Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 8.3s

Alternatives: 16

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. lift-tanh.f64 N/A

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

   7. lift-asinh.f64 N/A

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

   8. tanh-asinh N/A

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

   9. sin-atan-rev N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-sin.f64 N/A

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

7. Applied rewrites 99.8%

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

Alternative 2: 68.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ t_4 := \frac{\tan t}{ew}\\ t_5 := t\_4 \cdot eh\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-199}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_4 \cdot \left(eh \cdot eh\right), ew\right)}{\cosh \sinh^{-1} t\_5}\right|\\ \mathbf{elif}\;t\_3 \leq 10^{+149}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos t, ew, \frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \tan t\right)}{ew}\right)}{\sqrt{1 + {t\_5}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))))
        (t_4 (/ (tan t) ew))
        (t_5 (* t_4 eh)))
   (if (<= t_3 -4e+23)
     (fabs t_1)
     (if (<= t_3 2e-199)
       (fabs (/ (fma (sin t) (* t_4 (* eh eh)) ew) (cosh (asinh t_5))))
       (if (<= t_3 1e+149)
         (/
          (fma (cos t) ew (/ (* (* (sin t) eh) (* eh (tan t))) ew))
          (sqrt (+ 1.0 (pow t_5 2.0))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = (t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double t_4 = tan(t) / ew;
	double t_5 = t_4 * eh;
	double tmp;
	if (t_3 <= -4e+23) {
		tmp = fabs(t_1);
	} else if (t_3 <= 2e-199) {
		tmp = fabs((fma(sin(t), (t_4 * (eh * eh)), ew) / cosh(asinh(t_5))));
	} else if (t_3 <= 1e+149) {
		tmp = fma(cos(t), ew, (((sin(t) * eh) * (eh * tan(t))) / ew)) / sqrt((1.0 + pow(t_5, 2.0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	t_4 = Float64(tan(t) / ew)
	t_5 = Float64(t_4 * eh)
	tmp = 0.0
	if (t_3 <= -4e+23)
		tmp = abs(t_1);
	elseif (t_3 <= 2e-199)
		tmp = abs(Float64(fma(sin(t), Float64(t_4 * Float64(eh * eh)), ew) / cosh(asinh(t_5))));
	elseif (t_3 <= 1e+149)
		tmp = Float64(fma(cos(t), ew, Float64(Float64(Float64(sin(t) * eh) * Float64(eh * tan(t))) / ew)) / sqrt(Float64(1.0 + (t_5 ^ 2.0))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * eh), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+23], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[t$95$3, 2e-199], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$4 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 1e+149], N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.9999999999999997e23

   1. Initial program 99.8%

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

   3. Applied rewrites 60.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -3.9999999999999997e23 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1.99999999999999996e-199

   1. Initial program 99.8%

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

   3. Applied rewrites 83.6%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 67.5%

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

   if 1.99999999999999996e-199 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1.00000000000000005e149

   1. Initial program 99.8%

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

   3. Applied rewrites 78.1%

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

   5. Applied rewrites 72.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 79.4

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

   7. Applied rewrites 79.4%

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

   if 1.00000000000000005e149 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 53.7%

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 64.2

         \[\leadsto ew \cdot \cos t \]

