Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.1s

Alternatives: 14

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

Alternative 2: 94.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\\ t_2 := -\frac{eh}{ew} \cdot \tan t\\ \mathbf{if}\;eh \leq 7.2 \cdot 10^{+45}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} t\_2 \cdot \sin t}{ew}, -\frac{1}{\sqrt{1 + {t\_2}^{2}}} \cdot \cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) t) ew))) (t_2 (- (* (/ eh ew) (tan t)))))
   (if (<= eh 7.2e+45)
     (fabs
      (-
       (*
        (fma
         eh
         (/ (* (tanh (asinh t_2)) (sin t)) ew)
         (- (* (/ 1.0 (sqrt (+ 1.0 (pow t_2 2.0)))) (cos t))))
        ew)))
     (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * t) / ew));
	double t_2 = -((eh / ew) * tan(t));
	double tmp;
	if (eh <= 7.2e+45) {
		tmp = fabs(-(fma(eh, ((tanh(asinh(t_2)) * sin(t)) / ew), -((1.0 / sqrt((1.0 + pow(t_2, 2.0)))) * cos(t))) * ew));
	} else {
		tmp = fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * t) / ew))
	t_2 = Float64(-Float64(Float64(eh / ew) * tan(t)))
	tmp = 0.0
	if (eh <= 7.2e+45)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(t_2)) * sin(t)) / ew), Float64(-Float64(Float64(1.0 / sqrt(Float64(1.0 + (t_2 ^ 2.0)))) * cos(t)))) * ew)));
	else
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = (-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[eh, 7.2e+45], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[(N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 7.2e45

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 94.6%

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

   if 7.2e45 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 92.3%

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

Alternative 3: 93.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\\ \mathbf{if}\;eh \leq 2.9 \cdot 10^{+45}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot \sin t}{ew}, -\cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) t) ew))))
   (if (<= eh 2.9e+45)
     (fabs
      (-
       (*
        (fma
         eh
         (/ (* (tanh (asinh (- (* (/ eh ew) t)))) (sin t)) ew)
         (- (cos t)))
        ew)))
     (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * t) / ew));
	double tmp;
	if (eh <= 2.9e+45) {
		tmp = fabs(-(fma(eh, ((tanh(asinh(-((eh / ew) * t))) * sin(t)) / ew), -cos(t)) * ew));
	} else {
		tmp = fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * t) / ew))
	tmp = 0.0
	if (eh <= 2.9e+45)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * sin(t)) / ew), Float64(-cos(t))) * ew)));
	else
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, 2.9e+45], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 2.8999999999999997e45

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 84.1%

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

   11. Taylor expanded in eh around 0

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

       1. lift-cos.f64 93.2

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

   13. Applied rewrites 93.2%

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

   if 2.8999999999999997e45 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

   4. Applied rewrites 92.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 92.3%

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

Alternative 4: 91.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 2.5 \cdot 10^{+184}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot \sin t}{ew}, -\cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 2.5e+184)
   (fabs
    (-
     (*
      (fma
       eh
       (/ (* (tanh (asinh (- (* (/ eh ew) t)))) (sin t)) ew)
       (- (cos t)))
      ew)))
   (fabs (* (- eh) (* (tanh (asinh (- (* (/ eh ew) (tan t))))) (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 2.5e+184) {
		tmp = fabs(-(fma(eh, ((tanh(asinh(-((eh / ew) * t))) * sin(t)) / ew), -cos(t)) * ew));
	} else {
		tmp = fabs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 2.5e+184)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * sin(t)) / ew), Float64(-cos(t))) * ew)));
	else
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * tan(t))))) * sin(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 2.5e+184], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 2.4999999999999999e184

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 83.4%

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

   11. Taylor expanded in eh around 0

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

       1. lift-cos.f64 92.6

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

   13. Applied rewrites 92.6%

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

   if 2.4999999999999999e184 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

Alternative 5: 91.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 2.5 \cdot 10^{+184}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t}{ew}, -\cos t\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-eh\right) \cdot \left(\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot \tan t\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 2.5e+184)
   (fabs
    (-
     (*
      (fma eh (/ (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)) ew) (- (cos t)))
      ew)))
   (fabs (* (- eh) (* (tanh (asinh (- (* (/ eh ew) (tan t))))) (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 2.5e+184) {
		tmp = fabs(-(fma(eh, ((tanh((-1.0 * ((eh * t) / ew))) * sin(t)) / ew), -cos(t)) * ew));
	} else {
		tmp = fabs((-eh * (tanh(asinh(-((eh / ew) * tan(t)))) * sin(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 2.5e+184)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t)) / ew), Float64(-cos(t))) * ew)));
	else
		tmp = abs(Float64(Float64(-eh) * Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * tan(t))))) * sin(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 2.5e+184], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[((-eh) * N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 2.4999999999999999e184

