Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 11

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-tan.f64 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-tan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 99.8

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

7. Applied rewrites 99.8%

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

Alternative 2: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|\left(ew \cdot \sin t\right) \cdot \cos t\_2 + \left(eh \cdot \cos t\right) \cdot \sin t\_2\right| \leq 10^{+81}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<=
        (fabs (+ (* (* ew (sin t)) (cos t_2)) (* (* eh (cos t)) (sin t_2))))
        1e+81)
     (fabs
      (fma
       ew
       (sin t)
       (*
        t_1
        (tanh
         (asinh
          (/ (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew)) t))))))
     (fabs (fma ew (sin t) (* t_1 (tanh (asinh (/ eh (* ew t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs((((ew * sin(t)) * cos(t_2)) + ((eh * cos(t)) * sin(t_2)))) <= 1e+81) {
		tmp = fabs(fma(ew, sin(t), (t_1 * tanh(asinh((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))))));
	} else {
		tmp = fabs(fma(ew, sin(t), (t_1 * tanh(asinh((eh / (ew * t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(Float64(ew * sin(t)) * cos(t_2)) + Float64(Float64(eh * cos(t)) * sin(t_2)))) <= 1e+81)
		tmp = abs(fma(ew, sin(t), Float64(t_1 * tanh(asinh(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = abs(fma(ew, sin(t), Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e+81], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 9.99999999999999921e80

   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lift-sin.f64 98.0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 96.4

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

   11. Applied rewrites 96.4%

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

   if 9.99999999999999921e80 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lift-sin.f64 99.1

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in t around 0

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 91.4

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

   11. Applied rewrites 91.4%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $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 t around 0

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

4. Applied rewrites 99.2%

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

Alternative 4: 99.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.2

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

6. Applied rewrites 99.2%

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

Alternative 5: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lift-sin.f64 98.6

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

8. Applied rewrites 98.6%

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

   1. lift-fma.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-tanh.f64 N/A

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

   6. lift-asinh.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lower-+.f64 N/A

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

10. Applied rewrites 98.6%

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

Alternative 6: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lift-sin.f64 98.6

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

8. Applied rewrites 98.6%

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

Alternative 7: 94.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;eh \leq -2 \cdot 10^{+159}:\\ \;\;\;\;\left|eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)\right|\\ \mathbf{elif}\;eh \leq 3.6 \cdot 10^{+154}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, t\_1 \cdot \tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (<= eh -2e+159)
     (fabs (* eh (* (cos t) (sin (atan (/ (* eh (cos t)) (* ew (sin t))))))))
     (if (<= eh 3.6e+154)
       (fabs
        (fma
         ew
         (sin t)
         (*
          t_1
          (tanh
           (asinh
            (/
             (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew))
             t))))))
       (fabs (fma ew (sin t) (* t_1 (tanh (asinh (/ eh (* ew t)))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double tmp;
	if (eh <= -2e+159) {
		tmp = fabs((eh * (cos(t) * sin(atan(((eh * cos(t)) / (ew * sin(t))))))));
	} else if (eh <= 3.6e+154) {
		tmp = fabs(fma(ew, sin(t), (t_1 * tanh(asinh((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))))));
	} else {
		tmp = fabs(fma(ew, sin(t), (t_1 * tanh(asinh((eh / (ew * t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if (eh <= -2e+159)
		tmp = abs(Float64(eh * Float64(cos(t) * sin(atan(Float64(Float64(eh * cos(t)) / Float64(ew * sin(t))))))));
	elseif (eh <= 3.6e+154)
		tmp = abs(fma(ew, sin(t), Float64(t_1 * tanh(asinh(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = abs(fma(ew, sin(t), Float64(t_1 * tanh(asinh(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -2e+159], N[Abs[N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 3.6e+154], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(t$95$1 * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.9999999999999999e159

