Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.4s

Alternatives: 12

Speedup: 1.2×

Specification

Math

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

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

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

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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	return math.fabs((((math.cos(t) * ew) / math.cosh(math.asinh(((math.tan(t) * eh) / ew)))) - (math.tanh(math.asinh(((math.tan(t) / ew) * -eh))) * (math.sin(t) * eh))))

Julia

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

MATLAB

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

Wolfram

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

3. Applied rewrites 99.8%

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

Alternative 3: 95.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \tan^{-1} \left(\frac{-1 \cdot \left(\left|eh\right| \cdot t\right)}{ew}\right)\\ \mathbf{if}\;\left|eh\right| \leq 4.6 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{\cos t \cdot ew - \left(-\left(\sin t \cdot \left|eh\right|\right) \cdot \left(\left|eh\right| \cdot \frac{\tan t}{ew}\right)\right)}{\cosh \sinh^{-1} \left(\frac{\tan t \cdot \left|eh\right|}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(\left|eh\right| \cdot \sin t\right) \cdot \sin t\_1\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* -1.0 (* (fabs eh) t)) ew))))
   (if (<= (fabs eh) 4.6e+19)
     (fabs
      (/
       (-
        (* (cos t) ew)
        (- (* (* (sin t) (fabs eh)) (* (fabs eh) (/ (tan t) ew)))))
       (cosh (asinh (/ (* (tan t) (fabs eh)) ew)))))
     (fabs
      (- (* (* ew (cos t)) (cos t_1)) (* (* (fabs eh) (sin t)) (sin t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-1.0 * (fabs(eh) * t)) / ew));
	double tmp;
	if (fabs(eh) <= 4.6e+19) {
		tmp = fabs((((cos(t) * ew) - -((sin(t) * fabs(eh)) * (fabs(eh) * (tan(t) / ew)))) / cosh(asinh(((tan(t) * fabs(eh)) / ew)))));
	} else {
		tmp = fabs((((ew * cos(t)) * cos(t_1)) - ((fabs(eh) * sin(t)) * sin(t_1))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-1.0 * (math.fabs(eh) * t)) / ew))
	tmp = 0
	if math.fabs(eh) <= 4.6e+19:
		tmp = math.fabs((((math.cos(t) * ew) - -((math.sin(t) * math.fabs(eh)) * (math.fabs(eh) * (math.tan(t) / ew)))) / math.cosh(math.asinh(((math.tan(t) * math.fabs(eh)) / ew)))))
	else:
		tmp = math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((math.fabs(eh) * math.sin(t)) * math.sin(t_1))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(-1.0 * Float64(abs(eh) * t)) / ew))
	tmp = 0.0
	if (abs(eh) <= 4.6e+19)
		tmp = abs(Float64(Float64(Float64(cos(t) * ew) - Float64(-Float64(Float64(sin(t) * abs(eh)) * Float64(abs(eh) * Float64(tan(t) / ew))))) / cosh(asinh(Float64(Float64(tan(t) * abs(eh)) / ew)))));
	else
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(abs(eh) * sin(t)) * sin(t_1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((-1.0 * (abs(eh) * t)) / ew));
	tmp = 0.0;
	if (abs(eh) <= 4.6e+19)
		tmp = abs((((cos(t) * ew) - -((sin(t) * abs(eh)) * (abs(eh) * (tan(t) / ew)))) / cosh(asinh(((tan(t) * abs(eh)) / ew)))));
	else
		tmp = abs((((ew * cos(t)) * cos(t_1)) - ((abs(eh) * sin(t)) * sin(t_1))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(-1.0 * N[(N[Abs[eh], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[eh], $MachinePrecision], 4.6e+19], N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - (-N[(N[(N[Sin[t], $MachinePrecision] * N[Abs[eh], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[eh], $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] * N[Abs[eh], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[eh], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 4.6e19

