Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.8s

Alternatives: 14

Speedup: 1.0×

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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. cos-atan N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-tan.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Tanh[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $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. associate-*l* N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. cos-atan N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-tan.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lower-/.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-cos.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-sin.f64 97.3

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

11. Applied rewrites 97.3%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 97.2

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

7. Applied rewrites 97.2%

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

Alternative 5: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + t\_1 \cdot t\_1}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.3333333333333333, \left(t \cdot t\right) \cdot eh, eh\right)}{ew}}{t}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (fabs
    (+
     (* (* ew (sin t)) (/ 1.0 (sqrt (+ 1.0 (* t_1 t_1)))))
     (*
      (* eh (cos t))
      (sin
       (atan (/ (/ (fma -0.3333333333333333 (* (* t t) eh) eh) ew) t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	return fabs((((ew * sin(t)) * (1.0 / sqrt((1.0 + (t_1 * t_1))))) + ((eh * cos(t)) * sin(atan(((fma(-0.3333333333333333, ((t * t) * eh), eh) / ew) / t))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(Float64(ew * sin(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1))))) + 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_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(t$95$1 * t$95$1), $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 96.9

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

5. Applied rewrites 96.9%

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

   1. cos-atan N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-tan.f64 96.9

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

7. Applied rewrites 96.9%

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

Alternative 6: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. cos-atan N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-tan.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. lower-sin.f64 96.3

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

9. Applied rewrites 96.3%

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

Alternative 7: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-atan.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lower-tan.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. cos-atan N/A

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

   2. lower-/.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-tan.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. lower-tan.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. lower-sin.f64 96.3

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

9. Applied rewrites 96.3%

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

10. Taylor expanded in eh around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-cos.f64 N/A

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

    4. lower-*.f64 N/A

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

    5. lower-sin.f64 96.3

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

12. Applied rewrites 96.3%

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

Alternative 8: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.1 \cdot 10^{-63} \lor \neg \left(ew \leq 2.05 \cdot 10^{-111}\right):\\ \;\;\;\;\left|ew \cdot \left(\sin t + \frac{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.1e-63) (not (<= ew 2.05e-111)))
   (fabs
    (* ew (+ (sin t) (/ (* eh (* (cos t) (sin (atan (/ eh (* ew t)))))) ew))))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.1e-63) || !(ew <= 2.05e-111)) {
		tmp = fabs((ew * (sin(t) + ((eh * (cos(t) * sin(atan((eh / (ew * t)))))) / ew))));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-2.1d-63)) .or. (.not. (ew <= 2.05d-111))) then
        tmp = abs((ew * (sin(t) + ((eh * (cos(t) * sin(atan((eh / (ew * t)))))) / ew))))
    else
        tmp = abs((eh * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.1e-63) || !(ew <= 2.05e-111)) {
		tmp = Math.abs((ew * (Math.sin(t) + ((eh * (Math.cos(t) * Math.sin(Math.atan((eh / (ew * t)))))) / ew))));
	} else {
		tmp = Math.abs((eh * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.1e-63) or not (ew <= 2.05e-111):
		tmp = math.fabs((ew * (math.sin(t) + ((eh * (math.cos(t) * math.sin(math.atan((eh / (ew * t)))))) / ew))))
	else:
		tmp = math.fabs((eh * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.1e-63) || !(ew <= 2.05e-111))
		tmp = abs(Float64(ew * Float64(sin(t) + Float64(Float64(eh * Float64(cos(t) * sin(atan(Float64(eh / Float64(ew * t)))))) / ew))));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.1e-63) || ~((ew <= 2.05e-111)))
		tmp = abs((ew * (sin(t) + ((eh * (cos(t) * sin(atan((eh / (ew * t)))))) / ew))));
	else
		tmp = abs((eh * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.1e-63], N[Not[LessEqual[ew, 2.05e-111]], $MachinePrecision]], N[Abs[N[(ew * N[(N[Sin[t], $MachinePrecision] + N[(N[(eh * N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.1e-63 or 2.04999999999999984e-111 < ew

   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. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 99.7

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

   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 ew around inf

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

      1. pow2 N/A

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

      2. cos-atan-rev N/A

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

      3. sin-atan-rev N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 94.9%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 87.8

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

   10. Applied rewrites 87.8%

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

   if -2.1e-63 < ew < 2.04999999999999984e-111

   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 98.0

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

   5. Applied rewrites 98.0%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 17.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 90.3

