Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 11.7s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. unpow2 N/A

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

   8. lift-tan.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (+
   (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t)))))
   (*
    (* eh (cos t))
    (sin (atan (/ (/ (fma -0.3333333333333333 (* (* t t) eh) eh) ew) t)))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * sin(t)) * cos(atan(((eh / ew) / tan(t))))) + ((eh * cos(t)) * sin(atan(((fma(-0.3333333333333333, ((t * t) * eh), eh) / ew) / t))))));
}

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * sin(t)) * cos(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(t * t) * eh), eh) / ew) / 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] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $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. Add Preprocessing

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.0

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

5. Applied rewrites 99.0%

   \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)}\right| \]
6. Add Preprocessing

Alternative 3: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 98.5

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

7. Applied rewrites 98.5%

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

Alternative 4: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \left|t\_1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{if}\;t \leq -0.000225:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\left|t\_1 + \left(eh + -0.5 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2
         (fabs
          (+
           t_1
           (*
            (* eh (cos t))
            (sin (atan (* -0.3333333333333333 (/ (* eh t) ew)))))))))
   (if (<= t -0.000225)
     t_2
     (if (<= t 3.4e-41)
       (fabs
        (+
         t_1
         (*
          (+ eh (* -0.5 (* eh (* t t))))
          (sin (atan (/ (/ eh ew) (tan t)))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = fabs((t_1 + ((eh * cos(t)) * sin(atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	double tmp;
	if (t <= -0.000225) {
		tmp = t_2;
	} else if (t <= 3.4e-41) {
		tmp = fabs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * sin(atan(((eh / ew) / tan(t)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = abs((t_1 + ((eh * cos(t)) * sin(atan(((-0.3333333333333333d0) * ((eh * t) / ew)))))))
    if (t <= (-0.000225d0)) then
        tmp = t_2
    else if (t <= 3.4d-41) then
        tmp = abs((t_1 + ((eh + ((-0.5d0) * (eh * (t * t)))) * sin(atan(((eh / ew) / tan(t)))))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.abs((t_1 + ((eh * Math.cos(t)) * Math.sin(Math.atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	double tmp;
	if (t <= -0.000225) {
		tmp = t_2;
	} else if (t <= 3.4e-41) {
		tmp = Math.abs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.fabs((t_1 + ((eh * math.cos(t)) * math.sin(math.atan((-0.3333333333333333 * ((eh * t) / ew)))))))
	tmp = 0
	if t <= -0.000225:
		tmp = t_2
	elif t <= 3.4e-41:
		tmp = math.fabs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * math.sin(math.atan(((eh / ew) / math.tan(t)))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = abs(Float64(t_1 + Float64(Float64(eh * cos(t)) * sin(atan(Float64(-0.3333333333333333 * Float64(Float64(eh * t) / ew)))))))
	tmp = 0.0
	if (t <= -0.000225)
		tmp = t_2;
	elseif (t <= 3.4e-41)
		tmp = abs(Float64(t_1 + Float64(Float64(eh + Float64(-0.5 * Float64(eh * Float64(t * t)))) * sin(atan(Float64(Float64(eh / ew) / tan(t)))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = abs((t_1 + ((eh * cos(t)) * sin(atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	tmp = 0.0;
	if (t <= -0.000225)
		tmp = t_2;
	elseif (t <= 3.4e-41)
		tmp = abs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * sin(atan(((eh / ew) / tan(t)))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.000225], t$95$2, If[LessEqual[t, 3.4e-41], N[Abs[N[(t$95$1 + N[(N[(eh + N[(-0.5 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2499999999999999e-4 or 3.3999999999999998e-41 < t