   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%

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

   if 4.6e19 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8%

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8%

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

   7. Applied rewrites 89.8%

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

Alternative 4: 93.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot \left|eh\right|\\ t_2 := \tan^{-1} \left(\frac{-1 \cdot \left(\left|eh\right| \cdot t\right)}{ew}\right)\\ \mathbf{if}\;\left|eh\right| \leq 3 \cdot 10^{-42}:\\ \;\;\;\;\frac{\left|\left({t\_1}^{2} - -1\right) \cdot \left(\cos t \cdot ew\right)\right|}{\cosh \sinh^{-1} t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(\left|eh\right| \cdot \sin t\right) \cdot \sin t\_2\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) (fabs eh)))
        (t_2 (atan (/ (* -1.0 (* (fabs eh) t)) ew))))
   (if (<= (fabs eh) 3e-42)
     (/ (fabs (* (- (pow t_1 2.0) -1.0) (* (cos t) ew))) (cosh (asinh t_1)))
     (fabs
      (- (* (* ew (cos t)) (cos t_2)) (* (* (fabs eh) (sin t)) (sin t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * fabs(eh);
	double t_2 = atan(((-1.0 * (fabs(eh) * t)) / ew));
	double tmp;
	if (fabs(eh) <= 3e-42) {
		tmp = fabs(((pow(t_1, 2.0) - -1.0) * (cos(t) * ew))) / cosh(asinh(t_1));
	} else {
		tmp = fabs((((ew * cos(t)) * cos(t_2)) - ((fabs(eh) * sin(t)) * sin(t_2))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (math.tan(t) / ew) * math.fabs(eh)
	t_2 = math.atan(((-1.0 * (math.fabs(eh) * t)) / ew))
	tmp = 0
	if math.fabs(eh) <= 3e-42:
		tmp = math.fabs(((math.pow(t_1, 2.0) - -1.0) * (math.cos(t) * ew))) / math.cosh(math.asinh(t_1))
	else:
		tmp = math.fabs((((ew * math.cos(t)) * math.cos(t_2)) - ((math.fabs(eh) * math.sin(t)) * math.sin(t_2))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * abs(eh))
	t_2 = atan(Float64(Float64(-1.0 * Float64(abs(eh) * t)) / ew))
	tmp = 0.0
	if (abs(eh) <= 3e-42)
		tmp = Float64(abs(Float64(Float64((t_1 ^ 2.0) - -1.0) * Float64(cos(t) * ew))) / cosh(asinh(t_1)));
	else
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_2)) - Float64(Float64(abs(eh) * sin(t)) * sin(t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (tan(t) / ew) * abs(eh);
	t_2 = atan(((-1.0 * (abs(eh) * t)) / ew));
	tmp = 0.0;
	if (abs(eh) <= 3e-42)
		tmp = abs((((t_1 ^ 2.0) - -1.0) * (cos(t) * ew))) / cosh(asinh(t_1));
	else
		tmp = abs((((ew * cos(t)) * cos(t_2)) - ((abs(eh) * sin(t)) * sin(t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * N[Abs[eh], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(-1.0 * N[(N[Abs[eh], $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[eh], $MachinePrecision], 3e-42], N[(N[Abs[N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Abs[eh], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 3.00000000000000027e-42

   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 99.8%

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

   5. Applied rewrites 76.8%

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

   if 3.00000000000000027e-42 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8%