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

   10. Applied rewrites 90.3%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.1 \cdot 10^{-63} \lor \neg \left(ew \leq 2.05 \cdot 10^{-111}\right):\\ \;\;\;\;\left|ew \cdot \left(\sin t + \frac{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right)}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{\tan t}\\ \mathbf{if}\;ew \leq -3.1 \cdot 10^{-64}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \tanh \sinh^{-1} t\_1 \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{-151}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} t\_1 + eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) (tan t))))
   (if (<= ew -3.1e-64)
     (fabs (fma ew (sin t) (* (tanh (asinh t_1)) eh)))
     (if (<= ew 1.55e-151)
       (fabs (* eh (cos t)))
       (fabs (+ (* (* ew (sin t)) (cos (atan t_1))) eh))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / tan(t);
	double tmp;
	if (ew <= -3.1e-64) {
		tmp = fabs(fma(ew, sin(t), (tanh(asinh(t_1)) * eh)));
	} else if (ew <= 1.55e-151) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((((ew * sin(t)) * cos(atan(t_1))) + eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	tmp = 0.0
	if (ew <= -3.1e-64)
		tmp = abs(fma(ew, sin(t), Float64(tanh(asinh(t_1)) * eh)));
	elseif (ew <= 1.55e-151)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(Float64(Float64(ew * sin(t)) * cos(atan(t_1))) + eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -3.1e-64], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.55e-151], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.10000000000000025e-64

   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. associate-*l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-tan.f64 99.7

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in eh around 0

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

      1. lower-sin.f64 94.9

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

   9. Applied rewrites 94.9%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 88.3%

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

   if -3.10000000000000025e-64 < ew < 1.54999999999999992e-151

   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 97.9

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

   5. Applied rewrites 97.9%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 17.9%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 90.8

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

   10. Applied rewrites 90.8%

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

   if 1.54999999999999992e-151 < 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. 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 97.0

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

   5. Applied rewrites 97.0%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 42.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.8%

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

Alternative 10: 87.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.1 \cdot 10^{-64} \lor \neg \left(ew \leq 1.55 \cdot 10^{-151}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \tanh \sinh^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.1e-64) (not (<= ew 1.55e-151)))
   (fabs (fma ew (sin t) (* (tanh (asinh (/ (/ eh ew) (tan t)))) eh)))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.1e-64) || !(ew <= 1.55e-151)) {
		tmp = fabs(fma(ew, sin(t), (tanh(asinh(((eh / ew) / tan(t)))) * eh)));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.1e-64) || !(ew <= 1.55e-151))
		tmp = abs(fma(ew, sin(t), Float64(tanh(asinh(Float64(Float64(eh / ew) / tan(t)))) * eh)));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.1e-64], N[Not[LessEqual[ew, 1.55e-151]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision] + N[(N[Tanh[N[ArcSinh[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.10000000000000025e-64 or 1.54999999999999992e-151 < 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. associate-*l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-tan.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in eh around 0

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

      1. lower-sin.f64 95.6

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

   9. Applied rewrites 95.6%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 85.0%

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

   if -3.10000000000000025e-64 < ew < 1.54999999999999992e-151

   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 97.9

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

   5. Applied rewrites 97.9%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 17.9%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 90.8

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

   10. Applied rewrites 90.8%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.1 \cdot 10^{-64} \lor \neg \left(ew \leq 1.55 \cdot 10^{-151}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \sin t, \tanh \sinh^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.15 \cdot 10^{+52} \lor \neg \left(ew \leq 2.6 \cdot 10^{+89}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.15e+52) (not (<= ew 2.6e+89)))
   (fabs (* ew (sin t)))
   (fabs (* eh (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.15e+52) || !(ew <= 2.6e+89)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs((eh * cos(t)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-2.15d+52)) .or. (.not. (ew <= 2.6d+89))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs((eh * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.15e+52) || !(ew <= 2.6e+89)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs((eh * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.15e+52) or not (ew <= 2.6e+89):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs((eh * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.15e+52) || !(ew <= 2.6e+89))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(eh * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.15e+52) || ~((ew <= 2.6e+89)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs((eh * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.15e+52], N[Not[LessEqual[ew, 2.6e+89]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.15e52 or 2.6000000000000001e89 < ew

   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. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 99.7

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

   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. pow2 N/A

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

      2. cos-atan-rev N/A

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

      3. sin-atan-rev N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if -2.15e52 < ew < 2.6000000000000001e89

   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 96.4

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

   5. Applied rewrites 96.4%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 25.0%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 78.7

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

   10. Applied rewrites 78.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.15 \cdot 10^{+52} \lor \neg \left(ew \leq 2.6 \cdot 10^{+89}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.9% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot eh}{ew}\\ \mathbf{if}\;ew \leq 1.8 \cdot 10^{+202}:\\ \;\;\;\;\left|eh \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(0.5, t\_1, \left(t \cdot t\right) \cdot \left(ew + -0.4166666666666667 \cdot t\_1\right)\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh eh) ew)))
   (if (<= ew 1.8e+202)
     (fabs (* eh (cos t)))
     (fabs
      (/ (fma 0.5 t_1 (* (* t t) (+ ew (* -0.4166666666666667 t_1)))) t)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * eh) / ew;
	double tmp;
	if (ew <= 1.8e+202) {
		tmp = fabs((eh * cos(t)));
	} else {
		tmp = fabs((fma(0.5, t_1, ((t * t) * (ew + (-0.4166666666666667 * t_1)))) / t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * eh) / ew)
	tmp = 0.0
	if (ew <= 1.8e+202)
		tmp = abs(Float64(eh * cos(t)));
	else
		tmp = abs(Float64(fma(0.5, t_1, Float64(Float64(t * t) * Float64(ew + Float64(-0.4166666666666667 * t_1)))) / t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[ew, 1.8e+202], N[Abs[N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.5 * t$95$1 + N[(N[(t * t), $MachinePrecision] * N[(ew + N[(-0.4166666666666667 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < 1.80000000000000004e202

