Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 8.4s

Alternatives: 9

Speedup: N/A×

Specification

Math

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

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

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

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

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

FPCore

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

C

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

Python

def code(eh, ew, t):
	t_1 = eh / (math.tan(t) * ew)
	return math.fabs((((math.sin(t) * ew) / math.sqrt((math.pow(t_1, 2.0) - -1.0))) + (math.tanh(math.asinh(t_1)) * (math.cos(t) * eh))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.1% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((((sin(t) / sqrt(((((eh / (ew * t)) * eh) / (ew * t)) - (-1.0d0)))) * ew) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((Math.sin(t) / Math.sqrt(((((eh / (ew * t)) * eh) / (ew * t)) - -1.0))) * ew) + ((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((math.sin(t) / math.sqrt(((((eh / (ew * t)) * eh) / (ew * t)) - -1.0))) * ew) + ((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t)))))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((sin(t) / sqrt(((((eh / (ew * t)) * eh) / (ew * t)) - -1.0))) * ew) + ((eh * cos(t)) * sin(atan(((eh / ew) / tan(t)))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 99.1%

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

6. Applied rewrites 99.1%

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

   1. lift-pow.f64 N/A

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

   2. unpow2 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.1%

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

8. Applied rewrites 99.1%

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

Alternative 3: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in t around 0

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

   1. lower-*.f64 99.1%

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

6. Applied rewrites 99.1%

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

7. Taylor expanded in eh around 0

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

   1. lower-sin.f64 98.6%

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

9. Applied rewrites 98.6%

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

Alternative 4: 68.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := \left|\frac{t\_1 \cdot \left(\cos t \cdot eh\right) + \sin t \cdot ew}{\sqrt{{t\_1}^{2} - -1}}\right|\\ \mathbf{if}\;t \leq -3 \cdot 10^{-59}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;\left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1 (/ eh (* ew t)))
       (t_2
        (fabs
         (/
          (+ (* t_1 (* (cos t) eh)) (* (sin t) ew))
          (sqrt (- (pow t_1 2.0) -1.0))))))
  (if (<= t -3e-59)
    t_2
    (if (<= t 1.65e-109)
      (fabs (* eh (/ (* (/ eh (fabs eh)) (fabs ew)) ew)))
      t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = fabs((((t_1 * (cos(t) * eh)) + (sin(t) * ew)) / sqrt((pow(t_1, 2.0) - -1.0))));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_2;
	} else if (t <= 1.65e-109) {
		tmp = fabs((eh * (((eh / fabs(eh)) * fabs(ew)) / ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = eh / (ew * t)
    t_2 = abs((((t_1 * (cos(t) * eh)) + (sin(t) * ew)) / sqrt(((t_1 ** 2.0d0) - (-1.0d0)))))
    if (t <= (-3d-59)) then
        tmp = t_2
    else if (t <= 1.65d-109) then
        tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = Math.abs((((t_1 * (Math.cos(t) * eh)) + (Math.sin(t) * ew)) / Math.sqrt((Math.pow(t_1, 2.0) - -1.0))));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_2;
	} else if (t <= 1.65e-109) {
		tmp = Math.abs((eh * (((eh / Math.abs(eh)) * Math.abs(ew)) / ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / (ew * t)
	t_2 = math.fabs((((t_1 * (math.cos(t) * eh)) + (math.sin(t) * ew)) / math.sqrt((math.pow(t_1, 2.0) - -1.0))))
	tmp = 0
	if t <= -3e-59:
		tmp = t_2
	elif t <= 1.65e-109:
		tmp = math.fabs((eh * (((eh / math.fabs(eh)) * math.fabs(ew)) / ew)))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = abs(Float64(Float64(Float64(t_1 * Float64(cos(t) * eh)) + Float64(sin(t) * ew)) / sqrt(Float64((t_1 ^ 2.0) - -1.0))))
	tmp = 0.0
	if (t <= -3e-59)
		tmp = t_2;
	elseif (t <= 1.65e-109)
		tmp = abs(Float64(eh * Float64(Float64(Float64(eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / (ew * t);
	t_2 = abs((((t_1 * (cos(t) * eh)) + (sin(t) * ew)) / sqrt(((t_1 ^ 2.0) - -1.0))));
	tmp = 0.0;
	if (t <= -3e-59)
		tmp = t_2;
	elseif (t <= 1.65e-109)
		tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(t$95$1 * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3e-59], t$95$2, If[LessEqual[t, 1.65e-109], N[Abs[N[(eh * N[(N[(N[(eh / N[Abs[eh], $MachinePrecision]), $MachinePrecision] * N[Abs[ew], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0000000000000001e-59 or 1.65e-109 < t