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.8%

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

   7. Applied rewrites 89.8%

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

Alternative 5: 84.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{\tan \left(\left|t\right|\right)}{ew} \cdot eh\\ t_2 := \cos \left(\left|t\right|\right) \cdot ew\\ t_3 := \sinh^{-1} \left(\frac{\left|t\right|}{ew} \cdot eh\right)\\ \mathbf{if}\;\left|t\right| \leq 1960000:\\ \;\;\;\;\left|\frac{t\_2}{\cosh t\_3} - \tanh \left(-t\_3\right) \cdot \left(\mathsf{fma}\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot eh, -0.16666666666666666, eh\right) \cdot \left|t\right|\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left({t\_1}^{2} - -1\right) \cdot t\_2\right|}{\cosh \sinh^{-1} t\_1}\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan (fabs t)) ew) eh))
        (t_2 (* (cos (fabs t)) ew))
        (t_3 (asinh (* (/ (fabs t) ew) eh))))
   (if (<= (fabs t) 1960000.0)
     (fabs
      (-
       (/ t_2 (cosh t_3))
       (*
        (tanh (- t_3))
        (*
         (fma (* (* (fabs t) (fabs t)) eh) -0.16666666666666666 eh)
         (fabs t)))))
     (/ (fabs (* (- (pow t_1 2.0) -1.0) t_2)) (cosh (asinh t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(fabs(t)) / ew) * eh;
	double t_2 = cos(fabs(t)) * ew;
	double t_3 = asinh(((fabs(t) / ew) * eh));
	double tmp;
	if (fabs(t) <= 1960000.0) {
		tmp = fabs(((t_2 / cosh(t_3)) - (tanh(-t_3) * (fma(((fabs(t) * fabs(t)) * eh), -0.16666666666666666, eh) * fabs(t)))));
	} else {
		tmp = fabs(((pow(t_1, 2.0) - -1.0) * t_2)) / cosh(asinh(t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(abs(t)) / ew) * eh)
	t_2 = Float64(cos(abs(t)) * ew)
	t_3 = asinh(Float64(Float64(abs(t) / ew) * eh))
	tmp = 0.0
	if (abs(t) <= 1960000.0)
		tmp = abs(Float64(Float64(t_2 / cosh(t_3)) - Float64(tanh(Float64(-t_3)) * Float64(fma(Float64(Float64(abs(t) * abs(t)) * eh), -0.16666666666666666, eh) * abs(t)))));
	else
		tmp = Float64(abs(Float64(Float64((t_1 ^ 2.0) - -1.0) * t_2)) / cosh(asinh(t_1)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[N[Abs[t], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[Abs[t], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[ArcSinh[N[(N[(N[Abs[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[t], $MachinePrecision], 1960000.0], N[Abs[N[(N[(t$95$2 / N[Cosh[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(N[Tanh[(-t$95$3)], $MachinePrecision] * N[(N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * -0.16666666666666666 + eh), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[Power[t$95$1, 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if t < 1.96e6

   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. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \color{blue}{\left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \color{blue}{\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \color{blue}{\left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot \color{blue}{{t}^{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. lower-pow.f64 60.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{\color{blue}{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied rewrites 60.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{2}\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right|} \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 60.2%

         \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   9. Applied rewrites 60.2%

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 59.8%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   12. Applied rewrites 59.8%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   if 1.96e6 < t

