Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.0s

Alternatives: 14

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-flip-rev N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-tan.f64 N/A

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

   11. mult-flip N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-flip-rev N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-tan.f64 N/A

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

   11. mult-flip N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. tanh-asinh-rev N/A

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

   8. lift-asinh.f64 N/A

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

   9. lower-tanh.f64 99.8

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

10. Applied rewrites 99.8%

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

    1. lift-cos.f64 N/A

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

    2. lift-atan.f64 N/A

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

    3. cos-atan N/A

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

    4. +-commutative N/A

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

    5. cosh-asinh N/A

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

    6. lift-/.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. lift-tan.f64 N/A

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

    9. lower-/.f64 N/A

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

    10. lift-tan.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. lift-/.f64 N/A

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

    13. lift-asinh.f64 N/A

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

    14. lower-cosh.f64 99.8

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

12. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. fp-cancel-sign-sub-inv N/A

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

   7. fp-cancel-sub-sign-inv N/A

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

3. Applied rewrites 99.8%

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

Alternative 3: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{ew}}{t}\right)\\ t_2 := \frac{eh}{t \cdot ew}\\ \mathbf{if}\;eh \leq 7.8 \cdot 10^{-90}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \tanh \sinh^{-1} t\_2 \cdot \cos t, ew \cdot \left(\cos \tan^{-1} t\_2 \cdot \sin t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot eh, \tanh t\_1, \frac{\sin t \cdot ew}{\cosh t\_1}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (asinh (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) ew) t)))
        (t_2 (/ eh (* t ew))))
   (if (<= eh 7.8e-90)
     (fabs
      (fma
       eh
       (* (tanh (asinh t_2)) (cos t))
       (* ew (* (cos (atan t_2)) (sin t)))))
     (fabs (fma (* (cos t) eh) (tanh t_1) (/ (* (sin t) ew) (cosh t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = asinh(((fma(((t * t) * eh), -0.3333333333333333, eh) / ew) / t));
	double t_2 = eh / (t * ew);
	double tmp;
	if (eh <= 7.8e-90) {
		tmp = fabs(fma(eh, (tanh(asinh(t_2)) * cos(t)), (ew * (cos(atan(t_2)) * sin(t)))));
	} else {
		tmp = fabs(fma((cos(t) * eh), tanh(t_1), ((sin(t) * ew) / cosh(t_1))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = asinh(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / ew) / t))
	t_2 = Float64(eh / Float64(t * ew))
	tmp = 0.0
	if (eh <= 7.8e-90)
		tmp = abs(fma(eh, Float64(tanh(asinh(t_2)) * cos(t)), Float64(ew * Float64(cos(atan(t_2)) * sin(t)))));
	else
		tmp = abs(fma(Float64(cos(t) * eh), tanh(t_1), Float64(Float64(sin(t) * ew) / cosh(t_1))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcSinh[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 7.8e-90], N[Abs[N[(eh * N[(N[Tanh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[(N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[t$95$1], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Cosh[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 7.80000000000000009e-90

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around 0

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

      1. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip N/A

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

      12. associate-/r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. times-frac N/A

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

      5. associate-*l/ N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip N/A

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

      12. associate-/r* N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.8

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lift-asinh.f64 N/A

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

      9. lower-tanh.f64 99.8

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

   10. Applied rewrites 99.8%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 89.9%

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

   14. Taylor expanded in t around 0

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

   16. Applied rewrites 89.9%

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

   if 7.80000000000000009e-90 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

      5. lower-pow.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

      6. lower-/.f64 37.2

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

   6. Applied rewrites 37.2%

      \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right| \]

   7. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}}\right| \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{\color{blue}{t}}\right)}\right| \]

      2. lower-fma.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

      3. lower-/.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

      5. lower-pow.f64 N/A

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

      6. lower-/.f64 40.7

         \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

   9. Applied rewrites 40.7%

      \[\leadsto \left|\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}}\right| \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(\cos t \cdot eh, \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}}\right| \]

       2. lift-fma.f64 N/A

          \[\leadsto \left|\frac{\color{blue}{\left(\cos t \cdot eh\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t} + \sin t \cdot ew}}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]

       3. div-add N/A

          \[\leadsto \left|\color{blue}{\frac{\left(\cos t \cdot eh\right) \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)} + \frac{\sin t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \frac{eh \cdot {t}^{2}}{ew}, \frac{eh}{ew}\right)}{t}\right)}}\right| \]

