Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.0s

Alternatives: 12

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew (tan t)))))
   (fabs
    (fma
     (* (sin t) ew)
     (/ 1.0 (sqrt (+ 1.0 (pow t_1 2.0))))
     (* (* (cos t) eh) (tanh (asinh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * tan(t));
	return fabs(fma((sin(t) * ew), (1.0 / sqrt((1.0 + pow(t_1, 2.0)))), ((cos(t) * eh) * tanh(asinh(t_1)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * tan(t)))
	return abs(fma(Float64(sin(t) * ew), Float64(1.0 / sqrt(Float64(1.0 + (t_1 ^ 2.0)))), Float64(Float64(cos(t) * eh) * tanh(asinh(t_1)))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 92.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \sin t \cdot ew\\ t_3 := \left|\mathsf{fma}\left(t\_2, -1 \cdot \left(\frac{ew}{eh} \cdot \tan t\right), t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+119}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{t\_2}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (* (sin t) ew))
        (t_3
         (fabs
          (fma
           t_2
           (* -1.0 (* (/ ew eh) (tan t)))
           (* t_1 (tanh (asinh (/ eh (* ew (tan t))))))))))
   (if (<= eh -6.8e+119)
     t_3
     (if (<= eh 9.8e+20)
       (fabs
        (* (fma 1.0 (sin t) (/ (* t_1 (tanh (asinh (/ t_1 t_2)))) ew)) ew))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = sin(t) * ew;
	double t_3 = fabs(fma(t_2, (-1.0 * ((ew / eh) * tan(t))), (t_1 * tanh(asinh((eh / (ew * tan(t))))))));
	double tmp;
	if (eh <= -6.8e+119) {
		tmp = t_3;
	} else if (eh <= 9.8e+20) {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((t_1 / t_2)))) / ew)) * ew));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(sin(t) * ew)
	t_3 = abs(fma(t_2, Float64(-1.0 * Float64(Float64(ew / eh) * tan(t))), Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * tan(t))))))))
	tmp = 0.0
	if (eh <= -6.8e+119)
		tmp = t_3;
	elseif (eh <= 9.8e+20)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(t_1 / t_2)))) / ew)) * ew));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$2 * N[(-1.0 * N[(N[(ew / eh), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6.8e+119], t$95$3, If[LessEqual[eh, 9.8e+20], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.80000000000000027e119 or 9.8e20 < eh

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. tan-quot N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-tan.f64 86.8

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

   6. Applied rewrites 86.8%

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

   if -6.80000000000000027e119 < eh < 9.8e20

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 96.8%

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

Alternative 3: 92.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+119}:\\ \;\;\;\;\left|t\_2\right|\\ \mathbf{elif}\;eh \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_2}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew)))))))
   (if (<= eh -6.8e+119)
     (fabs t_2)
     (if (<= eh 9.8e+20)
       (fabs (* (fma 1.0 (sin t) (/ t_2 ew)) ew))
       (fabs
        (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = t_1 * tanh(asinh((t_1 / (sin(t) * ew))));
	double tmp;
	if (eh <= -6.8e+119) {
		tmp = fabs(t_2);
	} else if (eh <= 9.8e+20) {
		tmp = fabs((fma(1.0, sin(t), (t_2 / ew)) * ew));
	} else {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew)))))
	tmp = 0.0
	if (eh <= -6.8e+119)
		tmp = abs(t_2);
	elseif (eh <= 9.8e+20)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(t_2 / ew)) * ew));
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -6.8e+119], N[Abs[t$95$2], $MachinePrecision], If[LessEqual[eh, 9.8e+20], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(t$95$2 / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.80000000000000027e119

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 89.5

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

   4. Applied rewrites 89.5%

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

   if -6.80000000000000027e119 < eh < 9.8e20

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 96.8%

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

   if 9.8e20 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 4: 89.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\right|\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (fabs (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew))))))))
   (if (<= eh -6.8e+119)
     t_2
     (if (<= eh 9.8e+20)
       (fabs
        (*
         (fma
          1.0
          (sin t)
          (/
           (*
            t_1
            (tanh
             (asinh
              (/
               (fma
                (* t t)
                (- (* -0.5 (/ eh ew)) (* -0.16666666666666666 (/ eh ew)))
                (/ eh ew))
               t))))
           ew))
         ew))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((t_1 * tanh(asinh((t_1 / (sin(t) * ew))))));
	double tmp;
	if (eh <= -6.8e+119) {
		tmp = t_2;
	} else if (eh <= 9.8e+20) {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((fma((t * t), ((-0.5 * (eh / ew)) - (-0.16666666666666666 * (eh / ew))), (eh / ew)) / t)))) / ew)) * ew));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew))))))
	tmp = 0.0
	if (eh <= -6.8e+119)
		tmp = t_2;
	elseif (eh <= 9.8e+20)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * Float64(eh / ew)) - Float64(-0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t)))) / ew)) * ew));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6.8e+119], t$95$2, If[LessEqual[eh, 9.8e+20], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.80000000000000027e119 or 9.8e20 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 86.3

