Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.2s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-+.f64 N/A

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

8. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / tan(t))
	return abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(t * t) * eh), eh) / ew) / t)))) + Float64(Float64(ew * sin(t)) * Float64(1.0 / sqrt(Float64(1.0 + Float64(t_1 * t_1)))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. lift-tan.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. lift-tan.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. lower-+.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in t around 0

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

    1. lift-/.f64 N/A

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

    2. lift-*.f64 99.2

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

11. Applied rewrites 99.2%

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

Alternative 4: 98.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(t * t) * eh), eh) / ew) / t)))) + Float64(Float64(ew * sin(t)) * cos(atan(Float64(eh / Float64(ew * t)))))))
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. associate-*r/ N/A

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

   3. div-add-rev N/A

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

   4. lower-/.f64 N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 99.2

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

8. Applied rewrites 99.2%

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

9. Final simplification 99.2%

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

Alternative 5: 98.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. lift-tan.f64 N/A

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

   6. cos-atan N/A

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

   7. lower-/.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-tan.f64 N/A

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

   12. lift-/.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift-tan.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in eh around 0

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 98.6

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

9. Applied rewrites 98.6%

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

Alternative 6: 74.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{-21} \lor \neg \left(ew \leq 21500000000\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -9.5e-21) (not (<= ew 21500000000.0)))
   (fabs (* ew (sin t)))
   (fabs
    (*
     (* (cos t) eh)
     (tanh (asinh (* (/ (+ 1.0 (* -0.5 (* t t))) ew) (/ eh (sin t)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -9.5e-21) || !(ew <= 21500000000.0)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(((cos(t) * eh) * tanh(asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / sin(t)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -9.5e-21) or not (ew <= 21500000000.0):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(((math.cos(t) * eh) * math.tanh(math.asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / math.sin(t)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -9.5e-21) || !(ew <= 21500000000.0))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(t * t))) / ew) * Float64(eh / sin(t)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -9.5e-21) || ~((ew <= 21500000000.0)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(((cos(t) * eh) * tanh(asinh((((1.0 + (-0.5 * (t * t))) / ew) * (eh / sin(t)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -9.5e-21], N[Not[LessEqual[ew, 21500000000.0]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(N[(1.0 + N[(-0.5 * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -9.4999999999999994e-21 or 2.15e10 < ew

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 72.9

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

   7. Applied rewrites 72.9%

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

   if -9.4999999999999994e-21 < ew < 2.15e10

   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 eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in t around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 83.4

         \[\leadsto \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right| \]

   8. Applied rewrites 83.4%

      \[\leadsto \left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{-21} \lor \neg \left(ew \leq 21500000000\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{1 + -0.5 \cdot \left(t \cdot t\right)}{ew} \cdot \frac{eh}{\sin t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.7 \cdot 10^{-23} \lor \neg \left(ew \leq 14500000000\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.7e-23) (not (<= ew 14500000000.0)))
   (fabs (* ew (sin t)))
   (fabs (* (* (cos t) eh) (tanh (asinh (* (/ (cos t) ew) (/ eh t))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.7e-23) || !(ew <= 14500000000.0)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(((cos(t) * eh) * tanh(asinh(((cos(t) / ew) * (eh / t))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.7e-23) or not (ew <= 14500000000.0):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(((math.cos(t) * eh) * math.tanh(math.asinh(((math.cos(t) / ew) * (eh / t))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.7e-23) || !(ew <= 14500000000.0))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(Float64(Float64(cos(t) * eh) * tanh(asinh(Float64(Float64(cos(t) / ew) * Float64(eh / t))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.7e-23) || ~((ew <= 14500000000.0)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(((cos(t) * eh) * tanh(asinh(((cos(t) / ew) * (eh / t))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.7e-23], N[Not[LessEqual[ew, 14500000000.0]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Tanh[N[ArcSinh[N[(N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.69999999999999985e-23 or 1.45e10 < ew

   1. Initial program 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if -2.69999999999999985e-23 < ew < 1.45e10

   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 eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. sin-atan N/A

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

      7. tanh-asinh-rev N/A

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

      8. lower-tanh.f64 N/A

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

      9. lower-asinh.f64 N/A

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

      10. *-commutative N/A

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

      11. times-frac N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. lower-/.f64 N/A

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.5%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.7 \cdot 10^{-23} \lor \neg \left(ew \leq 14500000000\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \tanh \sinh^{-1} \left(\frac{\cos t}{ew} \cdot \frac{eh}{t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-7} \lor \neg \left(t \leq 5.5 \cdot 10^{-27}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -9e-7) (not (<= t 5.5e-27))) (fabs (* ew (sin t))) (fabs eh)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -9e-7) || !(t <= 5.5e-27)) {
		tmp = fabs((ew * sin(t)));
	} else {
		tmp = fabs(eh);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-9d-7)) .or. (.not. (t <= 5.5d-27))) then
        tmp = abs((ew * sin(t)))
    else
        tmp = abs(eh)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -9e-7) || !(t <= 5.5e-27)) {
		tmp = Math.abs((ew * Math.sin(t)));
	} else {
		tmp = Math.abs(eh);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -9e-7) or not (t <= 5.5e-27):
		tmp = math.fabs((ew * math.sin(t)))
	else:
		tmp = math.fabs(eh)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -9e-7) || !(t <= 5.5e-27))
		tmp = abs(Float64(ew * sin(t)));
	else
		tmp = abs(eh);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -9e-7) || ~((t <= 5.5e-27)))
		tmp = abs((ew * sin(t)));
	else
		tmp = abs(eh);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -9e-7], N[Not[LessEqual[t, 5.5e-27]], $MachinePrecision]], N[Abs[N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[eh], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -8.99999999999999959e-7 or 5.5000000000000002e-27 < t

   1. Initial program 99.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. lower-/.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. lift-tan.f64 N/A

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

      14. lift-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 61.0

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

   7. Applied rewrites 61.0%

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

   if -8.99999999999999959e-7 < t < 5.5000000000000002e-27

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.0

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

   8. Applied rewrites 76.0%

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

      1. lift-tanh.f64 N/A

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

      2. lift-asinh.f64 N/A

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

      3. tanh-asinh N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

   10. Applied rewrites 19.2%

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

   11. Taylor expanded in eh around inf

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

   13. Applied rewrites 76.2%

       \[\leadsto \left|eh\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{-7} \lor \neg \left(t \leq 5.5 \cdot 10^{-27}\right):\\ \;\;\;\;\left|ew \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.4% accurate, 290.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 42.1%

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

6. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 40.4

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

8. Applied rewrites 40.4%

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

   1. lift-tanh.f64 N/A

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

   2. lift-asinh.f64 N/A

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

   3. tanh-asinh N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

10. Applied rewrites 12.6%

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

11. Taylor expanded in eh around inf

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

13. Applied rewrites 42.6%

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

Reproduce

herbie shell --seed 2025073 
(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))))))))