   11. Applied rewrites 84.5%

       \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t \cdot eh, \tanh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{ew}}{t}\right), \frac{\sin t \cdot ew}{\cosh \sinh^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{ew}}{t}\right)}\right)}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in eh around 0

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

   1. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-flip-rev N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-tan.f64 N/A

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

   11. mult-flip N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*l/ N/A

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

   6. div-flip-rev N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. tan-quot N/A

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

   10. lift-tan.f64 N/A

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

   11. mult-flip N/A

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

   12. associate-/r* N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. tanh-asinh-rev N/A

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

   8. lift-asinh.f64 N/A

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

   9. lower-tanh.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Taylor expanded in t around 0

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

13. Applied rewrites 89.9%

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

14. Taylor expanded in t around 0

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

16. Applied rewrites 89.9%

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

Alternative 5: 68.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \sin t \cdot ew\\ t_3 := \frac{eh}{t \cdot ew}\\ t_4 := \sqrt{\mathsf{fma}\left(t\_3, t\_3, 1\right)}\\ \mathbf{if}\;t \leq -3.2 \cdot 10^{-73}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(eh, \frac{1}{ew \cdot t} \cdot t\_1, t\_2\right)}{\cosh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}\right|\\ \mathbf{elif}\;t \leq 3 \cdot 10^{-137}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \left(t \cdot \left(1 + -0.16666666666666666 \cdot {t}^{2}\right)\right)}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_3, \frac{t\_1}{t\_4}, \frac{t\_2}{t\_4}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2 (* (sin t) ew))
        (t_3 (/ eh (* t ew)))
        (t_4 (sqrt (fma t_3 t_3 1.0))))
   (if (<= t -3.2e-73)
     (fabs
      (/ (fma eh (* (/ 1.0 (* ew t)) t_1) t_2) (cosh (asinh (/ eh (* ew t))))))
     (if (<= t 3e-137)
       (fabs
        (*
         eh
         (sin
          (atan
           (/
            (* eh (cos t))
            (* ew (* t (+ 1.0 (* -0.16666666666666666 (pow t 2.0))))))))))
       (fabs (fma t_3 (/ t_1 t_4) (/ t_2 t_4)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = sin(t) * ew;
	double t_3 = eh / (t * ew);
	double t_4 = sqrt(fma(t_3, t_3, 1.0));
	double tmp;
	if (t <= -3.2e-73) {
		tmp = fabs((fma(eh, ((1.0 / (ew * t)) * t_1), t_2) / cosh(asinh((eh / (ew * t))))));
	} else if (t <= 3e-137) {
		tmp = fabs((eh * sin(atan(((eh * cos(t)) / (ew * (t * (1.0 + (-0.16666666666666666 * pow(t, 2.0))))))))));
	} else {
		tmp = fabs(fma(t_3, (t_1 / t_4), (t_2 / t_4)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = Float64(sin(t) * ew)
	t_3 = Float64(eh / Float64(t * ew))
	t_4 = sqrt(fma(t_3, t_3, 1.0))
	tmp = 0.0
	if (t <= -3.2e-73)
		tmp = abs(Float64(fma(eh, Float64(Float64(1.0 / Float64(ew * t)) * t_1), t_2) / cosh(asinh(Float64(eh / Float64(ew * t))))));
	elseif (t <= 3e-137)
		tmp = abs(Float64(eh * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * Float64(t * Float64(1.0 + Float64(-0.16666666666666666 * (t ^ 2.0))))))))));
	else
		tmp = abs(fma(t_3, Float64(t_1 / t_4), Float64(t_2 / t_4)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$3 * t$95$3 + 1.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.2e-73], N[Abs[N[(N[(eh * N[(N[(1.0 / N[(ew * t), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 3e-137], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[(t * N[(1.0 + N[(-0.16666666666666666 * N[Power[t, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$3 * N[(t$95$1 / t$95$4), $MachinePrecision] + N[(t$95$2 / t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3.19999999999999986e-73

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 61.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 61.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 57.5

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

   11. Applied rewrites 57.5%

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

   if -3.19999999999999986e-73 < t < 2.9999999999999998e-137

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 37.5

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

   7. Applied rewrites 37.5%

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

   if 2.9999999999999998e-137 < t

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 61.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 61.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 57.5

