Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.5s

Alternatives: 14

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

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(tan(t) / ew) * eh))
	return abs(fma(Float64(sin(t_1) * sin(t)), eh, Float64(cos(t_1) * Float64(ew * cos(t)))))
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[Sin[t$95$1], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[Cos[t$95$1], $MachinePrecision] * N[(ew * N[Cos[t], $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%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.8%

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

Alternative 2: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := \sin t \cdot eh\\ t_3 := eh \cdot t\_1\\ \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(t\_3 \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} t\_3\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{\frac{t\_2}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_2 \cdot {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-1} + \cos t \cdot ew}{{\cos \tan^{-1} \left(t\_1 \cdot eh\right)}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)) (t_2 (* (sin t) eh)) (t_3 (* eh t_1)))
   (if (<= ew -5.5e-51)
     (fabs
      (* (fma (- ew) (cos t) (* (* t_3 eh) (- (sin t)))) (cos (atan t_3))))
     (if (<= ew 8e-215)
       (fabs (* t_2 (sin (atan (/ (/ t_2 ew) (cos t))))))
       (fabs
        (/
         (+ (* t_2 (pow (/ ew (* eh (tan t))) -1.0)) (* (cos t) ew))
         (pow (cos (atan (* t_1 eh))) -1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = sin(t) * eh;
	double t_3 = eh * t_1;
	double tmp;
	if (ew <= -5.5e-51) {
		tmp = fabs((fma(-ew, cos(t), ((t_3 * eh) * -sin(t))) * cos(atan(t_3))));
	} else if (ew <= 8e-215) {
		tmp = fabs((t_2 * sin(atan(((t_2 / ew) / cos(t))))));
	} else {
		tmp = fabs((((t_2 * pow((ew / (eh * tan(t))), -1.0)) + (cos(t) * ew)) / pow(cos(atan((t_1 * eh))), -1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(sin(t) * eh)
	t_3 = Float64(eh * t_1)
	tmp = 0.0
	if (ew <= -5.5e-51)
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(t_3 * eh) * Float64(-sin(t)))) * cos(atan(t_3))));
	elseif (ew <= 8e-215)
		tmp = abs(Float64(t_2 * sin(atan(Float64(Float64(t_2 / ew) / cos(t))))));
	else
		tmp = abs(Float64(Float64(Float64(t_2 * (Float64(ew / Float64(eh * tan(t))) ^ -1.0)) + Float64(cos(t) * ew)) / (cos(atan(Float64(t_1 * eh))) ^ -1.0)));
	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[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$3 = N[(eh * t$95$1), $MachinePrecision]}, If[LessEqual[ew, -5.5e-51], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[(N[(t$95$3 * eh), $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8e-215], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$2 * N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.4999999999999997e-51

   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 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lower-*.f64 90.4

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

   6. Applied rewrites 90.4%

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

   if -5.4999999999999997e-51 < ew < 8.00000000000000033e-215

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if 8.00000000000000033e-215 < 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

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-*.f64 N/A

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

      5. clear-num N/A

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

      6. lift-/.f64 N/A

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

      7. lower-/.f64 81.5

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 81.5

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

   6. Applied rewrites 81.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-1} + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