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 85.7

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 97.1

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

   13. Applied rewrites 97.1%

       \[\leadsto \left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \color{blue}{\frac{eh \cdot t}{ew}}\right)\right| \]

   if -2.2499999999999999e-4 < t < 3.3999999999999998e-41

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 98.9

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

   10. Applied rewrites 98.9%

       \[\leadsto \left|ew \cdot \sin t + \color{blue}{\left(eh + -0.5 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := \left|t\_1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \frac{eh \cdot t}{ew}\right)\right|\\ \mathbf{if}\;t \leq -0.000225:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{-41}:\\ \;\;\;\;\left|t\_1 + \left(eh + -0.5 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2
         (fabs
          (+
           t_1
           (*
            (* eh (cos t))
            (sin (atan (* -0.3333333333333333 (/ (* eh t) ew)))))))))
   (if (<= t -0.000225)
     t_2
     (if (<= t 3.4e-41)
       (fabs
        (+
         t_1
         (* (+ eh (* -0.5 (* eh (* t t)))) (sin (atan (/ eh (* ew t)))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = fabs((t_1 + ((eh * cos(t)) * sin(atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	double tmp;
	if (t <= -0.000225) {
		tmp = t_2;
	} else if (t <= 3.4e-41) {
		tmp = fabs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * sin(atan((eh / (ew * t)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * sin(t)
    t_2 = abs((t_1 + ((eh * cos(t)) * sin(atan(((-0.3333333333333333d0) * ((eh * t) / ew)))))))
    if (t <= (-0.000225d0)) then
        tmp = t_2
    else if (t <= 3.4d-41) then
        tmp = abs((t_1 + ((eh + ((-0.5d0) * (eh * (t * t)))) * sin(atan((eh / (ew * t)))))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.sin(t);
	double t_2 = Math.abs((t_1 + ((eh * Math.cos(t)) * Math.sin(Math.atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	double tmp;
	if (t <= -0.000225) {
		tmp = t_2;
	} else if (t <= 3.4e-41) {
		tmp = Math.abs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * Math.sin(Math.atan((eh / (ew * t)))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.sin(t)
	t_2 = math.fabs((t_1 + ((eh * math.cos(t)) * math.sin(math.atan((-0.3333333333333333 * ((eh * t) / ew)))))))
	tmp = 0
	if t <= -0.000225:
		tmp = t_2
	elif t <= 3.4e-41:
		tmp = math.fabs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * math.sin(math.atan((eh / (ew * t)))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = abs(Float64(t_1 + Float64(Float64(eh * cos(t)) * sin(atan(Float64(-0.3333333333333333 * Float64(Float64(eh * t) / ew)))))))
	tmp = 0.0
	if (t <= -0.000225)
		tmp = t_2;
	elseif (t <= 3.4e-41)
		tmp = abs(Float64(t_1 + Float64(Float64(eh + Float64(-0.5 * Float64(eh * Float64(t * t)))) * sin(atan(Float64(eh / Float64(ew * t)))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * sin(t);
	t_2 = abs((t_1 + ((eh * cos(t)) * sin(atan((-0.3333333333333333 * ((eh * t) / ew)))))));
	tmp = 0.0;
	if (t <= -0.000225)
		tmp = t_2;
	elseif (t <= 3.4e-41)
		tmp = abs((t_1 + ((eh + (-0.5 * (eh * (t * t)))) * sin(atan((eh / (ew * t)))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(t$95$1 + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.000225], t$95$2, If[LessEqual[t, 3.4e-41], N[Abs[N[(t$95$1 + N[(N[(eh + N[(-0.5 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.2499999999999999e-4 or 3.3999999999999998e-41 < t

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.1

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

   7. Applied rewrites 98.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-/.f64 85.7

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

   10. Applied rewrites 85.7%

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

   11. Taylor expanded in t around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. lower-*.f64 97.1

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

   13. Applied rewrites 97.1%

       \[\leadsto \left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(-0.3333333333333333 \cdot \color{blue}{\frac{eh \cdot t}{ew}}\right)\right| \]

   if -2.2499999999999999e-4 < t < 3.3999999999999998e-41

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.9

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

   10. Applied rewrites 98.9%

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

   11. Taylor expanded in t around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 98.9

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

   13. Applied rewrites 98.9%

       \[\leadsto \left|ew \cdot \sin t + \color{blue}{\left(eh + -0.5 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right)} \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[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. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 89.0