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

   11. Applied rewrites 57.5%

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

       1. lift-/.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. div-add N/A

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

   13. Applied rewrites 57.2%

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

Alternative 6: 68.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin \tan^{-1} \left(\frac{eh \cdot \sin \left(\mathsf{fma}\left(\pi, 0.5, t\right)\right)}{ew \cdot \sin t}\right)\right|\\ \mathbf{if}\;eh \leq -2.7 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 270000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{t \cdot ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* eh (sin (atan (/ (* eh (sin (fma PI 0.5 t))) (* ew (sin t)))))))))
   (if (<= eh -2.7e+88)
     t_1
     (if (<= eh 270000.0)
       (fabs
        (/
         (fma (/ (* (cos t) eh) (* t ew)) eh (* (sin t) ew))
         (cosh (asinh (/ eh (* ew t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(atan(((eh * sin(fma(((double) M_PI), 0.5, t))) / (ew * sin(t)))))));
	double tmp;
	if (eh <= -2.7e+88) {
		tmp = t_1;
	} else if (eh <= 270000.0) {
		tmp = fabs((fma(((cos(t) * eh) / (t * ew)), eh, (sin(t) * ew)) / cosh(asinh((eh / (ew * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(atan(Float64(Float64(eh * sin(fma(pi, 0.5, t))) / Float64(ew * sin(t)))))))
	tmp = 0.0
	if (eh <= -2.7e+88)
		tmp = t_1;
	elseif (eh <= 270000.0)
		tmp = abs(Float64(fma(Float64(Float64(cos(t) * eh) / Float64(t * ew)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(eh / Float64(ew * t))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[N[ArcTan[N[(N[(eh * N[Sin[N[(Pi * 0.5 + t), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.7e+88], t$95$1, If[LessEqual[eh, 270000.0], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / N[(t * ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.70000000000000016e88 or 2.7e5 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-PI.f64 N/A

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

      10. metadata-eval 41.5

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

   6. Applied rewrites 41.5%

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

   if -2.70000000000000016e88 < eh < 2.7e5

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 61.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 61.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 57.5

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

   11. Applied rewrites 57.5%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 57.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. mult-flip-rev N/A

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

       8. lower-/.f64 57.9

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 7: 68.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right|\\ \mathbf{if}\;eh \leq -2.7 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 270000:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\cos t \cdot eh}{t \cdot ew}, eh, \sin t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (tanh (asinh (/ eh (* (tan t) ew)))) eh))))
   (if (<= eh -2.7e+88)
     t_1
     (if (<= eh 270000.0)
       (fabs
        (/
         (fma (/ (* (cos t) eh) (* t ew)) eh (* (sin t) ew))
         (cosh (asinh (/ eh (* ew t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((tanh(asinh((eh / (tan(t) * ew)))) * eh));
	double tmp;
	if (eh <= -2.7e+88) {
		tmp = t_1;
	} else if (eh <= 270000.0) {
		tmp = fabs((fma(((cos(t) * eh) / (t * ew)), eh, (sin(t) * ew)) / cosh(asinh((eh / (ew * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(tanh(asinh(Float64(eh / Float64(tan(t) * ew)))) * eh))
	tmp = 0.0
	if (eh <= -2.7e+88)
		tmp = t_1;
	elseif (eh <= 270000.0)
		tmp = abs(Float64(fma(Float64(Float64(cos(t) * eh) / Float64(t * ew)), eh, Float64(sin(t) * ew)) / cosh(asinh(Float64(eh / Float64(ew * t))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.7e+88], t$95$1, If[LessEqual[eh, 270000.0], N[Abs[N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / N[(t * ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.70000000000000016e88 or 2.7e5 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 41.5%

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

   if -2.70000000000000016e88 < eh < 2.7e5

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 61.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 61.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 57.5