6. Applied rewrites 99.8%

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

Alternative 4: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ t_2 := t\_1 \cdot eh\\ t_3 := \sin t \cdot eh\\ t_4 := eh \cdot t\_1\\ \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(t\_4 \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} t\_4\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\left|t\_3 \cdot \sin \tan^{-1} \left(\frac{\frac{t\_3}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_3 \cdot t\_2 + \cos t \cdot ew}{{\cos \tan^{-1} t\_2}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew))
        (t_2 (* t_1 eh))
        (t_3 (* (sin t) eh))
        (t_4 (* eh t_1)))
   (if (<= ew -5.5e-51)
     (fabs
      (* (fma (- ew) (cos t) (* (* t_4 eh) (- (sin t)))) (cos (atan t_4))))
     (if (<= ew 8e-215)
       (fabs (* t_3 (sin (atan (/ (/ t_3 ew) (cos t))))))
       (fabs
        (/ (+ (* t_3 t_2) (* (cos t) ew)) (pow (cos (atan t_2)) -1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double t_2 = t_1 * eh;
	double t_3 = sin(t) * eh;
	double t_4 = eh * t_1;
	double tmp;
	if (ew <= -5.5e-51) {
		tmp = fabs((fma(-ew, cos(t), ((t_4 * eh) * -sin(t))) * cos(atan(t_4))));
	} else if (ew <= 8e-215) {
		tmp = fabs((t_3 * sin(atan(((t_3 / ew) / cos(t))))));
	} else {
		tmp = fabs((((t_3 * t_2) + (cos(t) * ew)) / pow(cos(atan(t_2)), -1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	t_2 = Float64(t_1 * eh)
	t_3 = Float64(sin(t) * eh)
	t_4 = Float64(eh * t_1)
	tmp = 0.0
	if (ew <= -5.5e-51)
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(t_4 * eh) * Float64(-sin(t)))) * cos(atan(t_4))));
	elseif (ew <= 8e-215)
		tmp = abs(Float64(t_3 * sin(atan(Float64(Float64(t_3 / ew) / cos(t))))));
	else
		tmp = abs(Float64(Float64(Float64(t_3 * t_2) + Float64(cos(t) * ew)) / (cos(atan(t_2)) ^ -1.0)));
	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]}, Block[{t$95$3 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$4 = N[(eh * t$95$1), $MachinePrecision]}, If[LessEqual[ew, -5.5e-51], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[(N[(t$95$4 * eh), $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$4], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8e-215], N[Abs[N[(t$95$3 * N[Sin[N[ArcTan[N[(N[(t$95$3 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(t$95$3 * t$95$2), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -5.4999999999999997e-51

   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 90.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lower-*.f64 90.4

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

   6. Applied rewrites 90.4%

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

   if -5.4999999999999997e-51 < ew < 8.00000000000000033e-215

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 74.2

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

   7. Applied rewrites 74.2%

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

   if 8.00000000000000033e-215 < 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

   4. Applied rewrites 81.5%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{-215}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. lower-/.f64 98.9

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

9. Applied rewrites 98.9%

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

Alternative 6: 74.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ t_2 := \sin t \cdot eh\\ \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51} \lor \neg \left(ew \leq 8 \cdot 10^{-215}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(t\_1 \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2 \cdot \sin \tan^{-1} \left(\frac{\frac{t\_2}{ew}}{\cos t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))) (t_2 (* (sin t) eh)))
   (if (or (<= ew -5.5e-51) (not (<= ew 8e-215)))
     (fabs
      (* (fma (- ew) (cos t) (* (* t_1 eh) (- (sin t)))) (cos (atan t_1))))
     (fabs (* t_2 (sin (atan (/ (/ t_2 ew) (cos t)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double t_2 = sin(t) * eh;
	double tmp;
	if ((ew <= -5.5e-51) || !(ew <= 8e-215)) {
		tmp = fabs((fma(-ew, cos(t), ((t_1 * eh) * -sin(t))) * cos(atan(t_1))));
	} else {
		tmp = fabs((t_2 * sin(atan(((t_2 / ew) / cos(t))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	t_2 = Float64(sin(t) * eh)
	tmp = 0.0
	if ((ew <= -5.5e-51) || !(ew <= 8e-215))
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(Float64(t_1 * eh) * Float64(-sin(t)))) * cos(atan(t_1))));
	else
		tmp = abs(Float64(t_2 * sin(atan(Float64(Float64(t_2 / ew) / cos(t))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[ew, -5.5e-51], N[Not[LessEqual[ew, 8e-215]], $MachinePrecision]], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[(N[(t$95$1 * eh), $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.4999999999999997e-51 or 8.00000000000000033e-215 < 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