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

10. Applied rewrites 89.0%

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

Alternative 7: 81.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := eh \cdot \cos t\\ t_3 := \left|ew \cdot \sin t + t\_2 \cdot \frac{t\_1}{\sqrt{1 + t\_1 \cdot t\_1}}\right|\\ \mathbf{if}\;t \leq -0.058:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+26}:\\ \;\;\;\;\left|ew \cdot t + t\_2 \cdot \sin \tan^{-1} t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t)))
        (t_2 (* eh (cos t)))
        (t_3
         (fabs (+ (* ew (sin t)) (* t_2 (/ t_1 (sqrt (+ 1.0 (* t_1 t_1)))))))))
   (if (<= t -0.058)
     t_3
     (if (<= t 5e+26) (fabs (+ (* ew t) (* t_2 (sin (atan t_1))))) t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = eh * cos(t);
	double t_3 = fabs(((ew * sin(t)) + (t_2 * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	double tmp;
	if (t <= -0.058) {
		tmp = t_3;
	} else if (t <= 5e+26) {
		tmp = fabs(((ew * t) + (t_2 * sin(atan(t_1)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = eh / (ew * t)
    t_2 = eh * cos(t)
    t_3 = abs(((ew * sin(t)) + (t_2 * (t_1 / sqrt((1.0d0 + (t_1 * t_1)))))))
    if (t <= (-0.058d0)) then
        tmp = t_3
    else if (t <= 5d+26) then
        tmp = abs(((ew * t) + (t_2 * sin(atan(t_1)))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = eh * Math.cos(t);
	double t_3 = Math.abs(((ew * Math.sin(t)) + (t_2 * (t_1 / Math.sqrt((1.0 + (t_1 * t_1)))))));
	double tmp;
	if (t <= -0.058) {
		tmp = t_3;
	} else if (t <= 5e+26) {
		tmp = Math.abs(((ew * t) + (t_2 * Math.sin(Math.atan(t_1)))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	t_2 = eh * math.cos(t)
	t_3 = math.fabs(((ew * math.sin(t)) + (t_2 * (t_1 / math.sqrt((1.0 + (t_1 * t_1)))))))
	tmp = 0
	if t <= -0.058:
		tmp = t_3
	elif t <= 5e+26:
		tmp = math.fabs(((ew * t) + (t_2 * math.sin(math.atan(t_1)))))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = Float64(eh * cos(t))
	t_3 = abs(Float64(Float64(ew * sin(t)) + Float64(t_2 * Float64(t_1 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))))))
	tmp = 0.0
	if (t <= -0.058)
		tmp = t_3;
	elseif (t <= 5e+26)
		tmp = abs(Float64(Float64(ew * t) + Float64(t_2 * sin(atan(t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	t_2 = eh * cos(t);
	t_3 = abs(((ew * sin(t)) + (t_2 * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	tmp = 0.0;
	if (t <= -0.058)
		tmp = t_3;
	elseif (t <= 5e+26)
		tmp = abs(((ew * t) + (t_2 * sin(atan(t_1)))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(t$95$1 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.058], t$95$3, If[LessEqual[t, 5e+26], N[Abs[N[(N[(ew * t), $MachinePrecision] + N[(t$95$2 * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0580000000000000029 or 5.0000000000000001e26 < t

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.0

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

   7. Applied rewrites 98.0%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 78.3

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

   10. Applied rewrites 78.3%

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

       1. lift-sin.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. sin-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 66.0