   8. Applied rewrites 64.2%

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

Alternative 3: 72.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := t\_2 \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3\\ t_5 := \frac{\tan t}{ew}\\ t_6 := t\_5 \cdot eh\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{+51}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_5 \cdot \left(eh \cdot eh\right), t\_1\right)}{\sqrt{\mathsf{fma}\left(t\_6, t\_6, 1\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin t \cdot eh, eh \cdot t\_5, t\_1\right)}{\sqrt{1 + {t\_6}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew))
        (t_2 (* ew (cos t)))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (- (* t_2 (cos t_3)) (* (* eh (sin t)) (sin t_3))))
        (t_5 (/ (tan t) ew))
        (t_6 (* t_5 eh)))
   (if (<= t_4 -4e+51)
     (fabs t_2)
     (if (<= t_4 -5e-268)
       (fabs (/ (fma (sin t) (* t_5 (* eh eh)) t_1) (sqrt (fma t_6 t_6 1.0))))
       (/ (fma (* (sin t) eh) (* eh t_5) t_1) (sqrt (+ 1.0 (pow t_6 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = ew * cos(t);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = (t_2 * cos(t_3)) - ((eh * sin(t)) * sin(t_3));
	double t_5 = tan(t) / ew;
	double t_6 = t_5 * eh;
	double tmp;
	if (t_4 <= -4e+51) {
		tmp = fabs(t_2);
	} else if (t_4 <= -5e-268) {
		tmp = fabs((fma(sin(t), (t_5 * (eh * eh)), t_1) / sqrt(fma(t_6, t_6, 1.0))));
	} else {
		tmp = fma((sin(t) * eh), (eh * t_5), t_1) / sqrt((1.0 + pow(t_6, 2.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = Float64(Float64(t_2 * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3)))
	t_5 = Float64(tan(t) / ew)
	t_6 = Float64(t_5 * eh)
	tmp = 0.0
	if (t_4 <= -4e+51)
		tmp = abs(t_2);
	elseif (t_4 <= -5e-268)
		tmp = abs(Float64(fma(sin(t), Float64(t_5 * Float64(eh * eh)), t_1) / sqrt(fma(t_6, t_6, 1.0))));
	else
		tmp = Float64(fma(Float64(sin(t) * eh), Float64(eh * t_5), t_1) / sqrt(Float64(1.0 + (t_6 ^ 2.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * eh), $MachinePrecision]}, If[LessEqual[t$95$4, -4e+51], N[Abs[t$95$2], $MachinePrecision], If[LessEqual[t$95$4, -5e-268], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$5 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / N[Sqrt[N[(t$95$6 * t$95$6 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(eh * t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$6, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4e51

   1. Initial program 99.8%

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

   3. Applied rewrites 59.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 63.0

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

   6. Applied rewrites 63.0%

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

   if -4e51 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 82.2%

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

      1. lift-cosh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cosh-asinh N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-tan.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-*.f64 77.2

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

   5. Applied rewrites 77.2%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.1%

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

   5. Applied rewrites 64.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 76.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 4: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2\\ t_4 := \frac{\tan t}{ew}\\ t_5 := t\_4 \cdot eh\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+23}:\\ \;\;\;\;\left|t\_1\right|\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_4 \cdot \left(eh \cdot eh\right), ew\right)}{\cosh \sinh^{-1} t\_5}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sin t \cdot eh, eh \cdot t\_4, \cos t \cdot ew\right)}{\sqrt{1 + {t\_5}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t)))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))))
        (t_4 (/ (tan t) ew))
        (t_5 (* t_4 eh)))
   (if (<= t_3 -4e+23)
     (fabs t_1)
     (if (<= t_3 -5e-268)
       (fabs (/ (fma (sin t) (* t_4 (* eh eh)) ew) (cosh (asinh t_5))))
       (/
        (fma (* (sin t) eh) (* eh t_4) (* (cos t) ew))
        (sqrt (+ 1.0 (pow t_5 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = (t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2));
	double t_4 = tan(t) / ew;
	double t_5 = t_4 * eh;
	double tmp;
	if (t_3 <= -4e+23) {
		tmp = fabs(t_1);
	} else if (t_3 <= -5e-268) {
		tmp = fabs((fma(sin(t), (t_4 * (eh * eh)), ew) / cosh(asinh(t_5))));
	} else {
		tmp = fma((sin(t) * eh), (eh * t_4), (cos(t) * ew)) / sqrt((1.0 + pow(t_5, 2.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2)))
	t_4 = Float64(tan(t) / ew)
	t_5 = Float64(t_4 * eh)
	tmp = 0.0
	if (t_3 <= -4e+23)
		tmp = abs(t_1);
	elseif (t_3 <= -5e-268)
		tmp = abs(Float64(fma(sin(t), Float64(t_4 * Float64(eh * eh)), ew) / cosh(asinh(t_5))));
	else
		tmp = Float64(fma(Float64(sin(t) * eh), Float64(eh * t_4), Float64(cos(t) * ew)) / sqrt(Float64(1.0 + (t_5 ^ 2.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * eh), $MachinePrecision]}, If[LessEqual[t$95$3, -4e+23], N[Abs[t$95$1], $MachinePrecision], If[LessEqual[t$95$3, -5e-268], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$4 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[(eh * t$95$4), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.9999999999999997e23

   1. Initial program 99.8%

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

   3. Applied rewrites 60.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 62.7

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

   6. Applied rewrites 62.7%

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

   if -3.9999999999999997e23 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 83.8%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 66.8%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.1%