   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 99.8%

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

   5. Applied rewrites 76.8%

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

Alternative 6: 76.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_1 := \sin \left(\left|t\right|\right)\\ t_2 := \cos \left(\left|t\right|\right) \cdot ew\\ t_3 := \sinh^{-1} \left(\frac{\left|t\right|}{ew} \cdot eh\right)\\ t_4 := \tanh \left(-t\_3\right)\\ \mathbf{if}\;\left|t\right| \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\left|\frac{t\_2}{\cosh t\_3} - t\_4 \cdot \left(\mathsf{fma}\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot eh, -0.16666666666666666, eh\right) \cdot \left|t\right|\right)\right|\\ \mathbf{elif}\;\left|t\right| \leq 2.35 \cdot 10^{+269}:\\ \;\;\;\;\left|\left(t\_4 \cdot t\_1\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_2 - \left(-\left(t\_1 \cdot eh\right) \cdot \left(eh \cdot \frac{\tan \left(\left|t\right|\right)}{ew}\right)\right)}{1}\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sin (fabs t)))
        (t_2 (* (cos (fabs t)) ew))
        (t_3 (asinh (* (/ (fabs t) ew) eh)))
        (t_4 (tanh (- t_3))))
   (if (<= (fabs t) 1.15e+55)
     (fabs
      (-
       (/ t_2 (cosh t_3))
       (*
        t_4
        (*
         (fma (* (* (fabs t) (fabs t)) eh) -0.16666666666666666 eh)
         (fabs t)))))
     (if (<= (fabs t) 2.35e+269)
       (fabs (* (* t_4 t_1) (- eh)))
       (fabs
        (/ (- t_2 (- (* (* t_1 eh) (* eh (/ (tan (fabs t)) ew))))) 1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(fabs(t));
	double t_2 = cos(fabs(t)) * ew;
	double t_3 = asinh(((fabs(t) / ew) * eh));
	double t_4 = tanh(-t_3);
	double tmp;
	if (fabs(t) <= 1.15e+55) {
		tmp = fabs(((t_2 / cosh(t_3)) - (t_4 * (fma(((fabs(t) * fabs(t)) * eh), -0.16666666666666666, eh) * fabs(t)))));
	} else if (fabs(t) <= 2.35e+269) {
		tmp = fabs(((t_4 * t_1) * -eh));
	} else {
		tmp = fabs(((t_2 - -((t_1 * eh) * (eh * (tan(fabs(t)) / ew)))) / 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = sin(abs(t))
	t_2 = Float64(cos(abs(t)) * ew)
	t_3 = asinh(Float64(Float64(abs(t) / ew) * eh))
	t_4 = tanh(Float64(-t_3))
	tmp = 0.0
	if (abs(t) <= 1.15e+55)
		tmp = abs(Float64(Float64(t_2 / cosh(t_3)) - Float64(t_4 * Float64(fma(Float64(Float64(abs(t) * abs(t)) * eh), -0.16666666666666666, eh) * abs(t)))));
	elseif (abs(t) <= 2.35e+269)
		tmp = abs(Float64(Float64(t_4 * t_1) * Float64(-eh)));
	else
		tmp = abs(Float64(Float64(t_2 - Float64(-Float64(Float64(t_1 * eh) * Float64(eh * Float64(tan(abs(t)) / ew))))) / 1.0));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[Abs[t], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[ArcSinh[N[(N[(N[Abs[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Tanh[(-t$95$3)], $MachinePrecision]}, If[LessEqual[N[Abs[t], $MachinePrecision], 1.15e+55], N[Abs[N[(N[(t$95$2 / N[Cosh[t$95$3], $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * N[(N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * -0.16666666666666666 + eh), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.35e+269], N[Abs[N[(N[(t$95$4 * t$95$1), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$2 - (-N[(N[(t$95$1 * eh), $MachinePrecision] * N[(eh * N[(N[Tan[N[Abs[t], $MachinePrecision]], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / 1.0), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999994e55

   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. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \color{blue}{\left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \color{blue}{\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \color{blue}{\left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot \color{blue}{{t}^{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. lower-pow.f64 60.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{\color{blue}{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied rewrites 60.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{2}\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right|} \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 60.2%

         \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   9. Applied rewrites 60.2%

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 59.8%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   12. Applied rewrites 59.8%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   if 1.14999999999999994e55 < t < 2.35e269

   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. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 41.4%