   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 96.7

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

   5. Applied rewrites 96.7%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 33.3%

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

   8. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 62.5

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

   10. Applied rewrites 62.5%

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

   if 1.80000000000000004e202 < ew

   1. Initial program 99.9%

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

      1. associate-*l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-tan.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in eh around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 79.3%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \mathsf{fma}\left(-0.5, \frac{{\cos t}^{2}}{ew \cdot \sin t}, \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{-5}{12} \cdot \frac{{eh}^{2}}{ew}\right)}{\color{blue}{t}}\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{-5}{12} \cdot \frac{{eh}^{2}}{ew}\right)}{t}\right| \]

   12. Applied rewrites 37.1%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew}, \left(t \cdot t\right) \cdot \left(ew + -0.4166666666666667 \cdot \frac{eh \cdot eh}{ew}\right)\right)}{\color{blue}{t}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 42.9% accurate, 11.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot eh}{ew}\\ \mathbf{if}\;ew \leq 2.2 \cdot 10^{+185}:\\ \;\;\;\;\left|eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(0.5, t\_1, \left(t \cdot t\right) \cdot \left(ew + -0.4166666666666667 \cdot t\_1\right)\right)}{t}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh eh) ew)))
   (if (<= ew 2.2e+185)
     (fabs eh)
     (fabs
      (/ (fma 0.5 t_1 (* (* t t) (+ ew (* -0.4166666666666667 t_1)))) t)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * eh) / ew;
	double tmp;
	if (ew <= 2.2e+185) {
		tmp = fabs(eh);
	} else {
		tmp = fabs((fma(0.5, t_1, ((t * t) * (ew + (-0.4166666666666667 * t_1)))) / t));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * eh) / ew)
	tmp = 0.0
	if (ew <= 2.2e+185)
		tmp = abs(eh);
	else
		tmp = abs(Float64(fma(0.5, t_1, Float64(Float64(t * t) * Float64(ew + Float64(-0.4166666666666667 * t_1)))) / t));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * eh), $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[ew, 2.2e+185], N[Abs[eh], $MachinePrecision], N[Abs[N[(N[(0.5 * t$95$1 + N[(N[(t * t), $MachinePrecision] * N[(ew + N[(-0.4166666666666667 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < 2.2000000000000001e185

   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 96.7

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

   5. Applied rewrites 96.7%

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

      1. sin-atan N/A

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

      2. lower-/.f64 N/A

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

   7. Applied rewrites 32.9%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{eh}\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 42.4%

       \[\leadsto \left|\color{blue}{eh}\right| \]

   if 2.2000000000000001e185 < ew

   1. Initial program 99.9%

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

      1. associate-*l* N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-tan.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

      1. cos-atan N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-tan.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-tan.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in eh around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

   9. Applied rewrites 73.4%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \sin t, \left(eh \cdot eh\right) \cdot \mathsf{fma}\left(-0.5, \frac{{\cos t}^{2}}{ew \cdot \sin t}, \frac{{\cos t}^{2}}{ew \cdot \sin t}\right)\right)}\right| \]

   10. Taylor expanded in t around 0

       \[\leadsto \left|\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{-5}{12} \cdot \frac{{eh}^{2}}{ew}\right)}{\color{blue}{t}}\right| \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left|\frac{\frac{1}{2} \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(ew + \frac{-5}{12} \cdot \frac{{eh}^{2}}{ew}\right)}{t}\right| \]

   12. Applied rewrites 36.6%

       \[\leadsto \left|\frac{\mathsf{fma}\left(0.5, \frac{eh \cdot eh}{ew}, \left(t \cdot t\right) \cdot \left(ew + -0.4166666666666667 \cdot \frac{eh \cdot eh}{ew}\right)\right)}{\color{blue}{t}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 42.7% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[eh], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
2. 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 96.9

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

5. Applied rewrites 96.9%

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

   1. sin-atan N/A

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

   2. lower-/.f64 N/A

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

7. Applied rewrites 36.0%

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

8. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{eh}\right| \]
9. Step-by-step derivation

10. Applied rewrites 39.9%

    \[\leadsto \left|\color{blue}{eh}\right| \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025044 
(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))))))))