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

   5. Applied rewrites 64.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 76.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 5: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \tan^{-1} t\_1\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{eh \cdot \tan t}{ew}, \sin t \cdot eh, t\_4\right)}{-\cosh \sinh^{-1} t\_1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin t\_2 \cdot \sin t, eh, t\_4 \cdot \cos t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2 (atan t_1))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_3)) (* (* eh (sin t)) (sin t_3)))
        -5e-268)
     (/ (fma (/ (* eh (tan t)) ew) (* (sin t) eh) t_4) (- (cosh (asinh t_1))))
     (fma (* (sin t_2) (sin t)) eh (* t_4 (cos t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = atan(t_1);
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_3)) - ((eh * sin(t)) * sin(t_3))) <= -5e-268) {
		tmp = fma(((eh * tan(t)) / ew), (sin(t) * eh), t_4) / -cosh(asinh(t_1));
	} else {
		tmp = fma((sin(t_2) * sin(t)), eh, (t_4 * cos(t_2)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = atan(t_1)
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3))) <= -5e-268)
		tmp = Float64(fma(Float64(Float64(eh * tan(t)) / ew), Float64(sin(t) * eh), t_4) / Float64(-cosh(asinh(t_1))));
	else
		tmp = fma(Float64(sin(t_2) * sin(t)), eh, Float64(t_4 * cos(t_2)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-268], N[(N[(N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + t$95$4), $MachinePrecision] / (-N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sin[t$95$2], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(t$95$4 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

   5. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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. Step-by-step derivation

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

      4. rem-square-sqrt 98.7

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

      5. lift--.f64 N/A

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

      6. sub-negate1 N/A

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

   3. Applied rewrites 98.7%

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

Alternative 6: 81.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \sin t \cdot eh\\ t_3 := \cosh \sinh^{-1} t\_1\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{eh \cdot \tan t}{ew}, t\_2, t\_5\right)}{-t\_3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\frac{\mathsf{fma}\left(t\_1, t\_2, t\_5\right)}{t\_3}}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2 (* (sin t) eh))
        (t_3 (cosh (asinh t_1)))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4)))
        -5e-268)
     (/ (fma (/ (* eh (tan t)) ew) t_2 t_5) (- t_3))
     (pow (sqrt (/ (fma t_1 t_2 t_5) t_3)) 2.0))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = sin(t) * eh;
	double t_3 = cosh(asinh(t_1));
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4))) <= -5e-268) {
		tmp = fma(((eh * tan(t)) / ew), t_2, t_5) / -t_3;
	} else {
		tmp = pow(sqrt((fma(t_1, t_2, t_5) / t_3)), 2.0);
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = Float64(sin(t) * eh)
	t_3 = cosh(asinh(t_1))
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4))) <= -5e-268)
		tmp = Float64(fma(Float64(Float64(eh * tan(t)) / ew), t_2, t_5) / Float64(-t_3));
	else
		tmp = sqrt(Float64(fma(t_1, t_2, t_5) / t_3)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$3 = N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-268], N[(N[(N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * t$95$2 + t$95$5), $MachinePrecision] / (-t$95$3)), $MachinePrecision], N[Power[N[Sqrt[N[(N[(t$95$1 * t$95$2 + t$95$5), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

   5. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.1%

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

   5. Applied rewrites 80.4%

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

Alternative 7: 80.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ t_3 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_4 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_3 - \left(eh \cdot \sin t\right) \cdot \sin t\_3 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{eh \cdot \tan t}{ew}, t\_1, t\_4\right)}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_4 - t\_1 \cdot \left(\frac{-eh}{ew} \cdot \tan t\right)}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (cosh (asinh (* (/ (tan t) ew) eh))))
        (t_3 (atan (/ (* (- eh) (tan t)) ew)))
        (t_4 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_3)) (* (* eh (sin t)) (sin t_3)))
        -5e-268)
     (/ (fma (/ (* eh (tan t)) ew) t_1 t_4) (- t_2))
     (/ (- t_4 (* t_1 (* (/ (- eh) ew) (tan t)))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cosh(asinh(((tan(t) / ew) * eh)));
	double t_3 = atan(((-eh * tan(t)) / ew));
	double t_4 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_3)) - ((eh * sin(t)) * sin(t_3))) <= -5e-268) {
		tmp = fma(((eh * tan(t)) / ew), t_1, t_4) / -t_2;
	} else {
		tmp = (t_4 - (t_1 * ((-eh / ew) * tan(t)))) / t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = cosh(asinh(Float64(Float64(tan(t) / ew) * eh)))
	t_3 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_4 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_3)) - Float64(Float64(eh * sin(t)) * sin(t_3))) <= -5e-268)
		tmp = Float64(fma(Float64(Float64(eh * tan(t)) / ew), t_1, t_4) / Float64(-t_2));
	else
		tmp = Float64(Float64(t_4 - Float64(t_1 * Float64(Float64(Float64(-eh) / ew) * tan(t)))) / t_2);
	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[Cosh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-268], N[(N[(N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * t$95$1 + t$95$4), $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[(t$95$4 - N[(t$95$1 * N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