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

   11. Applied rewrites 57.5%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 57.5

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-/.f64 N/A

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

       7. mult-flip-rev N/A

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

       8. lower-/.f64 57.9

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

       9. lift-*.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 57.9

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

   13. Applied rewrites 57.9%

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

Alternative 8: 66.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right|\\ t_2 := \frac{eh}{t \cdot ew}\\ \mathbf{if}\;eh \leq -2.6 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 13500:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\frac{\cos t \cdot eh}{t \cdot ew}, eh, \sin t \cdot ew\right)\right|}{\sqrt{\mathsf{fma}\left(t\_2, t\_2, 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (tanh (asinh (/ eh (* (tan t) ew)))) eh)))
        (t_2 (/ eh (* t ew))))
   (if (<= eh -2.6e+88)
     t_1
     (if (<= eh 13500.0)
       (/
        (fabs (fma (/ (* (cos t) eh) (* t ew)) eh (* (sin t) ew)))
        (sqrt (fma t_2 t_2 1.0)))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((tanh(asinh((eh / (tan(t) * ew)))) * eh));
	double t_2 = eh / (t * ew);
	double tmp;
	if (eh <= -2.6e+88) {
		tmp = t_1;
	} else if (eh <= 13500.0) {
		tmp = fabs(fma(((cos(t) * eh) / (t * ew)), eh, (sin(t) * ew))) / sqrt(fma(t_2, t_2, 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(tanh(asinh(Float64(eh / Float64(tan(t) * ew)))) * eh))
	t_2 = Float64(eh / Float64(t * ew))
	tmp = 0.0
	if (eh <= -2.6e+88)
		tmp = t_1;
	elseif (eh <= 13500.0)
		tmp = Float64(abs(fma(Float64(Float64(cos(t) * eh) / Float64(t * ew)), eh, Float64(sin(t) * ew))) / sqrt(fma(t_2, t_2, 1.0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.6e+88], t$95$1, If[LessEqual[eh, 13500.0], N[(N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] / N[(t * ew), $MachinePrecision]), $MachinePrecision] * eh + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$2 * t$95$2 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.6000000000000001e88 or 13500 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 41.5%

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

   if -2.6000000000000001e88 < eh < 13500

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 61.8

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 61.8

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lift-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in t around 0

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

       1. lower-*.f64 57.5

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

   11. Applied rewrites 57.5%

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

       1. lift-fabs.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. div-fabs N/A

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

       4. lift-cosh.f64 N/A

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

       5. lift-asinh.f64 N/A

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

       6. cosh-asinh N/A

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

       7. sqrt-fabs-rev N/A

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

       8. cosh-asinh N/A

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

   13. Applied rewrites 54.9%

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

Alternative 9: 58.1% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.1 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.1e-58)
     t_1
     (if (<= ew 1.4e+58)
       (fabs (* (tanh (asinh (/ eh (* (tan t) ew)))) eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.1e-58) {
		tmp = t_1;
	} else if (ew <= 1.4e+58) {
		tmp = fabs((tanh(asinh((eh / (tan(t) * ew)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -2.1e-58:
		tmp = t_1
	elif ew <= 1.4e+58:
		tmp = math.fabs((math.tanh(math.asinh((eh / (math.tan(t) * ew)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.1e-58)
		tmp = t_1;
	elseif (ew <= 1.4e+58)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(tan(t) * ew)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -2.1e-58)
		tmp = t_1;
	elseif (ew <= 1.4e+58)
		tmp = abs((tanh(asinh((eh / (tan(t) * ew)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.1e-58], t$95$1, If[LessEqual[ew, 1.4e+58], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.09999999999999988e-58 or 1.3999999999999999e58 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.9

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

   6. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.9

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

   8. Applied rewrites 41.9%

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

   if -2.09999999999999988e-58 < ew < 1.3999999999999999e58

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 41.5%

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

Alternative 10: 57.1% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.1 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;\left|eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.1e-58)
     t_1
     (if (<= ew 1.4e+58) (fabs (* eh (sin (atan (/ eh (* ew t)))))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.1e-58) {
		tmp = t_1;
	} else if (ew <= 1.4e+58) {
		tmp = fabs((eh * sin(atan((eh / (ew * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (ew <= (-2.1d-58)) then
        tmp = t_1
    else if (ew <= 1.4d+58) then
        tmp = abs((eh * sin(atan((eh / (ew * t))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (ew <= -2.1e-58) {
		tmp = t_1;
	} else if (ew <= 1.4e+58) {
		tmp = Math.abs((eh * Math.sin(Math.atan((eh / (ew * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -2.1e-58:
		tmp = t_1
	elif ew <= 1.4e+58:
		tmp = math.fabs((eh * math.sin(math.atan((eh / (ew * t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.1e-58)
		tmp = t_1;
	elseif (ew <= 1.4e+58)
		tmp = abs(Float64(eh * sin(atan(Float64(eh / Float64(ew * t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -2.1e-58)
		tmp = t_1;
	elseif (ew <= 1.4e+58)
		tmp = abs((eh * sin(atan((eh / (ew * t))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.1e-58], t$95$1, If[LessEqual[ew, 1.4e+58], N[Abs[N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.09999999999999988e-58 or 1.3999999999999999e58 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.9