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 93.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 83.4%

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

   11. Taylor expanded in eh around 0

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

       1. lift-cos.f64 92.6

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

   13. Applied rewrites 92.6%

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

   14. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 92.6

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

   16. Applied rewrites 92.6%

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

   if 2.4999999999999999e184 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

   4. Applied rewrites 77.9%

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

Alternative 6: 90.6% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in ew around -inf

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

4. Applied rewrites 92.1%

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

5. Taylor expanded in t around 0

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

7. Applied rewrites 91.3%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 82.1%

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

11. Taylor expanded in eh around 0

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

    1. lift-cos.f64 90.6

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

13. Applied rewrites 90.6%

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

14. Taylor expanded in t around 0

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

    1. lower-*.f64 N/A

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

    2. lower-/.f64 N/A

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

    3. lower-*.f64 90.6

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

16. Applied rewrites 90.6%

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

Alternative 7: 70.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-\left(eh \cdot \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot \sin t}{ew} + -1\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.5e-87)
   (fabs (* ew (cos t)))
   (fabs
    (-
     (*
      (+ (* eh (/ (* (tanh (asinh (- (* (/ eh ew) t)))) (sin t)) ew)) -1.0)
      ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.5e-87) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(-(((eh * ((tanh(asinh(-((eh / ew) * t))) * sin(t)) / ew)) + -1.0) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= 5.5e-87:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs(-(((eh * ((math.tanh(math.asinh(-((eh / ew) * t))) * math.sin(t)) / ew)) + -1.0) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.5e-87)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(-Float64(Float64(Float64(eh * Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * sin(t)) / ew)) + -1.0) * ew)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= 5.5e-87)
		tmp = abs((ew * cos(t)));
	else
		tmp = abs(-(((eh * ((tanh(asinh(-((eh / ew) * t))) * sin(t)) / ew)) + -1.0) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.5e-87], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[(-N[(N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.5000000000000004e-87

   1. Initial program 99.8%

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

   3. Applied rewrites 36.6%

      \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 68.4

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

   6. Applied rewrites 68.4%

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

   if 5.5000000000000004e-87 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 79.1%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 74.3%

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

       1. lift-fma.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 74.3

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

   15. Applied rewrites 74.3%

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

Alternative 8: 70.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot \sin t}{ew}, -1\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.5e-87)
   (fabs (* ew (cos t)))
   (fabs
    (-
     (*
      (fma eh (/ (* (tanh (asinh (- (* (/ eh ew) t)))) (sin t)) ew) -1.0)
      ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.5e-87) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(-(fma(eh, ((tanh(asinh(-((eh / ew) * t))) * sin(t)) / ew), -1.0) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.5e-87)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * sin(t)) / ew), -1.0) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.5e-87], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + -1.0), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.5000000000000004e-87

   1. Initial program 99.8%

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

   3. Applied rewrites 36.6%

      \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 68.4

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

   6. Applied rewrites 68.4%

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

   if 5.5000000000000004e-87 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 79.1%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 74.3%

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

Alternative 9: 70.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \left(-1 \cdot \frac{eh \cdot t}{ew}\right) \cdot \sin t}{ew}, -1\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh 5.5e-87)
   (fabs (* ew (cos t)))
   (fabs
    (-
     (*
      (fma eh (/ (* (tanh (* -1.0 (/ (* eh t) ew))) (sin t)) ew) -1.0)
      ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= 5.5e-87) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs(-(fma(eh, ((tanh((-1.0 * ((eh * t) / ew))) * sin(t)) / ew), -1.0) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= 5.5e-87)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(Float64(-1.0 * Float64(Float64(eh * t) / ew))) * sin(t)) / ew), -1.0) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, 5.5e-87], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[(-1.0 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + -1.0), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < 5.5000000000000004e-87

   1. Initial program 99.8%

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

   3. Applied rewrites 36.6%

      \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 68.4

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

   6. Applied rewrites 68.4%

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

   if 5.5000000000000004e-87 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 87.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 79.1%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 74.3%

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

   14. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 74.3

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

   16. Applied rewrites 74.3%

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

Alternative 10: 62.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot \left(t \cdot t\right)\right)\right)}{ew}, -1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 4.1e-7)
   (fabs
    (-
     (*
      (fma
       eh
       (/
        (*
         (tanh (asinh (- (* (/ eh ew) t))))
         (* t (+ 1.0 (* -0.16666666666666666 (* t t)))))
        ew)
       -1.0)
      ew)))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 4.1e-7) {
		tmp = fabs(-(fma(eh, ((tanh(asinh(-((eh / ew) * t))) * (t * (1.0 + (-0.16666666666666666 * (t * t))))) / ew), -1.0) * ew));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 4.1e-7)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * Float64(t * Float64(1.0 + Float64(-0.16666666666666666 * Float64(t * t))))) / ew), -1.0) * ew)));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 4.1e-7], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * N[(t * N[(1.0 + N[(-0.16666666666666666 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + -1.0), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0999999999999999e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.7%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 79.1%