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

   12. Applied rewrites 66.0%

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

   if -0.0580000000000000029 < t < 5.0000000000000001e26

   1. Initial program 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.8

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

   10. Applied rewrites 98.8%

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

   11. Taylor expanded in t around 0

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

       1. lift-*.f64 96.4

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

   13. Applied rewrites 96.4%

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

Alternative 8: 69.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := ew \cdot \sin t\\ t_3 := eh \cdot \cos t\\ t_4 := \left|t\_2 + t\_3 \cdot \frac{t\_1}{\sqrt{1 + t\_1 \cdot t\_1}}\right|\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;\left|\tanh \left(\frac{t\_3}{t\_2}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t)))
        (t_2 (* ew (sin t)))
        (t_3 (* eh (cos t)))
        (t_4 (fabs (+ t_2 (* t_3 (/ t_1 (sqrt (+ 1.0 (* t_1 t_1)))))))))
   (if (<= t -4.8e-29)
     t_4
     (if (<= t 4.2e-72) (fabs (* (tanh (/ t_3 t_2)) eh)) t_4))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = ew * sin(t);
	double t_3 = eh * cos(t);
	double t_4 = fabs((t_2 + (t_3 * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	double tmp;
	if (t <= -4.8e-29) {
		tmp = t_4;
	} else if (t <= 4.2e-72) {
		tmp = fabs((tanh((t_3 / t_2)) * eh));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = eh / (ew * t)
    t_2 = ew * sin(t)
    t_3 = eh * cos(t)
    t_4 = abs((t_2 + (t_3 * (t_1 / sqrt((1.0d0 + (t_1 * t_1)))))))
    if (t <= (-4.8d-29)) then
        tmp = t_4
    else if (t <= 4.2d-72) then
        tmp = abs((tanh((t_3 / t_2)) * eh))
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = ew * Math.sin(t);
	double t_3 = eh * Math.cos(t);
	double t_4 = Math.abs((t_2 + (t_3 * (t_1 / Math.sqrt((1.0 + (t_1 * t_1)))))));
	double tmp;
	if (t <= -4.8e-29) {
		tmp = t_4;
	} else if (t <= 4.2e-72) {
		tmp = Math.abs((Math.tanh((t_3 / t_2)) * eh));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	t_2 = ew * math.sin(t)
	t_3 = eh * math.cos(t)
	t_4 = math.fabs((t_2 + (t_3 * (t_1 / math.sqrt((1.0 + (t_1 * t_1)))))))
	tmp = 0
	if t <= -4.8e-29:
		tmp = t_4
	elif t <= 4.2e-72:
		tmp = math.fabs((math.tanh((t_3 / t_2)) * eh))
	else:
		tmp = t_4
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = Float64(ew * sin(t))
	t_3 = Float64(eh * cos(t))
	t_4 = abs(Float64(t_2 + Float64(t_3 * Float64(t_1 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))))))
	tmp = 0.0
	if (t <= -4.8e-29)
		tmp = t_4;
	elseif (t <= 4.2e-72)
		tmp = abs(Float64(tanh(Float64(t_3 / t_2)) * eh));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	t_2 = ew * sin(t);
	t_3 = eh * cos(t);
	t_4 = abs((t_2 + (t_3 * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	tmp = 0.0;
	if (t <= -4.8e-29)
		tmp = t_4;
	elseif (t <= 4.2e-72)
		tmp = abs((tanh((t_3 / t_2)) * eh));
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(t$95$2 + N[(t$95$3 * N[(t$95$1 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.8e-29], t$95$4, If[LessEqual[t, 4.2e-72], N[Abs[N[(N[Tanh[N[(t$95$3 / t$95$2), $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.79999999999999984e-29 or 4.2e-72 < t

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.9

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

   10. Applied rewrites 81.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. sin-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 66.1

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

   12. Applied rewrites 66.1%

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

   if -4.79999999999999984e-29 < t < 4.2e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-*.f64 74.7

