Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 1.2min

Alternatives: 5

Speedup: N/A×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, N/A× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	return fabs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(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 * tan(t)) / -ew))
    code = abs((((eh * sin(t)) * sin(t_1)) - ((ew * cos(t)) * cos(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 98.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-eh}{ew} \cdot \tan t\\ t_2 := \cos \tan^{-1} t\_1\\ t_3 := \tanh \sinh^{-1} t\_1 \cdot \sin t\\ \mathbf{if}\;ew \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\left|\left(\frac{eh}{ew} \cdot \left(-\sin t\right) - \cos t \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right) \cdot \left(-ew\right)\right|\\ \mathbf{elif}\;ew \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;\left|\left(\frac{\left(\cos t \cdot ew\right) \cdot t\_2}{eh} - t\_3\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \frac{t\_3}{ew}, t\_2 \cdot \left(-\cos t\right)\right) \cdot \left(-ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (- eh) ew) (tan t)))
        (t_2 (cos (atan t_1)))
        (t_3 (* (tanh (asinh t_1)) (sin t))))
   (if (<= ew -5e+71)
     (fabs
      (*
       (-
        (* (/ eh ew) (- (sin t)))
        (* (cos t) (cos (atan (* (/ eh ew) (tan t))))))
       (- ew)))
     (if (<= ew 7.8e-100)
       (fabs (* (- (/ (* (* (cos t) ew) t_2) eh) t_3) eh))
       (fabs (* (fma eh (/ t_3 ew) (* t_2 (- (cos t)))) (- ew)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (-eh / ew) * tan(t);
	double t_2 = cos(atan(t_1));
	double t_3 = tanh(asinh(t_1)) * sin(t);
	double tmp;
	if (ew <= -5e+71) {
		tmp = fabs(((((eh / ew) * -sin(t)) - (cos(t) * cos(atan(((eh / ew) * tan(t)))))) * -ew));
	} else if (ew <= 7.8e-100) {
		tmp = fabs((((((cos(t) * ew) * t_2) / eh) - t_3) * eh));
	} else {
		tmp = fabs((fma(eh, (t_3 / ew), (t_2 * -cos(t))) * -ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(Float64(-eh) / ew) * tan(t))
	t_2 = cos(atan(t_1))
	t_3 = Float64(tanh(asinh(t_1)) * sin(t))
	tmp = 0.0
	if (ew <= -5e+71)
		tmp = abs(Float64(Float64(Float64(Float64(eh / ew) * Float64(-sin(t))) - Float64(cos(t) * cos(atan(Float64(Float64(eh / ew) * tan(t)))))) * Float64(-ew)));
	elseif (ew <= 7.8e-100)
		tmp = abs(Float64(Float64(Float64(Float64(Float64(cos(t) * ew) * t_2) / eh) - t_3) * eh));
	else
		tmp = abs(Float64(fma(eh, Float64(t_3 / ew), Float64(t_2 * Float64(-cos(t)))) * Float64(-ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tanh[N[ArcSinh[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -5e+71], N[Abs[N[(N[(N[(N[(eh / ew), $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] - N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-ew)), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 7.8e-100], N[Abs[N[(N[(N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * t$95$2), $MachinePrecision] / eh), $MachinePrecision] - t$95$3), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[(t$95$3 / ew), $MachinePrecision] + N[(t$95$2 * (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * (-ew)), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.99999999999999972e71

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 88.6%

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

   9. Taylor expanded in eh around inf

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

       1. lower-*.f64 N/A

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

       2. lift-sin.f64 99.9

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

   11. Applied rewrites 99.9%

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

   if -4.99999999999999972e71 < ew < 7.79999999999999955e-100

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if 7.79999999999999955e-100 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. *-commutative N/A