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

   4. Applied rewrites 86.3%

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

   if -6.80000000000000027e119 < eh < 9.8e20

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}{ew}\right) \cdot ew\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}{ew}\right) \cdot ew\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      3. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      10. lower-/.f64 91.5

          \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

   10. Applied rewrites 91.5%

       \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;eh \leq -6.8 \cdot 10^{+119}:\\ \;\;\;\;\left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\right|\\ \mathbf{elif}\;eh \leq 9.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= eh -6.8e+119)
     (fabs (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew))))))
     (if (<= eh 9.8e+20)
       (fabs
        (*
         (fma
          1.0
          (sin t)
          (/
           (*
            t_1
            (tanh
             (asinh
              (/
               (fma
                (* t t)
                (- (* -0.5 (/ eh ew)) (* -0.16666666666666666 (/ eh ew)))
                (/ eh ew))
               t))))
           ew))
         ew))
       (fabs
        (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (eh <= -6.8e+119) {
		tmp = fabs((t_1 * tanh(asinh((t_1 / (sin(t) * ew))))));
	} else if (eh <= 9.8e+20) {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((fma((t * t), ((-0.5 * (eh / ew)) - (-0.16666666666666666 * (eh / ew))), (eh / ew)) / t)))) / ew)) * ew));
	} else {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (eh <= -6.8e+119)
		tmp = abs(Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew))))));
	elseif (eh <= 9.8e+20)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(fma(Float64(t * t), Float64(Float64(-0.5 * Float64(eh / ew)) - Float64(-0.16666666666666666 * Float64(eh / ew))), Float64(eh / ew)) / t)))) / ew)) * ew));
	else
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -6.8e+119], N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 9.8e+20], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(N[(t * t), $MachinePrecision] * N[(N[(-0.5 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] - N[(-0.16666666666666666 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.80000000000000027e119

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 89.5

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

   4. Applied rewrites 89.5%

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

   if -6.80000000000000027e119 < eh < 9.8e20

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 96.8%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}{ew}\right) \cdot ew\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right)}{ew}\right) \cdot ew\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left({t}^{2}, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      3. pow2 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      4. lift-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      5. lower--.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      6. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      7. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      9. lower-/.f64 N/A

         \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, \frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

      10. lower-/.f64 91.5

          \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

   10. Applied rewrites 91.5%

       \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(t \cdot t, -0.5 \cdot \frac{eh}{ew} - -0.16666666666666666 \cdot \frac{eh}{ew}, \frac{eh}{ew}\right)}{t}\right)}{ew}\right) \cdot ew\right| \]

   if 9.8e20 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

Alternative 6: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|t\_1 \cdot \tanh \sinh^{-1} \left(\frac{t\_1}{\sin t \cdot ew}\right)\right|\\ \mathbf{if}\;eh \leq -1.04 \cdot 10^{+39}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 3.25 \cdot 10^{+19}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (fabs (* t_1 (tanh (asinh (/ t_1 (* (sin t) ew))))))))
   (if (<= eh -1.04e+39)
     t_2
     (if (<= eh 3.25e+19)
       (fabs
        (* (fma 1.0 (sin t) (/ (* t_1 (tanh (asinh (/ eh (* ew t))))) ew)) ew))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((t_1 * tanh(asinh((t_1 / (sin(t) * ew))))));
	double tmp;
	if (eh <= -1.04e+39) {
		tmp = t_2;
	} else if (eh <= 3.25e+19) {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((eh / (ew * t))))) / ew)) * ew));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(t_1 * tanh(asinh(Float64(t_1 / Float64(sin(t) * ew))))))
	tmp = 0.0
	if (eh <= -1.04e+39)
		tmp = t_2;
	elseif (eh <= 3.25e+19)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t))))) / ew)) * ew));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 * N[Tanh[N[ArcSinh[N[(t$95$1 / N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.04e+39], t$95$2, If[LessEqual[eh, 3.25e+19], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.04e39 or 3.25e19 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. lift-sin.f64 84.6