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

   8. Applied rewrites 74.7%

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

Alternative 9: 69.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := \left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \frac{t\_1}{\sqrt{1 + t\_1 \cdot t\_1}}\right|\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{-29}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-72}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t)))
        (t_2
         (fabs
          (+
           (* ew (sin t))
           (* (* eh (cos t)) (/ t_1 (sqrt (+ 1.0 (* t_1 t_1)))))))))
   (if (<= t -4.8e-29)
     t_2
     (if (<= t 4.2e-72)
       (fabs (* (tanh (asinh (* (/ 1.0 ew) (/ eh t)))) eh))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = fabs(((ew * sin(t)) + ((eh * cos(t)) * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	double tmp;
	if (t <= -4.8e-29) {
		tmp = t_2;
	} else if (t <= 4.2e-72) {
		tmp = fabs((tanh(asinh(((1.0 / ew) * (eh / t)))) * eh));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	t_2 = math.fabs(((ew * math.sin(t)) + ((eh * math.cos(t)) * (t_1 / math.sqrt((1.0 + (t_1 * t_1)))))))
	tmp = 0
	if t <= -4.8e-29:
		tmp = t_2
	elif t <= 4.2e-72:
		tmp = math.fabs((math.tanh(math.asinh(((1.0 / ew) * (eh / t)))) * eh))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = abs(Float64(Float64(ew * sin(t)) + Float64(Float64(eh * cos(t)) * Float64(t_1 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))))))
	tmp = 0.0
	if (t <= -4.8e-29)
		tmp = t_2;
	elseif (t <= 4.2e-72)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(1.0 / ew) * Float64(eh / t)))) * eh));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	t_2 = abs(((ew * sin(t)) + ((eh * cos(t)) * (t_1 / sqrt((1.0 + (t_1 * t_1)))))));
	tmp = 0.0;
	if (t <= -4.8e-29)
		tmp = t_2;
	elseif (t <= 4.2e-72)
		tmp = abs((tanh(asinh(((1.0 / ew) * (eh / t)))) * eh));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.8e-29], t$95$2, If[LessEqual[t, 4.2e-72], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(eh / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.79999999999999984e-29 or 4.2e-72 < t

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 81.9

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

   10. Applied rewrites 81.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-atan.f64 N/A

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

       3. sin-atan N/A

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

       4. lower-/.f64 N/A

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

       5. lower-sqrt.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 66.1

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

   12. Applied rewrites 66.1%

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

   if -4.79999999999999984e-29 < t < 4.2e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 74.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 74.7%

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

Alternative 10: 59.6% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;ew \leq -2.3 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.6 \cdot 10^{+24}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh + 0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= ew -2.3e+46)
     t_1
     (if (<= ew 1.6e+24)
       (fabs
        (*
         (tanh
          (asinh
           (* (/ 1.0 ew) (/ (+ eh (* 0.16666666666666666 (* eh (* t t)))) t))))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (ew <= -2.3e+46) {
		tmp = t_1;
	} else if (ew <= 1.6e+24) {
		tmp = fabs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if ew <= -2.3e+46:
		tmp = t_1
	elif ew <= 1.6e+24:
		tmp = math.fabs((math.tanh(math.asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (ew <= -2.3e+46)
		tmp = t_1;
	elseif (ew <= 1.6e+24)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(1.0 / ew) * Float64(Float64(eh + Float64(0.16666666666666666 * Float64(eh * Float64(t * t)))) / t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (ew <= -2.3e+46)
		tmp = t_1;
	elseif (ew <= 1.6e+24)
		tmp = abs((tanh(asinh(((1.0 / ew) * ((eh + (0.16666666666666666 * (eh * (t * t)))) / t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.3e+46], t$95$1, If[LessEqual[ew, 1.6e+24], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(N[(eh + N[(0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.3000000000000001e46 or 1.5999999999999999e24 < ew

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 66.5

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

   7. Applied rewrites 66.5%

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

   if -2.3000000000000001e46 < ew < 1.5999999999999999e24

   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. Add Preprocessing

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.8%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 54.0