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

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

      1. lower-*.f64 48.6%

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

   6. Applied rewrites 48.6%

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

   7. Taylor expanded in t around 0

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

      1. lower-*.f64 55.0%

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

   9. Applied rewrites 55.0%

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

   if -3.0000000000000001e-59 < t < 1.65e-109

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      5. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. lower-pow.f64 11.3%

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. associate-/l* N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      6. lower-/.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      8. *-commutative N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      9. lower-*.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      12. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      13. pow2 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. rem-sqrt-square N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      20. lower-fabs.f64 35.1%

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   8. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-fabs.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      7. fabs-div N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\frac{\left|eh\right|}{\left|ew\right|}}}{ew}\right| \]

      8. associate-/r/ N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      9. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      11. lower-fabs.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      12. lower-fabs.f64 42.6%

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   10. Applied rewrites 42.6%

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

Alternative 5: 60.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_1 := \left|\frac{\frac{eh}{\tan t \cdot ew} \cdot \left(\cos t \cdot eh\right) + \sin t \cdot ew}{\sqrt{1}}\right|\\ \mathbf{if}\;t \leq -3 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;\left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1
        (fabs
         (/
          (+ (* (/ eh (* (tan t) ew)) (* (cos t) eh)) (* (sin t) ew))
          (sqrt 1.0)))))
  (if (<= t -3e-59)
    t_1
    (if (<= t 1.65e-109)
      (fabs (* eh (/ (* (/ eh (fabs eh)) (fabs ew)) ew)))
      t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((((eh / (tan(t) * ew)) * (cos(t) * eh)) + (sin(t) * ew)) / sqrt(1.0)));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_1;
	} else if (t <= 1.65e-109) {
		tmp = fabs((eh * (((eh / fabs(eh)) * fabs(ew)) / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((((eh / (tan(t) * ew)) * (cos(t) * eh)) + (sin(t) * ew)) / sqrt(1.0d0)))
    if (t <= (-3d-59)) then
        tmp = t_1
    else if (t <= 1.65d-109) then
        tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((((eh / (Math.tan(t) * ew)) * (Math.cos(t) * eh)) + (Math.sin(t) * ew)) / Math.sqrt(1.0)));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_1;
	} else if (t <= 1.65e-109) {
		tmp = Math.abs((eh * (((eh / Math.abs(eh)) * Math.abs(ew)) / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((((eh / (math.tan(t) * ew)) * (math.cos(t) * eh)) + (math.sin(t) * ew)) / math.sqrt(1.0)))
	tmp = 0
	if t <= -3e-59:
		tmp = t_1
	elif t <= 1.65e-109:
		tmp = math.fabs((eh * (((eh / math.fabs(eh)) * math.fabs(ew)) / ew)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(Float64(eh / Float64(tan(t) * ew)) * Float64(cos(t) * eh)) + Float64(sin(t) * ew)) / sqrt(1.0)))
	tmp = 0.0
	if (t <= -3e-59)
		tmp = t_1;
	elseif (t <= 1.65e-109)
		tmp = abs(Float64(eh * Float64(Float64(Float64(eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((((eh / (tan(t) * ew)) * (cos(t) * eh)) + (sin(t) * ew)) / sqrt(1.0)));
	tmp = 0.0;
	if (t <= -3e-59)
		tmp = t_1;
	elseif (t <= 1.65e-109)
		tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[(N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3e-59], t$95$1, If[LessEqual[t, 1.65e-109], N[Abs[N[(eh * N[(N[(N[(eh / N[Abs[eh], $MachinePrecision]), $MachinePrecision] * N[Abs[ew], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0000000000000001e-59 or 1.65e-109 < t