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

   6. Applied rewrites 41.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 41.5%

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

   9. Applied rewrites 41.5%

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

   if 2.35e269 < t

   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%

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

   4. Taylor expanded in eh around 0

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

   6. Applied rewrites 62.4%

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

Alternative 7: 75.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|t\right|\right)\\ t_2 := \sinh^{-1} \left(\frac{\left|t\right|}{ew} \cdot eh\right)\\ t_3 := \tanh \left(-t\_2\right)\\ \mathbf{if}\;\left|t\right| \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\left|\frac{t\_1 \cdot ew}{\cosh t\_2} - t\_3 \cdot \left(\mathsf{fma}\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot eh, -0.16666666666666666, eh\right) \cdot \left|t\right|\right)\right|\\ \mathbf{elif}\;\left|t\right| \leq 2.35 \cdot 10^{+269}:\\ \;\;\;\;\left|\left(t\_3 \cdot \sin \left(\left|t\right|\right)\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\_1\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (cos (fabs t)))
        (t_2 (asinh (* (/ (fabs t) ew) eh)))
        (t_3 (tanh (- t_2))))
   (if (<= (fabs t) 1.15e+55)
     (fabs
      (-
       (/ (* t_1 ew) (cosh t_2))
       (*
        t_3
        (*
         (fma (* (* (fabs t) (fabs t)) eh) -0.16666666666666666 eh)
         (fabs t)))))
     (if (<= (fabs t) 2.35e+269)
       (fabs (* (* t_3 (sin (fabs t))) (- eh)))
       (fabs (* ew t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(fabs(t));
	double t_2 = asinh(((fabs(t) / ew) * eh));
	double t_3 = tanh(-t_2);
	double tmp;
	if (fabs(t) <= 1.15e+55) {
		tmp = fabs((((t_1 * ew) / cosh(t_2)) - (t_3 * (fma(((fabs(t) * fabs(t)) * eh), -0.16666666666666666, eh) * fabs(t)))));
	} else if (fabs(t) <= 2.35e+269) {
		tmp = fabs(((t_3 * sin(fabs(t))) * -eh));
	} else {
		tmp = fabs((ew * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = cos(abs(t))
	t_2 = asinh(Float64(Float64(abs(t) / ew) * eh))
	t_3 = tanh(Float64(-t_2))
	tmp = 0.0
	if (abs(t) <= 1.15e+55)
		tmp = abs(Float64(Float64(Float64(t_1 * ew) / cosh(t_2)) - Float64(t_3 * Float64(fma(Float64(Float64(abs(t) * abs(t)) * eh), -0.16666666666666666, eh) * abs(t)))));
	elseif (abs(t) <= 2.35e+269)
		tmp = abs(Float64(Float64(t_3 * sin(abs(t))) * Float64(-eh)));
	else
		tmp = abs(Float64(ew * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Cos[N[Abs[t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcSinh[N[(N[(N[Abs[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Tanh[(-t$95$2)], $MachinePrecision]}, If[LessEqual[N[Abs[t], $MachinePrecision], 1.15e+55], N[Abs[N[(N[(N[(t$95$1 * ew), $MachinePrecision] / N[Cosh[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * -0.16666666666666666 + eh), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.35e+269], N[Abs[N[(N[(t$95$3 * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999994e55

   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. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \color{blue}{\left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \color{blue}{\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \color{blue}{\left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot \color{blue}{{t}^{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. lower-pow.f64 60.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{\color{blue}{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied rewrites 60.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{2}\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right|} \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 60.2%

         \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   9. Applied rewrites 60.2%

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 59.8%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{t}{\color{blue}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   12. Applied rewrites 59.8%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\color{blue}{\frac{t}{ew}} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   if 1.14999999999999994e55 < t < 2.35e269

   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. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 41.4%