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

   6. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.9

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

   8. Applied rewrites 41.9%

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

   if -2.09999999999999988e-58 < ew < 1.3999999999999999e58

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 39.7

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

   7. Applied rewrites 39.7%

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

Alternative 11: 55.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.05 \cdot 10^{-58}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{\mathsf{fma}\left(\left(0.3333333333333333 \cdot ew\right) \cdot t, t, ew\right) \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.05e-58)
     t_1
     (if (<= ew 1.4e+58)
       (fabs
        (*
         (tanh (asinh (/ eh (* (fma (* (* 0.3333333333333333 ew) t) t ew) t))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.05e-58) {
		tmp = t_1;
	} else if (ew <= 1.4e+58) {
		tmp = fabs((tanh(asinh((eh / (fma(((0.3333333333333333 * ew) * t), t, ew) * t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.05e-58)
		tmp = t_1;
	elseif (ew <= 1.4e+58)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(fma(Float64(Float64(0.3333333333333333 * ew) * t), t, ew) * t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.05e-58], t$95$1, If[LessEqual[ew, 1.4e+58], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(N[(N[(N[(0.3333333333333333 * ew), $MachinePrecision] * t), $MachinePrecision] * t + ew), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.05000000000000014e-58 or 1.3999999999999999e58 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-atan.f64 N/A

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

      6. sin-atan N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-atan.f64 N/A

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

   3. Applied rewrites 62.2%

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

   4. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 41.9

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

   6. Applied rewrites 41.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 41.9

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

   8. Applied rewrites 41.9%

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

   if -2.05000000000000014e-58 < ew < 1.3999999999999999e58

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-atan.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 41.5

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

   4. Applied rewrites 41.5%

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

   6. Applied rewrites 20.5%

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

   7. Taylor expanded in t around 0

      \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

      2. lower-+.f64 N/A

         \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

      5. lower-pow.f64 16.8

         \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

   9. Applied rewrites 16.8%

      \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{\tan t \cdot ew}\right)}\right)\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]

       2. lower-+.f64 N/A

          \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]

       3. lower-*.f64 N/A

          \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]

       4. lower-*.f64 N/A

          \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + \frac{1}{3} \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]

       5. lower-pow.f64 17.8

          \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]

   12. Applied rewrites 17.8%

       \[\leadsto \left|eh \cdot \left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right)\right| \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 17.8

          \[\leadsto \left|\left(eh \cdot \frac{\frac{1}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}}{\cosh \sinh^{-1} \left(\frac{eh}{t \cdot \left(ew + 0.3333333333333333 \cdot \left(ew \cdot {t}^{2}\right)\right)}\right)}\right) \cdot \color{blue}{eh}\right| \]

   14. Applied rewrites 37.7%

       \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{eh}{\mathsf{fma}\left(\left(0.3333333333333333 \cdot ew\right) \cdot t, t, ew\right) \cdot t}\right) \cdot \color{blue}{eh}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 41.9% accurate, 6.7× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-atan.f64 N/A

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

3. Applied rewrites 62.2%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.9

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

6. Applied rewrites 41.9%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 41.9

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

8. Applied rewrites 41.9%

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

Alternative 13: 18.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \left|\mathsf{fma}\left(\left(-0.16666666666666666 \cdot ew\right) \cdot t, t, ew\right) \cdot t\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* (fma (* (* -0.16666666666666666 ew) t) t ew) t)))

C

double code(double eh, double ew, double t) {
	return fabs((fma(((-0.16666666666666666 * ew) * t), t, ew) * t));
}

Julia

function code(eh, ew, t)
	return abs(Float64(fma(Float64(Float64(-0.16666666666666666 * ew) * t), t, ew) * t))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[(-0.16666666666666666 * ew), $MachinePrecision] * t), $MachinePrecision] * t + ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-atan.f64 N/A

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

3. Applied rewrites 62.2%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.9

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

6. Applied rewrites 41.9%

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

7. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 18.3

      \[\leadsto \left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)\right| \]

9. Applied rewrites 18.3%

   \[\leadsto \left|t \cdot \color{blue}{\left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)}\right| \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

       \[\leadsto \left|\left(ew + \frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)\right) \cdot t\right| \]

    3. lower-*.f64 18.3

       \[\leadsto \left|\left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right) \cdot t\right| \]