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in eh around 0

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

   6. Applied rewrites 42.6%

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

   if -3.0000000000000001e-59 < t < 1.65e-109

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      5. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. lower-pow.f64 11.3%

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. associate-/l* N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      6. lower-/.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      8. *-commutative N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      9. lower-*.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      12. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      13. pow2 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. rem-sqrt-square N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      20. lower-fabs.f64 35.1%

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   8. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-fabs.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      7. fabs-div N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\frac{\left|eh\right|}{\left|ew\right|}}}{ew}\right| \]

      8. associate-/r/ N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      9. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      11. lower-fabs.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      12. lower-fabs.f64 42.6%

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   10. Applied rewrites 42.6%

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

Alternative 6: 53.5% accurate, 6.4× speedup

Math

\[\begin{array}{l} t_1 := \left|eh \cdot \frac{ew \cdot \sin t}{eh}\right|\\ \mathbf{if}\;t \leq -3 \cdot 10^{-59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.65 \cdot 10^{-109}:\\ \;\;\;\;\left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (let* ((t_1 (fabs (* eh (/ (* ew (sin t)) eh)))))
  (if (<= t -3e-59)
    t_1
    (if (<= t 1.65e-109)
      (fabs (* eh (/ (* (/ eh (fabs eh)) (fabs ew)) ew)))
      t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * ((ew * sin(t)) / eh)));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_1;
	} else if (t <= 1.65e-109) {
		tmp = fabs((eh * (((eh / fabs(eh)) * fabs(ew)) / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * ((ew * sin(t)) / eh)))
    if (t <= (-3d-59)) then
        tmp = t_1
    else if (t <= 1.65d-109) then
        tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * ((ew * Math.sin(t)) / eh)));
	double tmp;
	if (t <= -3e-59) {
		tmp = t_1;
	} else if (t <= 1.65e-109) {
		tmp = Math.abs((eh * (((eh / Math.abs(eh)) * Math.abs(ew)) / ew)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * ((ew * math.sin(t)) / eh)))
	tmp = 0
	if t <= -3e-59:
		tmp = t_1
	elif t <= 1.65e-109:
		tmp = math.fabs((eh * (((eh / math.fabs(eh)) * math.fabs(ew)) / ew)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * Float64(Float64(ew * sin(t)) / eh)))
	tmp = 0.0
	if (t <= -3e-59)
		tmp = t_1;
	elseif (t <= 1.65e-109)
		tmp = abs(Float64(eh * Float64(Float64(Float64(eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * ((ew * sin(t)) / eh)));
	tmp = 0.0;
	if (t <= -3e-59)
		tmp = t_1;
	elseif (t <= 1.65e-109)
		tmp = abs((eh * (((eh / abs(eh)) * abs(ew)) / ew)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3e-59], t$95$1, If[LessEqual[t, 1.65e-109], N[Abs[N[(eh * N[(N[(N[(eh / N[Abs[eh], $MachinePrecision]), $MachinePrecision] * N[Abs[ew], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.0000000000000001e-59 or 1.65e-109 < t

   1. Initial program 99.8%

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

   2. Taylor expanded in eh around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

   4. Applied rewrites 92.0%

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

   6. Applied rewrites 35.2%

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

   7. Taylor expanded in eh around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \left|eh \cdot \frac{ew \cdot \sin t}{eh}\right| \]

      2. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{ew \cdot \sin t}{eh}\right| \]

      3. lower-sin.f64 34.0%

         \[\leadsto \left|eh \cdot \frac{ew \cdot \sin t}{eh}\right| \]

   9. Applied rewrites 34.0%

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

   if -3.0000000000000001e-59 < t < 1.65e-109

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      5. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. lower-pow.f64 11.3%

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. associate-/l* N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      6. lower-/.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      8. *-commutative N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      9. lower-*.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      12. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      13. pow2 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. rem-sqrt-square N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      20. lower-fabs.f64 35.1%