   5. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 77.9%

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

Alternative 8: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ t_3 := \sin t \cdot eh\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{eh \cdot \tan t}{ew}, t\_3, t\_5\right)}{-\cosh \sinh^{-1} t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, eh \cdot t\_1, t\_5\right)}{\sqrt{1 + {t\_2}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* t_1 eh))
        (t_3 (* (sin t) eh))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4)))
        -5e-268)
     (/ (fma (/ (* eh (tan t)) ew) t_3 t_5) (- (cosh (asinh t_2))))
     (/ (fma t_3 (* eh t_1) t_5) (sqrt (+ 1.0 (pow t_2 2.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double t_3 = sin(t) * eh;
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4))) <= -5e-268) {
		tmp = fma(((eh * tan(t)) / ew), t_3, t_5) / -cosh(asinh(t_2));
	} else {
		tmp = fma(t_3, (eh * t_1), t_5) / sqrt((1.0 + pow(t_2, 2.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	t_3 = Float64(sin(t) * eh)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4))) <= -5e-268)
		tmp = Float64(fma(Float64(Float64(eh * tan(t)) / ew), t_3, t_5) / Float64(-cosh(asinh(t_2))));
	else
		tmp = Float64(fma(t_3, Float64(eh * t_1), t_5) / sqrt(Float64(1.0 + (t_2 ^ 2.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * eh), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-268], N[(N[(N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * t$95$3 + t$95$5), $MachinePrecision] / (-N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(t$95$3 * N[(eh * t$95$1), $MachinePrecision] + t$95$5), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

   5. Applied rewrites 82.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 82.4

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

   7. Applied rewrites 82.4%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.1%

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

   5. Applied rewrites 64.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 76.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 9: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ t_3 := \sin t \cdot eh\\ t_4 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_5 := \cos t \cdot ew\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_4 - \left(eh \cdot \sin t\right) \cdot \sin t\_4 \leq -5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_2, t\_3, t\_5\right)}{-\cosh \sinh^{-1} t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_3, eh \cdot t\_1, t\_5\right)}{\sqrt{1 + {t\_2}^{2}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* t_1 eh))
        (t_3 (* (sin t) eh))
        (t_4 (atan (/ (* (- eh) (tan t)) ew)))
        (t_5 (* (cos t) ew)))
   (if (<=
        (- (* (* ew (cos t)) (cos t_4)) (* (* eh (sin t)) (sin t_4)))
        -5e-268)
     (/ (fma t_2 t_3 t_5) (- (cosh (asinh t_2))))
     (/ (fma t_3 (* eh t_1) t_5) (sqrt (+ 1.0 (pow t_2 2.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double t_3 = sin(t) * eh;
	double t_4 = atan(((-eh * tan(t)) / ew));
	double t_5 = cos(t) * ew;
	double tmp;
	if ((((ew * cos(t)) * cos(t_4)) - ((eh * sin(t)) * sin(t_4))) <= -5e-268) {
		tmp = fma(t_2, t_3, t_5) / -cosh(asinh(t_2));
	} else {
		tmp = fma(t_3, (eh * t_1), t_5) / sqrt((1.0 + pow(t_2, 2.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	t_3 = Float64(sin(t) * eh)
	t_4 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_5 = Float64(cos(t) * ew)
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_4)) - Float64(Float64(eh * sin(t)) * sin(t_4))) <= -5e-268)
		tmp = Float64(fma(t_2, t_3, t_5) / Float64(-cosh(asinh(t_2))));
	else
		tmp = Float64(fma(t_3, Float64(eh * t_1), t_5) / sqrt(Float64(1.0 + (t_2 ^ 2.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * eh), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -5e-268], N[(N[(t$95$2 * t$95$3 + t$95$5), $MachinePrecision] / (-N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(t$95$3 * N[(eh * t$95$1), $MachinePrecision] + t$95$5), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.9999999999999999e-268