    4. lift-+.f64 N/A

       \[\leadsto \left|\left(ew + \frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right)\right) \cdot t\right| \]

    5. +-commutative N/A

       \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right) \cdot t\right| \]

    6. lift-*.f64 N/A

       \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right) \cdot t\right| \]

    7. lift-*.f64 N/A

       \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right) \cdot t\right| \]

    8. lift-pow.f64 N/A

       \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right) \cdot t\right| \]

    9. pow2 N/A

       \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot \left(t \cdot t\right)\right) + ew\right) \cdot t\right| \]

    10. lift-*.f64 N/A

        \[\leadsto \left|\left(\frac{-1}{6} \cdot \left(ew \cdot \left(t \cdot t\right)\right) + ew\right) \cdot t\right| \]

    11. associate-*r* N/A

        \[\leadsto \left|\left(\left(\frac{-1}{6} \cdot ew\right) \cdot \left(t \cdot t\right) + ew\right) \cdot t\right| \]

    12. lift-*.f64 N/A

        \[\leadsto \left|\left(\left(\frac{-1}{6} \cdot ew\right) \cdot \left(t \cdot t\right) + ew\right) \cdot t\right| \]

    13. associate-*r* N/A

        \[\leadsto \left|\left(\left(\left(\frac{-1}{6} \cdot ew\right) \cdot t\right) \cdot t + ew\right) \cdot t\right| \]

    14. lower-fma.f64 N/A

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

    15. lower-*.f64 N/A

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

    16. lower-*.f64 18.3

        \[\leadsto \left|\mathsf{fma}\left(\left(-0.16666666666666666 \cdot ew\right) \cdot t, t, ew\right) \cdot t\right| \]

11. Applied rewrites 18.3%

    \[\leadsto \left|\mathsf{fma}\left(\left(-0.16666666666666666 \cdot ew\right) \cdot t, t, ew\right) \cdot t\right| \]
12. Add Preprocessing

Alternative 14: 18.3% accurate, 15.3× speedup

Math

\[\begin{array}{l} \\ \left|t \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(t \cdot t\right), ew, ew\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs (* t (fma (* -0.16666666666666666 (* t t)) ew ew))))

C

double code(double eh, double ew, double t) {
	return fabs((t * fma((-0.16666666666666666 * (t * t)), ew, ew)));
}

Julia

function code(eh, ew, t)
	return abs(Float64(t * fma(Float64(-0.16666666666666666 * Float64(t * t)), ew, ew)))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(t * N[(N[(-0.16666666666666666 * N[(t * t), $MachinePrecision]), $MachinePrecision] * ew + ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. lift-atan.f64 N/A

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

   6. sin-atan N/A

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

   7. associate-*r/ N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-atan.f64 N/A

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

3. Applied rewrites 62.2%

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

4. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 41.9

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

6. Applied rewrites 41.9%

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

7. Taylor expanded in t around 0

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

   1. lower-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 18.3

      \[\leadsto \left|t \cdot \left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)\right| \]

9. Applied rewrites 18.3%

   \[\leadsto \left|t \cdot \color{blue}{\left(ew + -0.16666666666666666 \cdot \left(ew \cdot {t}^{2}\right)\right)}\right| \]
10. Step-by-step derivation

    1. lift-+.f64 N/A

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

    2. +-commutative N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right)\right| \]

    3. lift-*.f64 N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right)\right| \]

    4. lift-*.f64 N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right)\right| \]

    5. lift-pow.f64 N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot {t}^{2}\right) + ew\right)\right| \]

    6. pow2 N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot \left(t \cdot t\right)\right) + ew\right)\right| \]

    7. lift-*.f64 N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(ew \cdot \left(t \cdot t\right)\right) + ew\right)\right| \]

    8. *-commutative N/A

       \[\leadsto \left|t \cdot \left(\frac{-1}{6} \cdot \left(\left(t \cdot t\right) \cdot ew\right) + ew\right)\right| \]

    9. associate-*r* N/A

       \[\leadsto \left|t \cdot \left(\left(\frac{-1}{6} \cdot \left(t \cdot t\right)\right) \cdot ew + ew\right)\right| \]

    10. lower-fma.f64 N/A

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

    11. lower-*.f64 18.3

        \[\leadsto \left|t \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(t \cdot t\right), ew, ew\right)\right| \]

11. Applied rewrites 18.3%

    \[\leadsto \left|t \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(t \cdot t\right), ew, ew\right)\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))