Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.4s

Alternatives: 8

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

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 98.9

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 3: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -7.7 \cdot 10^{+136} \lor \neg \left(eh \leq 2.4 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(t\_1 \cdot eh\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot {\left({\left(eh \cdot t\_1\right)}^{2} + 1\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= eh -7.7e+136) (not (<= eh 2.4e+196)))
     (fabs
      (*
       (* (sin t) (- eh))
       (sin (atan (* (/ (- (sin t)) ew) (/ eh (cos t)))))))
     (*
      (fabs (+ (* (* (* t_1 eh) eh) (sin t)) (* (cos t) ew)))
      (pow (+ (pow (* eh t_1) 2.0) 1.0) -0.5)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((eh <= -7.7e+136) || !(eh <= 2.4e+196)) {
		tmp = fabs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))));
	} else {
		tmp = fabs(((((t_1 * eh) * eh) * sin(t)) + (cos(t) * ew))) * pow((pow((eh * t_1), 2.0) + 1.0), -0.5);
	}
	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) :: tmp
    t_1 = tan(t) / ew
    if ((eh <= (-7.7d+136)) .or. (.not. (eh <= 2.4d+196))) then
        tmp = abs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))))
    else
        tmp = abs(((((t_1 * eh) * eh) * sin(t)) + (cos(t) * ew))) * ((((eh * t_1) ** 2.0d0) + 1.0d0) ** (-0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) / ew;
	double tmp;
	if ((eh <= -7.7e+136) || !(eh <= 2.4e+196)) {
		tmp = Math.abs(((Math.sin(t) * -eh) * Math.sin(Math.atan(((-Math.sin(t) / ew) * (eh / Math.cos(t)))))));
	} else {
		tmp = Math.abs(((((t_1 * eh) * eh) * Math.sin(t)) + (Math.cos(t) * ew))) * Math.pow((Math.pow((eh * t_1), 2.0) + 1.0), -0.5);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.tan(t) / ew
	tmp = 0
	if (eh <= -7.7e+136) or not (eh <= 2.4e+196):
		tmp = math.fabs(((math.sin(t) * -eh) * math.sin(math.atan(((-math.sin(t) / ew) * (eh / math.cos(t)))))))
	else:
		tmp = math.fabs(((((t_1 * eh) * eh) * math.sin(t)) + (math.cos(t) * ew))) * math.pow((math.pow((eh * t_1), 2.0) + 1.0), -0.5)
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((eh <= -7.7e+136) || !(eh <= 2.4e+196))
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t)))))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(t_1 * eh) * eh) * sin(t)) + Float64(cos(t) * ew))) * (Float64((Float64(eh * t_1) ^ 2.0) + 1.0) ^ -0.5));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = tan(t) / ew;
	tmp = 0.0;
	if ((eh <= -7.7e+136) || ~((eh <= 2.4e+196)))
		tmp = abs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))));
	else
		tmp = abs(((((t_1 * eh) * eh) * sin(t)) + (cos(t) * ew))) * ((((eh * t_1) ^ 2.0) + 1.0) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -7.7e+136], N[Not[LessEqual[eh, 2.4e+196]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(t$95$1 * eh), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[Power[N[(eh * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.7000000000000003e136 or 2.4e196 < 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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-in N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 80.1%

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

   if -7.7000000000000003e136 < eh < 2.4e196

   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 99.8%

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

   6. Applied rewrites 79.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. inv-pow N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. pow2 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 91.5

          \[\leadsto \left|\cos t \cdot ew - \left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \left(-\sin t\right)\right| \cdot {\left({\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1\right)}^{\color{blue}{-0.5}} \]

   8. Applied rewrites 91.5%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.7 \cdot 10^{+136} \lor \neg \left(eh \leq 2.4 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\frac{\tan t}{ew} \cdot eh\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right| \cdot {\left({\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -1.2 \cdot 10^{+95} \lor \neg \left(eh \leq 2.1 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(eh \cdot t\_1\right) \cdot eh, \sin t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(t\_1 \cdot eh\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= eh -1.2e+95) (not (<= eh 2.1e+196)))
     (fabs
      (*
       (* (sin t) (- eh))
       (sin (atan (* (/ (- (sin t)) ew) (/ eh (cos t)))))))
     (*
      (fabs (fma (* (* eh t_1) eh) (sin t) (* (cos t) ew)))
      (cos (atan (* t_1 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((eh <= -1.2e+95) || !(eh <= 2.1e+196)) {
		tmp = fabs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))));
	} else {
		tmp = fabs(fma(((eh * t_1) * eh), sin(t), (cos(t) * ew))) * cos(atan((t_1 * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((eh <= -1.2e+95) || !(eh <= 2.1e+196))
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t)))))));
	else
		tmp = Float64(abs(fma(Float64(Float64(eh * t_1) * eh), sin(t), Float64(cos(t) * ew))) * cos(atan(Float64(t_1 * eh))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -1.2e+95], N[Not[LessEqual[eh, 2.1e+196]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(eh * t$95$1), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.2e95 or 2.10000000000000015e196 < 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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-in N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 78.0%