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   8. Applied rewrites 35.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lift-fabs.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      6. lift-/.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

      7. fabs-div N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\frac{\left|eh\right|}{\left|ew\right|}}}{ew}\right| \]

      8. associate-/r/ N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      9. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      10. lower-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      11. lower-fabs.f64 N/A

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

      12. lower-fabs.f64 42.6%

          \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   10. Applied rewrites 42.6%

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

Alternative 7: 42.6% accurate, 22.3× speedup

Math

\[\left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. sin-atan N/A

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

   8. associate-*l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-atan.f64 N/A

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

   12. cos-atan N/A

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

3. Applied rewrites 57.2%

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

4. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   2. lower-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   5. lower-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. lower-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   7. lower-pow.f64 11.3%

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

6. Applied rewrites 11.3%

   \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   2. lift-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   3. pow2 N/A

      \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   4. associate-/l* N/A

      \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   6. lower-/.f64 12.3%

      \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   8. *-commutative N/A

      \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

   9. lower-*.f64 12.3%

      \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   11. lift-/.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   12. lift-pow.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   13. pow2 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

   14. lift-pow.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

   15. unpow2 N/A

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

   16. times-frac N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. rem-sqrt-square N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   20. lower-fabs.f64 35.1%

       \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

8. Applied rewrites 35.1%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lift-fabs.f64 N/A

      \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

   6. lift-/.f64 N/A

      \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|\frac{eh}{ew}\right|}}{ew}\right| \]

   7. fabs-div N/A

      \[\leadsto \left|eh \cdot \frac{\frac{eh}{\frac{\left|eh\right|}{\left|ew\right|}}}{ew}\right| \]

   8. associate-/r/ N/A

      \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   9. lower-*.f64 N/A

      \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   10. lower-/.f64 N/A

       \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   11. lower-fabs.f64 N/A

       \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

   12. lower-fabs.f64 42.6%

       \[\leadsto \left|eh \cdot \frac{\frac{eh}{\left|eh\right|} \cdot \left|ew\right|}{ew}\right| \]

10. Applied rewrites 42.6%

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

Alternative 8: 36.1% accurate, 19.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;eh \leq -2 \cdot 10^{-13}:\\ \;\;\;\;\left|eh \cdot \frac{eh}{\frac{\left|eh\right| \cdot ew}{\left|ew\right|}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right|\\ \end{array} \]

FPCore

(FPCore (eh ew t)
  :precision binary64
  (if (<= eh -2e-13)
  (fabs (* eh (/ eh (/ (* (fabs eh) ew) (fabs ew)))))
  (fabs (* eh (/ eh (* (fabs (/ eh ew)) ew))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e-13) {
		tmp = fabs((eh * (eh / ((fabs(eh) * ew) / fabs(ew)))));
	} else {
		tmp = fabs((eh * (eh / (fabs((eh / ew)) * ew))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(eh, ew, t)
use fmin_fmax_functions
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-2d-13)) then
        tmp = abs((eh * (eh / ((abs(eh) * ew) / abs(ew)))))
    else
        tmp = abs((eh * (eh / (abs((eh / ew)) * ew))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -2e-13) {
		tmp = Math.abs((eh * (eh / ((Math.abs(eh) * ew) / Math.abs(ew)))));
	} else {
		tmp = Math.abs((eh * (eh / (Math.abs((eh / ew)) * ew))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -2e-13:
		tmp = math.fabs((eh * (eh / ((math.fabs(eh) * ew) / math.fabs(ew)))))
	else:
		tmp = math.fabs((eh * (eh / (math.fabs((eh / ew)) * ew))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -2e-13)
		tmp = abs(Float64(eh * Float64(eh / Float64(Float64(abs(eh) * ew) / abs(ew)))));
	else
		tmp = abs(Float64(eh * Float64(eh / Float64(abs(Float64(eh / ew)) * ew))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -2e-13)
		tmp = abs((eh * (eh / ((abs(eh) * ew) / abs(ew)))));
	else
		tmp = abs((eh * (eh / (abs((eh / ew)) * ew))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -2e-13], N[Abs[N[(eh * N[(eh / N[(N[(N[Abs[eh], $MachinePrecision] * ew), $MachinePrecision] / N[Abs[ew], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(eh / N[(N[Abs[N[(eh / ew), $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.0000000000000001e-13