   4. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. /-rgt-identity N/A

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

      5. lower-*.f64 84.9

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

   6. Applied rewrites 84.8%

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

   if -5.4999999999999997e-51 < ew < 8.00000000000000033e-215

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 74.2

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

   7. Applied rewrites 74.2%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.5 \cdot 10^{-51} \lor \neg \left(ew \leq 8 \cdot 10^{-215}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \left(-\sin t\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ {\left(\left|\frac{{ew}^{-1}}{{\left(\sqrt{{\left(\mathsf{fma}\left(-0.5, t \cdot t, 1\right) \cdot \frac{ew}{\sin t \cdot eh}\right)}^{-2} + 1}\right)}^{-1} \cdot \cos t}\right|\right)}^{-1} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (pow
  (fabs
   (/
    (pow ew -1.0)
    (*
     (pow
      (sqrt
       (+ (pow (* (fma -0.5 (* t t) 1.0) (/ ew (* (sin t) eh))) -2.0) 1.0))
      -1.0)
     (cos t))))
  -1.0))

C

double code(double eh, double ew, double t) {
	return pow(fabs((pow(ew, -1.0) / (pow(sqrt((pow((fma(-0.5, (t * t), 1.0) * (ew / (sin(t) * eh))), -2.0) + 1.0)), -1.0) * cos(t)))), -1.0);
}

Julia

function code(eh, ew, t)
	return abs(Float64((ew ^ -1.0) / Float64((sqrt(Float64((Float64(fma(-0.5, Float64(t * t), 1.0) * Float64(ew / Float64(sin(t) * eh))) ^ -2.0) + 1.0)) ^ -1.0) * cos(t)))) ^ -1.0
end

Wolfram

code[eh_, ew_, t_] := N[Power[N[Abs[N[(N[Power[ew, -1.0], $MachinePrecision] / N[(N[Power[N[Sqrt[N[(N[Power[N[(N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision] * N[(ew / N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $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 99.6%

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

5. Taylor expanded in eh around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-atan.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. lower-cos.f64 67.3

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

7. Applied rewrites 67.3%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 67.3%

    \[\leadsto \frac{1}{\left|\frac{\frac{1}{ew}}{\cos \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\mathsf{fma}\left(-0.5, t \cdot t, 1\right)}\right) \cdot \cos t}\right|} \]
11. Step-by-step derivation

12. Applied rewrites 67.2%

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

13. Final simplification 67.2%

    \[\leadsto {\left(\left|\frac{{ew}^{-1}}{{\left(\sqrt{{\left(\mathsf{fma}\left(-0.5, t \cdot t, 1\right) \cdot \frac{ew}{\sin t \cdot eh}\right)}^{-2} + 1}\right)}^{-1} \cdot \cos t}\right|\right)}^{-1} \]
14. Add Preprocessing