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

   if -1.2e95 < eh < 2.10000000000000015e196

   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 99.8%

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

   6. Applied rewrites 81.5%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. distribute-rgt-neg-out N/A

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

      6. remove-double-neg N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 81.5

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 81.5

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

   8. Applied rewrites 81.5%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.2 \cdot 10^{+95} \lor \neg \left(eh \leq 2.1 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh, \sin t, \cos t \cdot ew\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.2 \cdot 10^{+95} \lor \neg \left(eh \leq 2.1 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.2e+95) (not (<= eh 2.1e+196)))
   (fabs
    (* (* (sin t) (- eh)) (sin (atan (* (/ (- (sin t)) ew) (/ eh (cos t)))))))
   (fabs
    (/ (* (- ew) (cos t)) (pow (cos (atan (* (/ (tan t) ew) eh))) -1.0)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.2e+95) || !(eh <= 2.1e+196)) {
		tmp = fabs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))));
	} else {
		tmp = fabs(((-ew * cos(t)) / pow(cos(atan(((tan(t) / ew) * eh))), -1.0)));
	}
	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 <= (-1.2d+95)) .or. (.not. (eh <= 2.1d+196))) then
        tmp = abs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))))
    else
        tmp = abs(((-ew * cos(t)) / (cos(atan(((tan(t) / ew) * eh))) ** (-1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.2e+95) || !(eh <= 2.1e+196)) {
		tmp = Math.abs(((Math.sin(t) * -eh) * Math.sin(Math.atan(((-Math.sin(t) / ew) * (eh / Math.cos(t)))))));
	} else {
		tmp = Math.abs(((-ew * Math.cos(t)) / Math.pow(Math.cos(Math.atan(((Math.tan(t) / ew) * eh))), -1.0)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.2e+95) or not (eh <= 2.1e+196):
		tmp = math.fabs(((math.sin(t) * -eh) * math.sin(math.atan(((-math.sin(t) / ew) * (eh / math.cos(t)))))))
	else:
		tmp = math.fabs(((-ew * math.cos(t)) / math.pow(math.cos(math.atan(((math.tan(t) / ew) * eh))), -1.0)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.2e+95) || !(eh <= 2.1e+196))
		tmp = abs(Float64(Float64(sin(t) * Float64(-eh)) * sin(atan(Float64(Float64(Float64(-sin(t)) / ew) * Float64(eh / cos(t)))))));
	else
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / (cos(atan(Float64(Float64(tan(t) / ew) * eh))) ^ -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.2e+95) || ~((eh <= 2.1e+196)))
		tmp = abs(((sin(t) * -eh) * sin(atan(((-sin(t) / ew) * (eh / cos(t)))))));
	else
		tmp = abs(((-ew * cos(t)) / (cos(atan(((tan(t) / ew) * eh))) ^ -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.2e+95], N[Not[LessEqual[eh, 2.1e+196]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * (-eh)), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-N[Sin[t], $MachinePrecision]) / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.2e95 or 2.10000000000000015e196 < 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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

      15. distribute-lft-neg-in N/A

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 78.0%

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

   if -1.2e95 < eh < 2.10000000000000015e196

   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 81.5%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 79.8

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

   7. Applied rewrites 79.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.2 \cdot 10^{+95} \lor \neg \left(eh \leq 2.1 \cdot 10^{+196}\right):\\ \;\;\;\;\left|\left(\sin t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.7% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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)) / (cos(atan(((tan(t) / ew) * eh))) ** (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], -1.0], $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 68.6%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 66.7

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

7. Applied rewrites 66.7%

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

8. Final simplification 66.7%

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

Alternative 7: 61.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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((cos(t) * ew)) * cos(atan(((tan(t) / ew) * eh)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 68.6%

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

7. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 66.7

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

9. Applied rewrites 66.7%

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

Alternative 8: 42.4% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

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 / 1.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $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 t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 44.3%

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

10. Applied rewrites 43.4%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 45.6%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024317 
(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)))))))