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      5. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. lower-pow.f64 11.3%

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. associate-/l* N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      6. lower-/.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      8. *-commutative N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      9. lower-*.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      12. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      13. pow2 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. rem-sqrt-square N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      20. lower-fabs.f64 35.1%

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   8. Applied rewrites 35.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fabs.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      4. fabs-div N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\frac{\left|eh\right|}{\left|ew\right|} \cdot ew}\right| \]

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-fabs.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\frac{\left|eh\right| \cdot ew}{\left|ew\right|}}\right| \]

      9. lower-fabs.f64 31.2%

         \[\leadsto \left|eh \cdot \frac{eh}{\frac{\left|eh\right| \cdot ew}{\left|ew\right|}}\right| \]

   10. Applied rewrites 31.2%

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

   if -2.0000000000000001e-13 < eh

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-atan.f64 N/A

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

      7. sin-atan N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-atan.f64 N/A

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

      12. cos-atan N/A

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

   3. Applied rewrites 57.2%

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

   4. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      5. lower-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      6. lower-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      7. lower-pow.f64 11.3%

         \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. Applied rewrites 11.3%

      \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      3. pow2 N/A

         \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

      4. associate-/l* N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      5. lower-*.f64 N/A

         \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      6. lower-/.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      7. lift-*.f64 N/A

         \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

      8. *-commutative N/A

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      9. lower-*.f64 12.3%

         \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      11. lift-/.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      12. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

      13. pow2 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      14. lift-pow.f64 N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

      15. unpow2 N/A

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

      16. times-frac N/A

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

      17. lift-/.f64 N/A

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

      18. lift-/.f64 N/A

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

      19. rem-sqrt-square N/A

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

      20. lower-fabs.f64 35.1%

          \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   8. Applied rewrites 35.1%

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

Alternative 9: 35.1% accurate, 23.5× speedup

Math

\[\left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-sin.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. sin-atan N/A

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

   8. associate-*l/ N/A

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

   9. lift-*.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-atan.f64 N/A

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

   12. cos-atan N/A

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

3. Applied rewrites 57.2%

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

4. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   2. lower-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   5. lower-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   6. lower-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   7. lower-pow.f64 11.3%

      \[\leadsto \left|\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

6. Applied rewrites 11.3%

   \[\leadsto \left|\color{blue}{\frac{{eh}^{2}}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   2. lift-pow.f64 N/A

      \[\leadsto \left|\frac{{eh}^{2}}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   3. pow2 N/A

      \[\leadsto \left|\frac{eh \cdot eh}{\color{blue}{ew} \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}\right| \]

   4. associate-/l* N/A

      \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|eh \cdot \color{blue}{\frac{eh}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   6. lower-/.f64 12.3%

      \[\leadsto \left|eh \cdot \frac{eh}{\color{blue}{ew \cdot \sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   7. lift-*.f64 N/A

      \[\leadsto \left|eh \cdot \frac{eh}{ew \cdot \color{blue}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}}}}\right| \]

   8. *-commutative N/A

      \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

   9. lower-*.f64 12.3%

      \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot \color{blue}{ew}}\right| \]

   10. lift-sqrt.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   11. lift-/.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   12. lift-pow.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{{eh}^{2}}{{ew}^{2}}} \cdot ew}\right| \]

   13. pow2 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

   14. lift-pow.f64 N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\sqrt{\frac{eh \cdot eh}{{ew}^{2}}} \cdot ew}\right| \]

   15. unpow2 N/A

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

   16. times-frac N/A

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

   17. lift-/.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. rem-sqrt-square N/A

       \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

   20. lower-fabs.f64 35.1%

       \[\leadsto \left|eh \cdot \frac{eh}{\left|\frac{eh}{ew}\right| \cdot ew}\right| \]

8. Applied rewrites 35.1%

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

Reproduce

herbie shell --seed 2025258 
(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))))))))