Alternative 8: 72.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cos t \cdot ew\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50}:\\ \;\;\;\;\left|t\_2 \cdot \cos \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right)\right|\\ \mathbf{elif}\;ew \leq 1.52 \cdot 10^{-161}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_2 + t\_1 \cdot \left(\frac{\tan t}{ew} \cdot eh\right)}{\frac{-1}{{\left({\left(\frac{t}{ew} \cdot eh\right)}^{2} + 1\right)}^{-0.5}}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (* (cos t) ew)))
   (if (<= ew -1.65e-50)
     (fabs (* t_2 (cos (atan (/ (/ t_1 ew) (cos t))))))
     (if (<= ew 1.52e-161)
       (fabs (* t_1 (sin (atan (* (/ eh (cos t)) (/ (sin t) ew))))))
       (fabs
        (/
         (+ t_2 (* t_1 (* (/ (tan t) ew) eh)))
         (/ -1.0 (pow (+ (pow (* (/ t ew) eh) 2.0) 1.0) -0.5))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cos(t) * ew;
	double tmp;
	if (ew <= -1.65e-50) {
		tmp = fabs((t_2 * cos(atan(((t_1 / ew) / cos(t))))));
	} else if (ew <= 1.52e-161) {
		tmp = fabs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))));
	} else {
		tmp = fabs(((t_2 + (t_1 * ((tan(t) / ew) * eh))) / (-1.0 / pow((pow(((t / ew) * eh), 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) :: t_2
    real(8) :: tmp
    t_1 = sin(t) * eh
    t_2 = cos(t) * ew
    if (ew <= (-1.65d-50)) then
        tmp = abs((t_2 * cos(atan(((t_1 / ew) / cos(t))))))
    else if (ew <= 1.52d-161) then
        tmp = abs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))))
    else
        tmp = abs(((t_2 + (t_1 * ((tan(t) / ew) * eh))) / ((-1.0d0) / (((((t / ew) * eh) ** 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.sin(t) * eh;
	double t_2 = Math.cos(t) * ew;
	double tmp;
	if (ew <= -1.65e-50) {
		tmp = Math.abs((t_2 * Math.cos(Math.atan(((t_1 / ew) / Math.cos(t))))));
	} else if (ew <= 1.52e-161) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan(((eh / Math.cos(t)) * (Math.sin(t) / ew))))));
	} else {
		tmp = Math.abs(((t_2 + (t_1 * ((Math.tan(t) / ew) * eh))) / (-1.0 / Math.pow((Math.pow(((t / ew) * eh), 2.0) + 1.0), -0.5))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.cos(t) * ew
	tmp = 0
	if ew <= -1.65e-50:
		tmp = math.fabs((t_2 * math.cos(math.atan(((t_1 / ew) / math.cos(t))))))
	elif ew <= 1.52e-161:
		tmp = math.fabs((t_1 * math.sin(math.atan(((eh / math.cos(t)) * (math.sin(t) / ew))))))
	else:
		tmp = math.fabs(((t_2 + (t_1 * ((math.tan(t) / ew) * eh))) / (-1.0 / math.pow((math.pow(((t / ew) * eh), 2.0) + 1.0), -0.5))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(cos(t) * ew)
	tmp = 0.0
	if (ew <= -1.65e-50)
		tmp = abs(Float64(t_2 * cos(atan(Float64(Float64(t_1 / ew) / cos(t))))));
	elseif (ew <= 1.52e-161)
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew))))));
	else
		tmp = abs(Float64(Float64(t_2 + Float64(t_1 * Float64(Float64(tan(t) / ew) * eh))) / Float64(-1.0 / (Float64((Float64(Float64(t / ew) * eh) ^ 2.0) + 1.0) ^ -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = cos(t) * ew;
	tmp = 0.0;
	if (ew <= -1.65e-50)
		tmp = abs((t_2 * cos(atan(((t_1 / ew) / cos(t))))));
	elseif (ew <= 1.52e-161)
		tmp = abs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))));
	else
		tmp = abs(((t_2 + (t_1 * ((tan(t) / ew) * eh))) / (-1.0 / (((((t / ew) * eh) ^ 2.0) + 1.0) ^ -0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[ew, -1.65e-50], N[Abs[N[(t$95$2 * N[Cos[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.52e-161], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$2 + N[(t$95$1 * N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Power[N[(N[Power[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.6499999999999999e-50

   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%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if -1.6499999999999999e-50 < ew < 1.52000000000000002e-161

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. *-commutative N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-cos.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-sin.f64 71.2

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

   11. Applied rewrites 71.2%

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

   if 1.52000000000000002e-161 < 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