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

   6. Applied rewrites 41.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 41.5%

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

   9. Applied rewrites 41.5%

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

   if 2.35e269 < t

   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 99.8%

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

   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. lower-cos.f64 61.9%

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

   6. Applied rewrites 61.9%

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

Alternative 8: 75.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_1 := \cos \left(\left|t\right|\right)\\ t_2 := \frac{eh \cdot \left|t\right|}{ew}\\ \mathbf{if}\;\left|t\right| \leq 1.15 \cdot 10^{+55}:\\ \;\;\;\;\left|\frac{t\_1 \cdot ew}{\cosh t\_2} - \tanh \left(-t\_2\right) \cdot \left(\mathsf{fma}\left(\left(\left|t\right| \cdot \left|t\right|\right) \cdot eh, -0.16666666666666666, eh\right) \cdot \left|t\right|\right)\right|\\ \mathbf{elif}\;\left|t\right| \leq 2.35 \cdot 10^{+269}:\\ \;\;\;\;\left|\left(\tanh \left(-\sinh^{-1} \left(\frac{\left|t\right|}{ew} \cdot eh\right)\right) \cdot \sin \left(\left|t\right|\right)\right) \cdot \left(-eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\_1\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (cos (fabs t))) (t_2 (/ (* eh (fabs t)) ew)))
   (if (<= (fabs t) 1.15e+55)
     (fabs
      (-
       (/ (* t_1 ew) (cosh t_2))
       (*
        (tanh (- t_2))
        (*
         (fma (* (* (fabs t) (fabs t)) eh) -0.16666666666666666 eh)
         (fabs t)))))
     (if (<= (fabs t) 2.35e+269)
       (fabs
        (*
         (* (tanh (- (asinh (* (/ (fabs t) ew) eh)))) (sin (fabs t)))
         (- eh)))
       (fabs (* ew t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(fabs(t));
	double t_2 = (eh * fabs(t)) / ew;
	double tmp;
	if (fabs(t) <= 1.15e+55) {
		tmp = fabs((((t_1 * ew) / cosh(t_2)) - (tanh(-t_2) * (fma(((fabs(t) * fabs(t)) * eh), -0.16666666666666666, eh) * fabs(t)))));
	} else if (fabs(t) <= 2.35e+269) {
		tmp = fabs(((tanh(-asinh(((fabs(t) / ew) * eh))) * sin(fabs(t))) * -eh));
	} else {
		tmp = fabs((ew * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = cos(abs(t))
	t_2 = Float64(Float64(eh * abs(t)) / ew)
	tmp = 0.0
	if (abs(t) <= 1.15e+55)
		tmp = abs(Float64(Float64(Float64(t_1 * ew) / cosh(t_2)) - Float64(tanh(Float64(-t_2)) * Float64(fma(Float64(Float64(abs(t) * abs(t)) * eh), -0.16666666666666666, eh) * abs(t)))));
	elseif (abs(t) <= 2.35e+269)
		tmp = abs(Float64(Float64(tanh(Float64(-asinh(Float64(Float64(abs(t) / ew) * eh)))) * sin(abs(t))) * Float64(-eh)));
	else
		tmp = abs(Float64(ew * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Cos[N[Abs[t], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(eh * N[Abs[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[N[Abs[t], $MachinePrecision], 1.15e+55], N[Abs[N[(N[(N[(t$95$1 * ew), $MachinePrecision] / N[Cosh[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[Tanh[(-t$95$2)], $MachinePrecision] * N[(N[(N[(N[(N[Abs[t], $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * -0.16666666666666666 + eh), $MachinePrecision] * N[Abs[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[Abs[t], $MachinePrecision], 2.35e+269], N[Abs[N[(N[(N[Tanh[(-N[ArcSinh[N[(N[(N[Abs[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[Sin[N[Abs[t], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t$95$1), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 1.14999999999999994e55

   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. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \color{blue}{\left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \color{blue}{\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \color{blue}{\left(eh \cdot {t}^{2}\right)}\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot \color{blue}{{t}^{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. lower-pow.f64 60.2%

         \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{\color{blue}{2}}\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   4. Applied rewrites 60.2%

      \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \color{blue}{\left(t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot {t}^{2}\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. Applied rewrites 60.2%

      \[\leadsto \color{blue}{\left|\frac{\cos t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right|} \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \color{blue}{\left(\frac{eh \cdot t}{ew}\right)}} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 60.0%

         \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \left(\frac{eh \cdot t}{ew}\right)} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   9. Applied rewrites 60.0%

      \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \color{blue}{\left(\frac{eh \cdot t}{ew}\right)}} - \tanh \left(-\sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \left(\frac{eh \cdot t}{ew}\right)} - \tanh \left(-\color{blue}{\frac{eh \cdot t}{ew}}\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \left(\frac{eh \cdot t}{ew}\right)} - \tanh \left(-\frac{eh \cdot t}{\color{blue}{ew}}\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, \frac{-1}{6}, eh\right) \cdot t\right)\right| \]