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

   11. Applied rewrites 54.0%

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

Alternative 11: 61.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -0.0066:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-72}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{1}{ew} \cdot \frac{eh}{t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -0.0066)
     t_1
     (if (<= t 4.3e-72)
       (fabs (* (tanh (asinh (* (/ 1.0 ew) (/ eh t)))) eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -0.0066) {
		tmp = t_1;
	} else if (t <= 4.3e-72) {
		tmp = fabs((tanh(asinh(((1.0 / ew) * (eh / t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -0.0066:
		tmp = t_1
	elif t <= 4.3e-72:
		tmp = math.fabs((math.tanh(math.asinh(((1.0 / ew) * (eh / t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -0.0066)
		tmp = t_1;
	elseif (t <= 4.3e-72)
		tmp = abs(Float64(tanh(asinh(Float64(Float64(1.0 / ew) * Float64(eh / t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -0.0066)
		tmp = t_1;
	elseif (t <= 4.3e-72)
		tmp = abs((tanh(asinh(((1.0 / ew) * (eh / t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0066], t$95$1, If[LessEqual[t, 4.3e-72], N[Abs[N[(N[Tanh[N[ArcSinh[N[(N[(1.0 / ew), $MachinePrecision] * N[(eh / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0066 or 4.2999999999999999e-72 < t

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 51.4

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

   7. Applied rewrites 51.4%

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

   if -0.0066 < t < 4.2999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 73.6%

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

Alternative 12: 61.2% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \sin t\right|\\ \mathbf{if}\;t \leq -0.0066:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.3 \cdot 10^{-72}:\\ \;\;\;\;\left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (sin t)))))
   (if (<= t -0.0066)
     t_1
     (if (<= t 4.3e-72) (fabs (* (tanh (asinh (/ eh (* ew t)))) eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * sin(t)));
	double tmp;
	if (t <= -0.0066) {
		tmp = t_1;
	} else if (t <= 4.3e-72) {
		tmp = fabs((tanh(asinh((eh / (ew * t)))) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.sin(t)))
	tmp = 0
	if t <= -0.0066:
		tmp = t_1
	elif t <= 4.3e-72:
		tmp = math.fabs((math.tanh(math.asinh((eh / (ew * t)))) * eh))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * sin(t)))
	tmp = 0.0
	if (t <= -0.0066)
		tmp = t_1;
	elseif (t <= 4.3e-72)
		tmp = abs(Float64(tanh(asinh(Float64(eh / Float64(ew * t)))) * eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * sin(t)));
	tmp = 0.0;
	if (t <= -0.0066)
		tmp = t_1;
	elseif (t <= 4.3e-72)
		tmp = abs((tanh(asinh((eh / (ew * t)))) * eh));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -0.0066], t$95$1, If[LessEqual[t, 4.3e-72], N[Abs[N[(N[Tanh[N[ArcSinh[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -0.0066 or 4.2999999999999999e-72 < t

   1. Initial program 99.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. unpow2 N/A

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

      2. cos-atan N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 51.4

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

   7. Applied rewrites 51.4%

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

   if -0.0066 < t < 4.2999999999999999e-72

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

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

   5. Applied rewrites 73.6%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\tanh \sinh^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot eh\right| \]
   7. 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 73.6

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

   8. Applied rewrites 73.6%

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

Alternative 13: 41.9% accurate, 8.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Sin[t], $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. Add Preprocessing
3. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 41.9

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

7. Applied rewrites 41.9%

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

Alternative 14: 18.8% accurate, 108.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-tan.f64 N/A

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

   14. lift-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around 0

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

   1. unpow2 N/A

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

   2. cos-atan N/A

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

   3. lift-sin.f64 N/A

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

   4. lift-*.f64 41.9

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

7. Applied rewrites 41.9%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 18.8%

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

Reproduce

herbie shell --seed 2025088 
(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))))))))