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 58.9%

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

   5. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-*.f64 92.5

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

   7. Applied rewrites 92.5%

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

   if -1.9999999999999999e159 < eh < 3.6000000000000001e154

   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lift-sin.f64 98.4

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

   8. Applied rewrites 98.4%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-*.f64 N/A

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

       7. lift-/.f64 95.6

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

   11. Applied rewrites 95.6%

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

   if 3.6000000000000001e154 < 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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. lift-sin.f64 98.9

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

   8. Applied rewrites 98.9%

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

   9. Taylor expanded in t around 0

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 87.8

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

   11. Applied rewrites 87.8%

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

Alternative 8: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lift-sin.f64 98.6

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

8. Applied rewrites 98.6%

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

9. Taylor expanded in eh around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lift-cos.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lift-sin.f64 98.6

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

11. Applied rewrites 98.6%

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

Alternative 9: 89.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $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. 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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-sin.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-sin.f64 N/A

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

   10. lift-atan.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

3. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in eh around 0

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

   1. cos-atan-rev N/A

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

   2. lift-sin.f64 98.6

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

8. Applied rewrites 98.6%

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

9. Taylor expanded in t around 0

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 89.4

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

11. Applied rewrites 89.4%

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

Alternative 10: 59.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -15500000000000:\\ \;\;\;\;\left|eh\right|\\ \mathbf{elif}\;eh \leq 1.52 \cdot 10^{-58}:\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -15500000000000.0)
   (fabs eh)
   (if (<= eh 1.52e-58) (fabs (* ew (sin t))) (fabs eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -15500000000000.0) {
		tmp = fabs(eh);
	} else if (eh <= 1.52e-58) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-15500000000000.0d0)) then
        tmp = abs(eh)
    else if (eh <= 1.52d-58) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -15500000000000.0) {
		tmp = Math.abs(eh);
	} else if (eh <= 1.52e-58) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -15500000000000.0:
		tmp = math.fabs(eh)
	elif eh <= 1.52e-58:
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -15500000000000.0)
		tmp = abs(eh);
	elseif (eh <= 1.52e-58)
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -15500000000000.0)
		tmp = abs(eh);
	elseif (eh <= 1.52e-58)
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -15500000000000.0], N[Abs[eh], $MachinePrecision], If[LessEqual[eh, 1.52e-58], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.55e13 or 1.51999999999999993e-58 < eh

   1. Initial program 99.8%

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

   2. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 53.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.7

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

   7. Applied rewrites 51.7%

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

      1. lift-tanh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. tanh-asinh N/A

         \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

      4. lower-/.f64 N/A

         \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

      6. lower-+.f64 N/A

         \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

   9. Applied rewrites 12.3%

      \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

   10. Taylor expanded in eh around inf

       \[\leadsto \left|eh\right| \]
   11. Step-by-step derivation

   12. Applied rewrites 54.2%

       \[\leadsto \left|eh\right| \]

   if -1.55e13 < eh < 1.51999999999999993e-58

   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. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-atan.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

   3. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-tan.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. lift-tan.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in eh around 0

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

      1. cos-atan-rev N/A

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

      2. tanh-asinh N/A

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

      3. sin-atan N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 66.2

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

   8. Applied rewrites 66.2%

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

Alternative 11: 42.4% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

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(eh)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

4. Applied rewrites 42.0%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 40.1

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

7. Applied rewrites 40.1%

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

   1. lift-tanh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. tanh-asinh N/A

      \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

   4. lower-/.f64 N/A

      \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

   6. lower-+.f64 N/A

      \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

9. Applied rewrites 12.5%

   \[\leadsto \left|\frac{\frac{eh}{ew \cdot t}}{\sqrt{1 + \frac{eh}{ew \cdot t} \cdot \frac{eh}{ew \cdot t}}} \cdot eh\right| \]

10. Taylor expanded in eh around inf

    \[\leadsto \left|eh\right| \]
11. Step-by-step derivation

12. Applied rewrites 42.4%

    \[\leadsto \left|eh\right| \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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))))))))