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

   4. Applied rewrites 84.6%

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

   if -1.04e39 < eh < 3.25e19

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 90.3

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

   10. Applied rewrites 90.3%

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

Alternative 7: 82.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{if}\;t \leq -0.00078:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-23}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t \cdot ew, 1, \left(eh + -0.5 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right) \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right)\right|\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+219}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right)}{ew}\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2
         (fabs
          (*
           (fma 1.0 (sin t) (/ (* t_1 (tanh (asinh (/ eh (* ew t))))) ew))
           ew))))
   (if (<= t -0.00078)
     t_2
     (if (<= t 2e-23)
       (fabs
        (fma
         (* (sin t) ew)
         1.0
         (*
          (+ eh (* -0.5 (* eh (* t t))))
          (tanh (asinh (/ eh (* ew (tan t))))))))
       (if (<= t 2.7e+219)
         t_2
         (fabs
          (*
           (fma
            1.0
            (sin t)
            (/ (* t_1 (tanh (+ (log (* 2.0 (/ eh ew))) (* -1.0 (log t))))) ew))
           ew)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((eh / (ew * t))))) / ew)) * ew));
	double tmp;
	if (t <= -0.00078) {
		tmp = t_2;
	} else if (t <= 2e-23) {
		tmp = fabs(fma((sin(t) * ew), 1.0, ((eh + (-0.5 * (eh * (t * t)))) * tanh(asinh((eh / (ew * tan(t))))))));
	} else if (t <= 2.7e+219) {
		tmp = t_2;
	} else {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t))))) / ew)) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t))))) / ew)) * ew))
	tmp = 0.0
	if (t <= -0.00078)
		tmp = t_2;
	elseif (t <= 2e-23)
		tmp = abs(fma(Float64(sin(t) * ew), 1.0, Float64(Float64(eh + Float64(-0.5 * Float64(eh * Float64(t * t)))) * tanh(asinh(Float64(eh / Float64(ew * tan(t))))))));
	elseif (t <= 2.7e+219)
		tmp = t_2;
	else
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(Float64(log(Float64(2.0 * Float64(eh / ew))) + Float64(-1.0 * log(t))))) / ew)) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.00078], t$95$2, If[LessEqual[t, 2e-23], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * 1.0 + N[(N[(eh + N[(-0.5 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.7e+219], t$95$2, N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[(N[Log[N[(2.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.79999999999999986e-4 or 1.99999999999999992e-23 < t < 2.6999999999999999e219

   1. Initial program 99.6%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 87.7%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 86.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.4

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

   10. Applied rewrites 70.4%

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

   if -7.79999999999999986e-4 < t < 1.99999999999999992e-23

   1. Initial program 100.0%

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

   3. Applied rewrites 100.0%

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

   4. Taylor expanded in eh around -inf

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

      1. lower-*.f64 N/A

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

      2. times-frac N/A

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

      3. tan-quot N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-tan.f64 72.7

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

   6. Applied rewrites 72.7%

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

   7. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 72.7

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

   9. Applied rewrites 72.7%

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

   10. Taylor expanded in eh around 0

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

   12. Applied rewrites 98.7%

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

   if 2.6999999999999999e219 < t

   1. Initial program 99.7%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right)}{ew}\right) \cdot ew\right| \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 46.4

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

   10. Applied rewrites 46.4%

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

Alternative 8: 77.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;t \leq 2.7 \cdot 10^{+219}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(1, \sin t, \frac{t\_1 \cdot \tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right)}{ew}\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= t 2.7e+219)
     (fabs
      (* (fma 1.0 (sin t) (/ (* t_1 (tanh (asinh (/ eh (* ew t))))) ew)) ew))
     (fabs
      (*
       (fma
        1.0
        (sin t)
        (/ (* t_1 (tanh (+ (log (* 2.0 (/ eh ew))) (* -1.0 (log t))))) ew))
       ew)))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (t <= 2.7e+219) {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh(asinh((eh / (ew * t))))) / ew)) * ew));
	} else {
		tmp = fabs((fma(1.0, sin(t), ((t_1 * tanh((log((2.0 * (eh / ew))) + (-1.0 * log(t))))) / ew)) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (t <= 2.7e+219)
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t))))) / ew)) * ew));
	else
		tmp = abs(Float64(fma(1.0, sin(t), Float64(Float64(t_1 * tanh(Float64(log(Float64(2.0 * Float64(eh / ew))) + Float64(-1.0 * log(t))))) / ew)) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[t, 2.7e+219], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(t$95$1 * N[Tanh[N[(N[Log[N[(2.0 * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-1.0 * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 2.6999999999999999e219

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 86.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.1

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

   10. Applied rewrites 79.1%

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

   if 2.6999999999999999e219 < t

   1. Initial program 99.7%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 87.1%

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

   5. Taylor expanded in eh around 0

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

   7. Applied rewrites 86.5%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(1, \sin t, \frac{\left(\cos t \cdot eh\right) \cdot \tanh \left(\log \left(2 \cdot \frac{eh}{ew}\right) + -1 \cdot \log t\right)}{ew}\right) \cdot ew\right| \]
   9. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 46.4

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

   10. Applied rewrites 46.4%

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

Alternative 9: 76.6% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (fma 1.0 (sin t) (/ (* (* (cos t) eh) (tanh (asinh (/ eh (* ew t))))) ew))
   ew)))

