)), Float64((sin(t) ^ 2.0) / cos(t)), Float64(t_1 / Float64(-1.0 * eh)))) * eh));
	else
		tmp = abs(Float64(Float64(Float64(-1.0 * eh) * Float64(Float64(t_1 / eh) + Float64(Float64(sin(t) / ew) * Float64(Float64(t_2 - t_3) / Float64(t_2 + t_3))))) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(-1.0 * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(-1.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[Log[N[(N[Sin[t], $MachinePrecision] / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[Log[N[Power[N[(1.0 / N[(-1.0 * eh), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, -1.0], $MachinePrecision]}, If[Or[LessEqual[ew, -3.5e-116], N[Not[LessEqual[ew, 2.05e-177]], $MachinePrecision]], N[Abs[N[(N[(ew * N[(N[(eh / N[(ew * ew), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / N[(-1.0 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(-1.0 * eh), $MachinePrecision] * N[(N[(t$95$1 / eh), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(N[(t$95$2 - t$95$3), $MachinePrecision] / N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.49999999999999984e-116 or 2.05e-177 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in ew around inf

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. lower-fma.f64 N/A

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

   8. Applied rewrites 52.5%

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

   if -3.49999999999999984e-116 < ew < 2.05e-177

   1. Initial program 99.7%

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

   3. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 83.0%

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

   6. Taylor expanded in eh around -inf

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

   8. Applied rewrites 24.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.5 \cdot 10^{-116} \lor \neg \left(ew \leq 2.05 \cdot 10^{-177}\right):\\ \;\;\;\;\left|\left(ew \cdot \mathsf{fma}\left(\frac{eh}{ew \cdot ew}, \frac{{\sin t}^{2}}{\cos t}, \frac{\left(-1 \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-1 \cdot \frac{eh}{ew}\right) \cdot \tan t\right)}{-1 \cdot eh}\right)\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-1 \cdot eh\right) \cdot \left(\frac{\left(-1 \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-1 \cdot \frac{eh}{ew}\right) \cdot \tan t\right)}{eh} + \frac{\sin t}{ew} \cdot \frac{e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)} - {\left(e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)}\right)}^{-1}}{e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)} + {\left(e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)}\right)}^{-1}}\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 15.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{\log 2 + \log \left(\frac{\sin t}{ew \cdot \cos t}\right)} \cdot e^{\log \left({\left(\frac{1}{-1 \cdot eh}\right)}^{-1}\right)}\\ t_2 := {t\_1}^{-1}\\ \left|\left(\left(-1 \cdot eh\right) \cdot \left(\frac{\left(-1 \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\left(-1 \cdot \frac{eh}{ew}\right) \cdot \tan t\right)}{eh} + \frac{\sin t}{ew} \cdot \frac{t\_1 - t\_2}{t\_1 + t\_2}\right)\right) \cdot ew\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (*
          (exp (+ (log 2.0) (log (/ (sin t) (* ew (cos t))))))
          (exp (log (pow (/ 1.0 (* -1.0 eh)) -1.0)))))
        (t_2 (pow t_1 -1.0)))
   (fabs
    (*
     (*
      (* -1.0 eh)
      (+
       (/ (* (* -1.0 (cos t)) (cos (atan (* (* -1.0 (/ eh ew)) (tan t))))) eh)
       (* (/ (sin t) ew) (/ (- t_1 t_2) (+ t_1 t_2)))))
     ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = exp((log(2.0) + log((sin(t) / (ew * cos(t)))))) * exp(log(pow((1.0 / (-1.0 * eh)), -1.0)));
	double t_2 = pow(t_1, -1.0);
	return fabs((((-1.0 * eh) * ((((-1.0 * cos(t)) * cos(atan(((-1.0 * (eh / ew)) * tan(t))))) / eh) + ((sin(t) / ew) * ((t_1 - t_2) / (t_1 + t_2))))) * 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
    real(8) :: t_1
    real(8) :: t_2
    t_1 = exp((log(2.0d0) + log((sin(t) / (ew * cos(t)))))) * exp(log(((1.0d0 / ((-1.0d0) * eh)) ** (-1.0d0))))
    t_2 = t_1 ** (-1.0d0)
    code = abs(((((-1.0d0) * eh) * (((((-1.0d0) * cos(t)) * cos(atan((((-1.0d0) * (eh / ew)) * tan(t))))) / eh) + ((sin(t) / ew) * ((t_1 - t_2) / (t_1 + t_2))))) * ew))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.exp((Math.log(2.0) + Math.log((Math.sin(t) / (ew * Math.cos(t)))))) * Math.exp(Math.log(Math.pow((1.0 / (-1.0 * eh)), -1.0)));
	double t_2 = Math.pow(t_1, -1.0);
	return Math.abs((((-1.0 * eh) * ((((-1.0 * Math.cos(t)) * Math.cos(Math.atan(((-1.0 * (eh / ew)) * Math.tan(t))))) / eh) + ((Math.sin(t) / ew) * ((t_1 - t_2) / (t_1 + t_2))))) * ew));
}

Python

def code(eh, ew, t):
	t_1 = math.exp((math.log(2.0) + math.log((math.sin(t) / (ew * math.cos(t)))))) * math.exp(math.log(math.pow((1.0 / (-1.0 * eh)), -1.0)))
	t_2 = math.pow(t_1, -1.0)
	return math.fabs((((-1.0 * eh) * ((((-1.0 * math.cos(t)) * math.cos(math.atan(((-1.0 * (eh / ew)) * math.tan(t))))) / eh) + ((math.sin(t) / ew) * ((t_1 - t_2) / (t_1 + t_2))))) * ew))

Julia

function code(eh, ew, t)
	t_1 = Float64(exp(Float64(log(2.0) + log(Float64(sin(t) / Float64(ew * cos(t)))))) * exp(log((Float64(1.0 / Float64(-1.0 * eh)) ^ -1.0))))
	t_2 = t_1 ^ -1.0
	return abs(Float64(Float64(Float64(-1.0 * eh) * Float64(Float64(Float64(Float64(-1.0 * cos(t)) * cos(atan(Float64(Float64(-1.0 * Float64(eh / ew)) * tan(t))))) / eh) + Float64(Float64(sin(t) / ew) * Float64(Float64(t_1 - t_2) / Float64(t_1 + t_2))))) * ew))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = exp((log(2.0) + log((sin(t) / (ew * cos(t)))))) * exp(log(((1.0 / (-1.0 * eh)) ^ -1.0)));
	t_2 = t_1 ^ -1.0;
	tmp = abs((((-1.0 * eh) * ((((-1.0 * cos(t)) * cos(atan(((-1.0 * (eh / ew)) * tan(t))))) / eh) + ((sin(t) / ew) * ((t_1 - t_2) / (t_1 + t_2))))) * ew));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Exp[N[(N[Log[2.0], $MachinePrecision] + N[Log[N[(N[Sin[t], $MachinePrecision] / N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[Log[N[Power[N[(1.0 / N[(-1.0 * eh), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, -1.0], $MachinePrecision]}, N[Abs[N[(N[(N[(-1.0 * eh), $MachinePrecision] * N[(N[(N[(N[(-1.0 * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(-1.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(N[(t$95$1 - t$95$2), $MachinePrecision] / N[(t$95$1 + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 92.5%

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

6. Taylor expanded in eh around -inf

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

8. Applied rewrites 14.1%

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

9. Final simplification 14.1%

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

Reproduce

herbie shell --seed 2025064 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))