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

   14. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 65.1

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

   16. Applied rewrites 65.1%

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

   if 4.0999999999999999e-7 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 34.6%

      \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 11: 61.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{-7}:\\ \;\;\;\;\left|-\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(-\frac{eh}{ew} \cdot t\right) \cdot t}{ew}, -1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t 4.1e-7)
   (fabs
    (- (* (fma eh (/ (* (tanh (asinh (- (* (/ eh ew) t)))) t) ew) -1.0) ew)))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= 4.1e-7) {
		tmp = fabs(-(fma(eh, ((tanh(asinh(-((eh / ew) * t))) * t) / ew), -1.0) * ew));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= 4.1e-7)
		tmp = abs(Float64(-Float64(fma(eh, Float64(Float64(tanh(asinh(Float64(-Float64(Float64(eh / ew) * t)))) * t) / ew), -1.0) * ew)));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, 4.1e-7], N[Abs[(-N[(N[(eh * N[(N[(N[Tanh[N[ArcSinh[(-N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision])], $MachinePrecision]], $MachinePrecision] * t), $MachinePrecision] / ew), $MachinePrecision] + -1.0), $MachinePrecision] * ew), $MachinePrecision])], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0999999999999999e-7

   1. Initial program 99.9%

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

   2. Taylor expanded in ew around -inf

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.7%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 79.1%

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

   14. Taylor expanded in t around 0

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

   16. Applied rewrites 66.3%

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

   if 4.0999999999999999e-7 < t

   1. Initial program 99.6%

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

   3. Applied rewrites 34.6%

      \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 50.9

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

   6. Applied rewrites 50.9%

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

Alternative 12: 61.4% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 37.1%

   \[\leadsto \left|\color{blue}{\frac{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{3} - {\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{3}}{{\left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right)}^{2} + \left({\left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)}^{2} + \left(\left(\cos t \cdot ew\right) \cdot \frac{1}{\sqrt{1 + {\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}^{2}}}\right) \cdot \left(\left(\sin t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)}}\right| \]

4. Taylor expanded in eh around 0

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 61.8

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

6. Applied rewrites 61.8%

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

Alternative 13: 33.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \mathbf{if}\;t\_1 \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -4 \cdot 10^{-243}:\\ \;\;\;\;\sqrt{ew} \cdot \sqrt{ew}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (atan (/ (* (- eh) (tan t)) ew))))
   (if (<= (- (* t_1 (cos t_2)) (* (* eh (sin t)) (sin t_2))) -4e-243)
     (* (sqrt ew) (sqrt ew))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double tmp;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -4e-243) {
		tmp = sqrt(ew) * sqrt(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = atan(((-eh * tan(t)) / ew))
    if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= (-4d-243)) then
        tmp = sqrt(ew) * sqrt(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.atan(((-eh * Math.tan(t)) / ew));
	double tmp;
	if (((t_1 * Math.cos(t_2)) - ((eh * Math.sin(t)) * Math.sin(t_2))) <= -4e-243) {
		tmp = Math.sqrt(ew) * Math.sqrt(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	tmp = 0
	if ((t_1 * math.cos(t_2)) - ((eh * math.sin(t)) * math.sin(t_2))) <= -4e-243:
		tmp = math.sqrt(ew) * math.sqrt(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	tmp = 0.0
	if (Float64(Float64(t_1 * cos(t_2)) - Float64(Float64(eh * sin(t)) * sin(t_2))) <= -4e-243)
		tmp = Float64(sqrt(ew) * sqrt(ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = atan(((-eh * tan(t)) / ew));
	tmp = 0.0;
	if (((t_1 * cos(t_2)) - ((eh * sin(t)) * sin(t_2))) <= -4e-243)
		tmp = sqrt(ew) * sqrt(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(t$95$1 * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -4e-243], N[(N[Sqrt[ew], $MachinePrecision] * N[Sqrt[ew], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999998e-243

   1. Initial program 99.8%

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

   3. Applied rewrites 0.0%

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

   4. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 0.0

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

   6. Applied rewrites 0.0%

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

   7. Taylor expanded in t around 0

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

      1. lower-sqrt.f64 3.5

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

   9. Applied rewrites 3.5%

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

   if -3.99999999999999998e-243 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Applied rewrites 96.5%

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

   4. Taylor expanded in eh around 0

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 60.8

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

   6. Applied rewrites 60.8%

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

Alternative 14: 22.8% accurate, 246.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 49.7%

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

4. Taylor expanded in t around 0

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

6. Applied rewrites 22.8%

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

Reproduce

herbie shell --seed 2025120 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))