   1. Initial program 99.8%

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

   3. Applied rewrites 1.2%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. distribute-frac-neg2 N/A

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

      9. distribute-neg-frac N/A

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

   5. Applied rewrites 82.4%

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

   if -4.9999999999999999e-268 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.1%

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

   5. Applied rewrites 64.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*r* N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 76.9

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 76.9

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

   7. Applied rewrites 76.9%

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

Alternative 10: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq 5 \cdot 10^{-215}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) 5e-215)
     (fabs ew)
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 5e-215) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((-eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 5d-215) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= 5e-215) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= 5e-215:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= 5e-215)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= 5e-215)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-215], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 4.99999999999999956e-215

   1. Initial program 99.8%

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

   3. Applied rewrites 69.2%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 43.0%

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

   if 4.99999999999999956e-215 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 68.2%

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 61.8

         \[\leadsto ew \cdot \cos t \]

   8. Applied rewrites 61.8%

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

Alternative 11: 76.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := eh \cdot \sin t\\ t_3 := -1 \cdot t\_2\\ \mathbf{if}\;eh \leq -3 \cdot 10^{+270}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 3.4 \cdot 10^{+162}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \mathbf{elif}\;eh \leq 9.2 \cdot 10^{+210}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* eh (sin t))) (t_3 (* -1.0 t_2)))
   (if (<= eh -3e+270)
     t_3
     (if (<= eh -1.35e+154)
       t_2
       (if (<= eh 3.4e+162)
         (fabs
          (/
           (fma (sin t) (* t_1 (* eh eh)) (* (cos t) ew))
           (cosh (asinh (* t_1 eh)))))
         (if (<= eh 9.2e+210) t_3 t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = eh * sin(t);
	double t_3 = -1.0 * t_2;
	double tmp;
	if (eh <= -3e+270) {
		tmp = t_3;
	} else if (eh <= -1.35e+154) {
		tmp = t_2;
	} else if (eh <= 3.4e+162) {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), (cos(t) * ew)) / cosh(asinh((t_1 * eh)))));
	} else if (eh <= 9.2e+210) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(eh * sin(t))
	t_3 = Float64(-1.0 * t_2)
	tmp = 0.0
	if (eh <= -3e+270)
		tmp = t_3;
	elseif (eh <= -1.35e+154)
		tmp = t_2;
	elseif (eh <= 3.4e+162)
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), Float64(cos(t) * ew)) / cosh(asinh(Float64(t_1 * eh)))));
	elseif (eh <= 9.2e+210)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(-1.0 * t$95$2), $MachinePrecision]}, If[LessEqual[eh, -3e+270], t$95$3, If[LessEqual[eh, -1.35e+154], t$95$2, If[LessEqual[eh, 3.4e+162], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 9.2e+210], t$95$3, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.00000000000000014e270 or 3.40000000000000003e162 < eh < 9.1999999999999995e210

   1. Initial program 99.8%

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

   3. Applied rewrites 1.9%

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

   5. Applied rewrites 1.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 1.2

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

   7. Applied rewrites 1.2%

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

   8. Taylor expanded in eh around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(eh \cdot \sin t\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(eh \cdot \color{blue}{\sin t}\right) \]

      3. lift-sin.f64 35.4

         \[\leadsto -1 \cdot \left(eh \cdot \sin t\right) \]

   10. Applied rewrites 35.4%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]

   if -3.00000000000000014e270 < eh < -1.35000000000000003e154 or 9.1999999999999995e210 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 1.9%

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

   5. Applied rewrites 1.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 1.1

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

   7. Applied rewrites 1.1%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 38.6

         \[\leadsto eh \cdot \sin t \]

   10. Applied rewrites 38.6%

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

   if -1.35000000000000003e154 < eh < 3.40000000000000003e162

   1. Initial program 99.8%

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

   3. Applied rewrites 89.4%

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

Alternative 12: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.25 \cdot 10^{-124}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 4.7 \cdot 10^{-129}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (fabs (* ew (cos t)))))
   (if (<= ew -1.25e-124)
     t_2
     (if (<= ew 4.7e-129)
       (fabs (/ (fma (sin t) (* t_1 (* eh eh)) ew) (cosh (asinh (* t_1 eh)))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.25e-124) {
		tmp = t_2;
	} else if (ew <= 4.7e-129) {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), ew) / cosh(asinh((t_1 * eh)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.25e-124)
		tmp = t_2;
	elseif (ew <= 4.7e-129)
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), ew) / cosh(asinh(Float64(t_1 * eh)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.25e-124], t$95$2, If[LessEqual[ew, 4.7e-129], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + ew), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.2500000000000001e-124 or 4.7000000000000002e-129 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 74.6%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 75.3