       2. lower-*.f64 59.6%

          \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \left(\frac{eh \cdot t}{ew}\right)} - \tanh \left(-\frac{eh \cdot t}{ew}\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   12. Applied rewrites 59.6%

       \[\leadsto \left|\frac{\cos t \cdot ew}{\cosh \left(\frac{eh \cdot t}{ew}\right)} - \tanh \left(-\color{blue}{\frac{eh \cdot t}{ew}}\right) \cdot \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.16666666666666666, eh\right) \cdot t\right)\right| \]

   if 1.14999999999999994e55 < t < 2.35e269

   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. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 41.4%

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

   6. Applied rewrites 41.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 41.5%

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

   9. Applied rewrites 41.5%

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

   if 2.35e269 < t

   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 99.8%

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

   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. lower-cos.f64 61.9%

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

   6. Applied rewrites 61.9%

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

Alternative 9: 74.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|eh\right| \leq 1.05 \cdot 10^{+36}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\tanh \left(-\sinh^{-1} \left(\frac{t}{ew} \cdot \left|eh\right|\right)\right) \cdot \sin t\right) \cdot \left(-\left|eh\right|\right)\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= (fabs eh) 1.05e+36)
   (fabs (* ew (cos t)))
   (fabs
    (* (* (tanh (- (asinh (* (/ t ew) (fabs eh))))) (sin t)) (- (fabs eh))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (fabs(eh) <= 1.05e+36) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(((tanh(-asinh(((t / ew) * fabs(eh)))) * sin(t)) * -fabs(eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if math.fabs(eh) <= 1.05e+36:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(((math.tanh(-math.asinh(((t / ew) * math.fabs(eh)))) * math.sin(t)) * -math.fabs(eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (abs(eh) <= 1.05e+36)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(Float64(tanh(Float64(-asinh(Float64(Float64(t / ew) * abs(eh))))) * sin(t)) * Float64(-abs(eh))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (abs(eh) <= 1.05e+36)
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(((tanh(-asinh(((t / ew) * abs(eh)))) * sin(t)) * -abs(eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[N[Abs[eh], $MachinePrecision], 1.05e+36], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Tanh[(-N[ArcSinh[N[(N[(t / ew), $MachinePrecision] * N[Abs[eh], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * (-N[Abs[eh], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.05000000000000002e36

   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 99.8%

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

   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. lower-cos.f64 61.9%

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

   6. Applied rewrites 61.9%

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

   if 1.05000000000000002e36 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 41.4%

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

   6. Applied rewrites 41.3%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 41.5%

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

   9. Applied rewrites 41.5%

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

Alternative 10: 61.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

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. lower-cos.f64 61.9%

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

6. Applied rewrites 61.9%

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

Alternative 11: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left|ew\right|}\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{\left|ew\right|}\right)\\ \mathbf{if}\;\left(\left|ew\right| \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -2 \cdot 10^{-239}:\\ \;\;\;\;t\_1 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot \left|ew\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (fabs ew))) (t_2 (atan (/ (* (- eh) (tan t)) (fabs ew)))))
   (if (<=
        (- (* (* (fabs ew) (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2)))
        -2e-239)
     (* t_1 t_1)
     (* (cos t) (fabs ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = sqrt(fabs(ew));
	double t_2 = atan(((-eh * tan(t)) / fabs(ew)));
	double tmp;
	if ((((fabs(ew) * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-239) {
		tmp = t_1 * t_1;
	} else {
		tmp = cos(t) * fabs(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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = sqrt(abs(ew))
    t_2 = atan(((-eh * tan(t)) / abs(ew)))
    if ((((abs(ew) * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-2d-239)) then
        tmp = t_1 * t_1
    else
        tmp = cos(t) * abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sqrt(Math.abs(ew));
	double t_2 = Math.atan(((-eh * Math.tan(t)) / Math.abs(ew)));
	double tmp;
	if ((((Math.abs(ew) * Math.cos(t)) * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -2e-239) {
		tmp = t_1 * t_1;
	} else {
		tmp = Math.cos(t) * Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sqrt(math.fabs(ew))
	t_2 = math.atan(((-eh * math.tan(t)) / math.fabs(ew)))
	tmp = 0
	if (((math.fabs(ew) * math.cos(t)) * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -2e-239:
		tmp = t_1 * t_1
	else:
		tmp = math.cos(t) * math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sqrt(abs(ew))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / abs(ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(abs(ew) * cos(t)) * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -2e-239)
		tmp = Float64(t_1 * t_1);
	else
		tmp = Float64(cos(t) * abs(ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sqrt(abs(ew));
	t_2 = atan(((-eh * tan(t)) / abs(ew)));
	tmp = 0.0;
	if ((((abs(ew) * cos(t)) * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -2e-239)
		tmp = t_1 * t_1;
	else
		tmp = cos(t) * abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sqrt[N[Abs[ew], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[Abs[ew], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(N[Abs[ew], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2e-239], N[(t$95$1 * t$95$1), $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * N[Abs[ew], $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))))) < -2.0000000000000002e-239

