Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.6s

Alternatives: 9

Speedup: 1.0×

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

real(8) function code(eh, ew, t)
    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

real(8) function code(eh, ew, t)
    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, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 3: 96.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;ew \leq -9.6 \cdot 10^{-129} \lor \neg \left(ew \leq 7.5 \cdot 10^{-154}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\tanh \left(\frac{eh \cdot t}{ew}\right)}{-ew}, \sin t \cdot eh, \cos \tan^{-1} \left(eh \cdot t\_1\right) \cdot \left(-\cos t\right)\right) \cdot \left(-ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(t\_1 \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= ew -9.6e-129) (not (<= ew 7.5e-154)))
     (fabs
      (*
       (fma
        (/ (tanh (/ (* eh t) ew)) (- ew))
        (* (sin t) eh)
        (* (cos (atan (* eh t_1))) (- (cos t))))
       (- ew)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (* (* (cos (atan (* t_1 eh))) 1.0) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((ew <= -9.6e-129) || !(ew <= 7.5e-154)) {
		tmp = fabs((fma((tanh(((eh * t) / ew)) / -ew), (sin(t) * eh), (cos(atan((eh * t_1))) * -cos(t))) * -ew));
	} else {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan((t_1 * eh))) * 1.0) * ew)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((ew <= -9.6e-129) || !(ew <= 7.5e-154))
		tmp = abs(Float64(fma(Float64(tanh(Float64(Float64(eh * t) / ew)) / Float64(-ew)), Float64(sin(t) * eh), Float64(cos(atan(Float64(eh * t_1))) * Float64(-cos(t)))) * Float64(-ew)));
	else
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(Float64(t_1 * eh))) * 1.0) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[ew, -9.6e-129], N[Not[LessEqual[ew, 7.5e-154]], $MachinePrecision]], N[Abs[N[(N[(N[(N[Tanh[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision] / (-ew)), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(eh * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * (-ew)), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[ArcTan[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -9.59999999999999954e-129 or 7.5e-154 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

   7. Applied rewrites 98.2%

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

   9. Applied rewrites 98.2%

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

   10. Taylor expanded in t around 0

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

   12. Applied rewrites 97.7%

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

   if -9.59999999999999954e-129 < ew < 7.5e-154

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 99.8

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 95.4%

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

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.6 \cdot 10^{-129} \lor \neg \left(ew \leq 7.5 \cdot 10^{-154}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\tanh \left(\frac{eh \cdot t}{ew}\right)}{-ew}, \sin t \cdot eh, \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right)\right) \cdot \left(-ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ \mathbf{if}\;eh \leq -1.75 \cdot 10^{+131} \lor \neg \left(eh \leq 2.45 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} t\_2 \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\sin t \cdot t\_1\right) \cdot eh, eh, \cos t \cdot ew\right)}{\cosh \sinh^{-1} t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* t_1 eh)))
   (if (or (<= eh -1.75e+131) (not (<= eh 2.45e+64)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (* (* (cos (atan t_2)) 1.0) ew)))
     (fabs
      (/ (fma (* (* (sin t) t_1) eh) eh (* (cos t) ew)) (cosh (asinh t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double tmp;
	if ((eh <= -1.75e+131) || !(eh <= 2.45e+64)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(t_2)) * 1.0) * ew)));
	} else {
		tmp = fabs((fma(((sin(t) * t_1) * eh), eh, (cos(t) * ew)) / cosh(asinh(t_2))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	tmp = 0.0
	if ((eh <= -1.75e+131) || !(eh <= 2.45e+64))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(t_2)) * 1.0) * ew)));
	else
		tmp = abs(Float64(fma(Float64(Float64(sin(t) * t_1) * eh), eh, Float64(cos(t) * ew)) / cosh(asinh(t_2))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * eh), $MachinePrecision]}, If[Or[LessEqual[eh, -1.75e+131], N[Not[LessEqual[eh, 2.45e+64]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * t$95$1), $MachinePrecision] * eh), $MachinePrecision] * eh + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.7499999999999999e131 or 2.4500000000000001e64 < eh