C

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

Julia

function code(eh, ew, t)
	return abs(Float64(fma(1.0, sin(t), Float64(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(eh / Float64(ew * t))))) / ew)) * ew))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(1.0 * N[Sin[t], $MachinePrecision] + N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in ew around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 88.0%

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

5. Taylor expanded in eh around 0

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

7. Applied rewrites 86.6%

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

8. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 77.4

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

10. Applied rewrites 77.4%

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

Alternative 10: 61.3% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -110:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{-11}:\\ \;\;\;\;\left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -110.0)
     t_1
     (if (<= t 1.15e-11) (fabs (* (tanh (/ eh (* ew t))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -110.0) {
		tmp = t_1;
	} else if (t <= 1.15e-11) {
		tmp = fabs((tanh((eh / (ew * t))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (t <= (-110.0d0)) then
        tmp = t_1
    else if (t <= 1.15d-11) then
        tmp = abs((tanh((eh / (ew * t))) * eh))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (t <= -110.0) {
		tmp = t_1;
	} else if (t <= 1.15e-11) {
		tmp = Math.abs((Math.tanh((eh / (ew * t))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if t <= -110.0:
		tmp = t_1
	elif t <= 1.15e-11:
		tmp = math.fabs((math.tanh((eh / (ew * t))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -110.0)
		tmp = t_1;
	elseif (t <= 1.15e-11)
		tmp = abs(Float64(tanh(Float64(eh / Float64(ew * t))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (t <= -110.0)
		tmp = t_1;
	elseif (t <= 1.15e-11)
		tmp = abs((tanh((eh / (ew * t))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -110.0], t$95$1, If[LessEqual[t, 1.15e-11], N[Abs[N[(N[Tanh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -110 or 1.15000000000000007e-11 < t

   1. Initial program 99.6%

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

   2. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 87.5%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\sin t \cdot ew\right| \]
   6. Step-by-step derivation

      1. lift-sin.f64 51.4

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

   7. Applied rewrites 51.4%

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

   if -110 < t < 1.15000000000000007e-11

   1. Initial program 100.0%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      3. lift-cos.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      4. lift-sin.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      5. lift-*.f64 71.4

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

   7. Applied rewrites 71.4%

      \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]

      2. lower-*.f64 71.4

         \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]

   10. Applied rewrites 71.4%

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

Alternative 11: 41.6% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 1.65 \cdot 10^{+217}:\\ \;\;\;\;\left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 1.65e+217) (fabs (* (tanh (/ eh (* ew t))) eh)) (fabs (* ew t))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.65e+217) {
		tmp = fabs((tanh((eh / (ew * t))) * eh));
	} else {
		tmp = fabs((ew * t));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= 1.65d+217) then
        tmp = abs((tanh((eh / (ew * t))) * eh))
    else
        tmp = abs((ew * t))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 1.65e+217) {
		tmp = Math.abs((Math.tanh((eh / (ew * t))) * eh));
	} else {
		tmp = Math.abs((ew * t));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 1.65e+217:
		tmp = math.fabs((math.tanh((eh / (ew * t))) * eh))
	else:
		tmp = math.fabs((ew * t))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 1.65e+217)
		tmp = abs(Float64(tanh(Float64(eh / Float64(ew * t))) * eh));
	else
		tmp = abs(Float64(ew * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 1.65e+217)
		tmp = abs((tanh((eh / (ew * t))) * eh));
	else
		tmp = abs((ew * t));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 1.65e+217], N[Abs[N[(N[Tanh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * t), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.65e217

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 43.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      3. lift-cos.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      4. lift-sin.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

      5. lift-*.f64 43.6

         \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

   7. Applied rewrites 43.6%

      \[\leadsto \left|\tanh \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right) \cdot eh\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]

      2. lower-*.f64 41.6

         \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]

   10. Applied rewrites 41.6%

       \[\leadsto \left|\tanh \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]

   if 1.65e217 < ew

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 82.5%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 40.4%

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

   8. Taylor expanded in eh around 0

      \[\leadsto \left|\left(1 \cdot ew\right) \cdot t\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 40.7%

       \[\leadsto \left|\left(1 \cdot ew\right) \cdot t\right| \]

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 40.7%

       \[\leadsto \left|ew \cdot t\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 19.0% accurate, 47.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 41.2%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 18.2%

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

8. Taylor expanded in eh around 0

   \[\leadsto \left|\left(1 \cdot ew\right) \cdot t\right| \]
9. Step-by-step derivation

10. Applied rewrites 19.0%

    \[\leadsto \left|\left(1 \cdot ew\right) \cdot t\right| \]

11. Taylor expanded in eh around 0

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

13. Applied rewrites 19.0%

    \[\leadsto \left|ew \cdot t\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025110 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))