   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 99.8%

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

   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. lower-cos.f64 61.9%

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

   6. Applied rewrites 61.9%

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

      1. lift-fabs.f64 N/A

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

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

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. sqrt-prod N/A

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

      4. lower-unsound-*.f64 N/A

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

   8. Applied rewrites 30.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-sqrt.f64 19.6%

          \[\leadsto \sqrt{ew} \cdot \sqrt{\cos t \cdot ew} \]

   11. Applied rewrites 19.6%

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

   12. Taylor expanded in t around 0

       \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]
   13. Step-by-step derivation

       1. lower-sqrt.f64 21.0%

          \[\leadsto \sqrt{ew} \cdot \sqrt{ew} \]

   14. Applied rewrites 21.0%

       \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]

   if -2.0000000000000002e-239 < (-.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 99.8%

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

   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. lower-cos.f64 61.9%

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

   6. Applied rewrites 61.9%

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

      1. lift-fabs.f64 N/A

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

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

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

      3. sqrt-prod N/A

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

      4. lower-unsound-*.f64 N/A

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

   8. Applied rewrites 30.7%

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

   10. Applied rewrites 31.7%

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

Alternative 12: 42.2% accurate, 25.0× speedup

Math

\[\begin{array}{l} t_1 := \sqrt{\left|ew\right|}\\ t\_1 \cdot t\_1 \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (fabs ew)))) (* t_1 t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = sqrt(fabs(ew));
	return t_1 * 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 = sqrt(abs(ew))
    code = t_1 * t_1
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sqrt(Math.abs(ew));
	return t_1 * t_1;
}

Python

def code(eh, ew, t):
	t_1 = math.sqrt(math.fabs(ew))
	return t_1 * t_1

Julia

function code(eh, ew, t)
	t_1 = sqrt(abs(ew))
	return Float64(t_1 * t_1)
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = sqrt(abs(ew));
	tmp = t_1 * t_1;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Sqrt[N[Abs[ew], $MachinePrecision]], $MachinePrecision]}, N[(t$95$1 * t$95$1), $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 99.8%

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

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. lower-cos.f64 61.9%

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

6. Applied rewrites 61.9%

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

   1. lift-fabs.f64 N/A

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

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

      \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \left(ew \cdot \cos t\right)}} \]

   3. sqrt-prod N/A

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

   4. lower-unsound-*.f64 N/A

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

8. Applied rewrites 30.7%

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

9. Taylor expanded in t around 0

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

    1. lower-sqrt.f64 19.6%

       \[\leadsto \sqrt{ew} \cdot \sqrt{\cos t \cdot ew} \]

11. Applied rewrites 19.6%

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

12. Taylor expanded in t around 0

    \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]
13. Step-by-step derivation

    1. lower-sqrt.f64 21.0%

       \[\leadsto \sqrt{ew} \cdot \sqrt{ew} \]

14. Applied rewrites 21.0%

    \[\leadsto \sqrt{ew} \cdot \color{blue}{\sqrt{ew}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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)))))))