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

   6. Applied rewrites 75.3%

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

   if -1.2500000000000001e-124 < ew < 4.7000000000000002e-129

   1. Initial program 99.8%

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

   3. Applied rewrites 54.7%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 48.1%

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

Alternative 13: 64.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-223}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.9e-142) t_1 (if (<= ew 8e-223) (* eh (sin t)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.9e-142) {
		tmp = t_1;
	} else if (ew <= 8e-223) {
		tmp = eh * sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-1.9d-142)) then
        tmp = t_1
    else if (ew <= 8d-223) then
        tmp = eh * sin(t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -1.9e-142) {
		tmp = t_1;
	} else if (ew <= 8e-223) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -1.9e-142:
		tmp = t_1
	elif ew <= 8e-223:
		tmp = eh * math.sin(t)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.9e-142)
		tmp = t_1;
	elseif (ew <= 8e-223)
		tmp = Float64(eh * sin(t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -1.9e-142)
		tmp = t_1;
	elseif (ew <= 8e-223)
		tmp = eh * sin(t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.9e-142], t$95$1, If[LessEqual[ew, 8e-223], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.89999999999999986e-142 or 7.9999999999999998e-223 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 73.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lift-cos.f64 71.0

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

   6. Applied rewrites 71.0%

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

   if -1.89999999999999986e-142 < ew < 7.9999999999999998e-223

   1. Initial program 99.8%

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

   3. Applied rewrites 21.6%

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

   5. Applied rewrites 16.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 10.5

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 40.0

         \[\leadsto eh \cdot \sin t \]

   10. Applied rewrites 40.0%

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

Alternative 14: 45.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{-142}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{-214}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.9e-142) (fabs ew) (if (<= ew 1.05e-214) (* eh (sin t)) ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.9e-142) {
		tmp = fabs(ew);
	} else if (ew <= 1.05e-214) {
		tmp = eh * sin(t);
	} else {
		tmp = 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 (ew <= (-1.9d-142)) then
        tmp = abs(ew)
    else if (ew <= 1.05d-214) then
        tmp = eh * sin(t)
    else
        tmp = ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.9e-142) {
		tmp = Math.abs(ew);
	} else if (ew <= 1.05e-214) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.9e-142:
		tmp = math.fabs(ew)
	elif ew <= 1.05e-214:
		tmp = eh * math.sin(t)
	else:
		tmp = ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.9e-142)
		tmp = abs(ew);
	elseif (ew <= 1.05e-214)
		tmp = Float64(eh * sin(t));
	else
		tmp = ew;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.9e-142)
		tmp = abs(ew);
	elseif (ew <= 1.05e-214)
		tmp = eh * sin(t);
	else
		tmp = ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.9e-142], N[Abs[ew], $MachinePrecision], If[LessEqual[ew, 1.05e-214], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], ew]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.89999999999999986e-142

   1. Initial program 99.8%

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

   3. Applied rewrites 73.9%

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

   4. Taylor expanded in t around 0

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

   6. Applied rewrites 48.7%

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

   if -1.89999999999999986e-142 < ew < 1.04999999999999996e-214

   1. Initial program 99.8%

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

   3. Applied rewrites 22.2%

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

   5. Applied rewrites 17.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-prod-down N/A

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

      6. pow2 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-pow.f64 11.0

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

   7. Applied rewrites 11.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 39.9

         \[\leadsto eh \cdot \sin t \]

   10. Applied rewrites 39.9%

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

   if 1.04999999999999996e-214 < ew

   1. Initial program 99.8%

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

   3. Applied rewrites 54.9%

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

   5. Applied rewrites 65.2%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{ew} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.4%

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

Alternative 15: 42.4% accurate, 287.3× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(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(ew)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 68.7%

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

4. Taylor expanded in t around 0

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

6. Applied rewrites 42.4%

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

Alternative 16: 22.3% accurate, 862.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 35.3%

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

5. Applied rewrites 40.9%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{ew} \]
7. Step-by-step derivation

8. Applied rewrites 22.3%

   \[\leadsto \color{blue}{ew} \]
9. Add Preprocessing

Reproduce

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