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 95.3%

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

   if -1.7499999999999999e131 < eh < 2.4500000000000001e64

   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

   4. Applied rewrites 94.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. unpow1 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-fma.f64 N/A

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

   6. Applied rewrites 95.8%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\left(\sin t \cdot \frac{\tan t}{ew}\right) \cdot eh, eh, \cos t \cdot ew\right)}}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.75 \cdot 10^{+131} \lor \neg \left(eh \leq 2.45 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\left(\sin t \cdot \frac{\tan t}{ew}\right) \cdot eh, eh, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ \mathbf{if}\;eh \leq -1.8 \cdot 10^{+132} \lor \neg \left(eh \leq 2.45 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} t\_2 \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot t\_1\right) \cdot eh\right) \cdot \sin t\right)}{\cosh \sinh^{-1} t\_2}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* t_1 eh)))
   (if (or (<= eh -1.8e+132) (not (<= eh 2.45e+64)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
       (* (* (cos (atan t_2)) 1.0) ew)))
     (fabs
      (/ (fma (cos t) ew (* (* (* eh t_1) eh) (sin t))) (cosh (asinh t_2)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double tmp;
	if ((eh <= -1.8e+132) || !(eh <= 2.45e+64)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(t_2)) * 1.0) * ew)));
	} else {
		tmp = fabs((fma(cos(t), ew, (((eh * t_1) * eh) * sin(t))) / cosh(asinh(t_2))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	tmp = 0.0
	if ((eh <= -1.8e+132) || !(eh <= 2.45e+64))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(t_2)) * 1.0) * ew)));
	else
		tmp = abs(Float64(fma(cos(t), ew, Float64(Float64(Float64(eh * t_1) * eh) * sin(t))) / cosh(asinh(t_2))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * eh), $MachinePrecision]}, If[Or[LessEqual[eh, -1.8e+132], N[Not[LessEqual[eh, 2.45e+64]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(N[(eh * t$95$1), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.80000000000000008e132 or 2.4500000000000001e64 < eh

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 99.3

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

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 95.3%

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

   if -1.80000000000000008e132 < eh < 2.4500000000000001e64

   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

   4. Applied rewrites 94.2%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 94.2

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 95.8

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 95.8

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

   6. Applied rewrites 95.8%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{+132} \lor \neg \left(eh \leq 2.45 \cdot 10^{+64}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\cos t, ew, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -8500000000 \lor \neg \left(eh \leq 5 \cdot 10^{+60}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -8500000000.0) (not (<= eh 5e+60)))
   (fabs
    (-
     (* (* eh (sin t)) (sin (atan (/ (* (- eh) t) ew))))
     (* (* (cos (atan (* (/ (tan t) ew) eh))) 1.0) ew)))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8500000000.0) || !(eh <= 5e+60)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * 1.0) * ew)));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-8500000000.0d0)) .or. (.not. (eh <= 5d+60))) then
        tmp = abs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * 1.0d0) * ew)))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -8500000000.0) || !(eh <= 5e+60)) {
		tmp = Math.abs((((eh * Math.sin(t)) * Math.sin(Math.atan(((-eh * t) / ew)))) - ((Math.cos(Math.atan(((Math.tan(t) / ew) * eh))) * 1.0) * ew)));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -8500000000.0) or not (eh <= 5e+60):
		tmp = math.fabs((((eh * math.sin(t)) * math.sin(math.atan(((-eh * t) / ew)))) - ((math.cos(math.atan(((math.tan(t) / ew) * eh))) * 1.0) * ew)))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -8500000000.0) || !(eh <= 5e+60))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-eh) * t) / ew)))) - Float64(Float64(cos(atan(Float64(Float64(tan(t) / ew) * eh))) * 1.0) * ew)));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -8500000000.0) || ~((eh <= 5e+60)))
		tmp = abs((((eh * sin(t)) * sin(atan(((-eh * t) / ew)))) - ((cos(atan(((tan(t) / ew) * eh))) * 1.0) * ew)));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -8500000000.0], N[Not[LessEqual[eh, 5e+60]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.5e9 or 4.99999999999999975e60 < eh