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\left|\left(\frac{eh}{ew} \cdot \left(-\sin t\right) - \cos t \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right) \cdot \left(-ew\right)\right|\\ \mathbf{elif}\;ew \leq 7.8 \cdot 10^{-100}:\\ \;\;\;\;\left|\left(\frac{\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right)}{eh} - \tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \frac{\tanh \sinh^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t}{ew}, \cos \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \left(-\cos t\right)\right) \cdot \left(-ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, N/A× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

5. Applied rewrites 90.3%

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

6. Final simplification 90.3%

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

Alternative 4: 86.8% accurate, N/A× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((((eh / ew) * -sin(t)) - (cos(t) * cos(atan(((eh / ew) * tan(t)))))) * -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 / ew) * -sin(t)) - (cos(t) * cos(atan(((eh / ew) * tan(t)))))) * -ew))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

5. Applied rewrites 90.3%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 60.5%

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

9. Taylor expanded in eh around inf

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

    1. lower-*.f64 N/A

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

    2. lift-sin.f64 84.0

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

11. Applied rewrites 84.0%

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

12. Final simplification 84.0%

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

Alternative 5: 28.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)\\ t_2 := e^{t\_1}\\ t_3 := e^{-t\_1}\\ \left|-\mathsf{fma}\left(-1, ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right), \frac{eh \cdot \left(\sin t \cdot \left(t\_2 - t\_3\right)\right)}{t\_2 + t\_3}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (log (* ew (* 0.5 (/ (cos t) (* eh (sin t)))))))
        (t_2 (exp t_1))
        (t_3 (exp (- t_1))))
   (fabs
    (-
     (fma
      -1.0
      (* ew (* (cos t) (cos (atan (* (/ eh ew) (tan t))))))
      (/ (* eh (* (sin t) (- t_2 t_3))) (+ t_2 t_3)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = log((ew * (0.5 * (cos(t) / (eh * sin(t))))));
	double t_2 = exp(t_1);
	double t_3 = exp(-t_1);
	return fabs(-fma(-1.0, (ew * (cos(t) * cos(atan(((eh / ew) * tan(t)))))), ((eh * (sin(t) * (t_2 - t_3))) / (t_2 + t_3))));
}

Julia

function code(eh, ew, t)
	t_1 = log(Float64(ew * Float64(0.5 * Float64(cos(t) / Float64(eh * sin(t))))))
	t_2 = exp(t_1)
	t_3 = exp(Float64(-t_1))
	return abs(Float64(-fma(-1.0, Float64(ew * Float64(cos(t) * cos(atan(Float64(Float64(eh / ew) * tan(t)))))), Float64(Float64(eh * Float64(sin(t) * Float64(t_2 - t_3))) / Float64(t_2 + t_3)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Log[N[(ew * N[(0.5 * N[(N[Cos[t], $MachinePrecision] / N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Exp[(-t$95$1)], $MachinePrecision]}, N[Abs[(-N[(-1.0 * N[(ew * N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(eh * N[(N[Sin[t], $MachinePrecision] * N[(t$95$2 - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$2 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]]]]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in ew around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. *-commutative N/A

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

5. Applied rewrites 90.3%

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

6. Taylor expanded in ew around 0

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

8. Applied rewrites 32.9%

   \[\leadsto \left|-\mathsf{fma}\left(-1, ew \cdot \left(\cos t \cdot \cos \left(-\tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right)\right), \frac{eh \cdot \left(\sin t \cdot \left(e^{\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)} - e^{-\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)}\right)\right)}{e^{\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)} + e^{-\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)}}\right)\right| \]

9. Final simplification 32.9%

   \[\leadsto \left|-\mathsf{fma}\left(-1, ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \tan t\right)\right), \frac{eh \cdot \left(\sin t \cdot \left(e^{\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)} - e^{-\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)}\right)\right)}{e^{\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)} + e^{-\log \left(ew \cdot \left(0.5 \cdot \frac{\cos t}{eh \cdot \sin t}\right)\right)}}\right)\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))