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 91.8%

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

   if -8.5e9 < eh < 4.99999999999999975e60

   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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 85.5

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

   7. Applied rewrites 85.5%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8500000000 \lor \neg \left(eh \leq 5 \cdot 10^{+60}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot 1\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left(-eh\right) \cdot \sin t\\ \mathbf{if}\;eh \leq -8500000000000 \lor \neg \left(eh \leq 10^{+117}\right):\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{t\_2}{t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (* (- eh) (sin t))))
   (if (or (<= eh -8500000000000.0) (not (<= eh 1e+117)))
     (fabs (* t_2 (sin (atan (/ t_2 t_1)))))
     (fabs t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = -eh * sin(t);
	double tmp;
	if ((eh <= -8500000000000.0) || !(eh <= 1e+117)) {
		tmp = fabs((t_2 * sin(atan((t_2 / t_1)))));
	} else {
		tmp = fabs(t_1);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = -eh * sin(t)
    if ((eh <= (-8500000000000.0d0)) .or. (.not. (eh <= 1d+117))) then
        tmp = abs((t_2 * sin(atan((t_2 / t_1)))))
    else
        tmp = abs(t_1)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = -eh * Math.sin(t);
	double tmp;
	if ((eh <= -8500000000000.0) || !(eh <= 1e+117)) {
		tmp = Math.abs((t_2 * Math.sin(Math.atan((t_2 / t_1)))));
	} else {
		tmp = Math.abs(t_1);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = -eh * math.sin(t)
	tmp = 0
	if (eh <= -8500000000000.0) or not (eh <= 1e+117):
		tmp = math.fabs((t_2 * math.sin(math.atan((t_2 / t_1)))))
	else:
		tmp = math.fabs(t_1)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(Float64(-eh) * sin(t))
	tmp = 0.0
	if ((eh <= -8500000000000.0) || !(eh <= 1e+117))
		tmp = abs(Float64(t_2 * sin(atan(Float64(t_2 / t_1)))));
	else
		tmp = abs(t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = -eh * sin(t);
	tmp = 0.0;
	if ((eh <= -8500000000000.0) || ~((eh <= 1e+117)))
		tmp = abs((t_2 * sin(atan((t_2 / t_1)))));
	else
		tmp = abs(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[eh, -8500000000000.0], N[Not[LessEqual[eh, 1e+117]], $MachinePrecision]], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(t$95$2 / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[t$95$1], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.5e12 or 1.00000000000000005e117 < eh

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if -8.5e12 < eh < 1.00000000000000005e117

   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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in eh around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 84.3

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

   7. Applied rewrites 84.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8500000000000 \lor \neg \left(eh \leq 10^{+117}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{ew \cdot \cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 69.1%

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

5. Taylor expanded in eh around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 61.2

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

7. Applied rewrites 61.2%

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

Alternative 9: 42.6% accurate, 107.8× speedup

Math

\[\begin{array}{l} \\ \left|1 \cdot ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* 1.0 ew)))

C

double code(double eh, double ew, double t) {
	return fabs((1.0 * ew));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((1.0d0 * ew))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((1.0 * ew));
}

Python

def code(eh, ew, t):
	return math.fabs((1.0 * ew))

Julia

function code(eh, ew, t)
	return abs(Float64(1.0 * ew))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((1.0 * ew));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(1.0 * 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

4. Applied rewrites 69.1%

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

5. Taylor expanded in t around 0

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

   1. lower-+.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 30.7

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

7. Applied rewrites 30.7%

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

8. Taylor expanded in ew around inf

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

10. Applied rewrites 35.8%

    \[\leadsto \left|\mathsf{fma}\left(t \cdot t, -0.5, 1\right) \cdot \color{blue}{ew}\right| \]

11. Taylor expanded in t around 0

    \[\leadsto \left|1 \cdot ew\right| \]
12. Step-by-step derivation

13. Applied rewrites 39.6%

    \[\leadsto \left|1 \cdot ew\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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)))))))