   4. Applied rewrites 84.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 77.2

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

   7. Applied rewrites 77.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. inv-pow N/A

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

      5. sqrt-pow2 N/A

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

      6. metadata-eval N/A

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

      7. lower-pow.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 81.7

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

   9. Applied rewrites 81.7%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \mathbf{elif}\;ew \leq 1.52 \cdot 10^{-161}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\cos t \cdot ew + \left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right)}{\frac{-1}{{\left({\left(\frac{t}{ew} \cdot eh\right)}^{2} + 1\right)}^{-0.5}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50} \lor \neg \left(ew \leq 1.7 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (or (<= ew -1.65e-50) (not (<= ew 1.7e-161)))
     (fabs (* (* (cos t) ew) (cos (atan (/ (/ t_1 ew) (cos t))))))
     (fabs (* t_1 (sin (atan (* (/ eh (cos t)) (/ (sin t) ew)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161)) {
		tmp = fabs(((cos(t) * ew) * cos(atan(((t_1 / ew) / cos(t))))));
	} else {
		tmp = fabs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))));
	}
	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 = sin(t) * eh
    if ((ew <= (-1.65d-50)) .or. (.not. (ew <= 1.7d-161))) then
        tmp = abs(((cos(t) * ew) * cos(atan(((t_1 / ew) / cos(t))))))
    else
        tmp = abs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double tmp;
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161)) {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(((t_1 / ew) / Math.cos(t))))));
	} else {
		tmp = Math.abs((t_1 * Math.sin(Math.atan(((eh / Math.cos(t)) * (Math.sin(t) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if (ew <= -1.65e-50) or not (ew <= 1.7e-161):
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(((t_1 / ew) / math.cos(t))))))
	else:
		tmp = math.fabs((t_1 * math.sin(math.atan(((eh / math.cos(t)) * (math.sin(t) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161))
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(Float64(t_1 / ew) / cos(t))))));
	else
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(eh / cos(t)) * Float64(sin(t) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if ((ew <= -1.65e-50) || ~((ew <= 1.7e-161)))
		tmp = abs(((cos(t) * ew) * cos(atan(((t_1 / ew) / cos(t))))));
	else
		tmp = abs((t_1 * sin(atan(((eh / cos(t)) * (sin(t) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[ew, -1.65e-50], N[Not[LessEqual[ew, 1.7e-161]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6499999999999999e-50 or 1.69999999999999991e-161 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -1.6499999999999999e-50 < ew < 1.69999999999999991e-161

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around inf

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. *-commutative N/A

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

       9. times-frac N/A

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

       10. lower-*.f64 N/A

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

       11. lower-/.f64 N/A

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

       12. lower-cos.f64 N/A

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

       13. lower-/.f64 N/A

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

       14. lower-sin.f64 71.2

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

   11. Applied rewrites 71.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50} \lor \neg \left(ew \leq 1.7 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{\sin t}{ew}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right)\\ \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50} \lor \neg \left(ew \leq 1.7 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 \cdot \sin t\_2\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (atan (/ (/ t_1 ew) (cos t)))))
   (if (or (<= ew -1.65e-50) (not (<= ew 1.7e-161)))
     (fabs (* (* (cos t) ew) (cos t_2)))
     (fabs (* t_1 (sin t_2))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = atan(((t_1 / ew) / cos(t)));
	double tmp;
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161)) {
		tmp = fabs(((cos(t) * ew) * cos(t_2)));
	} else {
		tmp = fabs((t_1 * sin(t_2)));
	}
	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 = sin(t) * eh
    t_2 = atan(((t_1 / ew) / cos(t)))
    if ((ew <= (-1.65d-50)) .or. (.not. (ew <= 1.7d-161))) then
        tmp = abs(((cos(t) * ew) * cos(t_2)))
    else
        tmp = abs((t_1 * sin(t_2)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double t_2 = Math.atan(((t_1 / ew) / Math.cos(t)));
	double tmp;
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161)) {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(t_2)));
	} else {
		tmp = Math.abs((t_1 * Math.sin(t_2)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.atan(((t_1 / ew) / math.cos(t)))
	tmp = 0
	if (ew <= -1.65e-50) or not (ew <= 1.7e-161):
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(t_2)))
	else:
		tmp = math.fabs((t_1 * math.sin(t_2)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = atan(Float64(Float64(t_1 / ew) / cos(t)))
	tmp = 0.0
	if ((ew <= -1.65e-50) || !(ew <= 1.7e-161))
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(t_2)));
	else
		tmp = abs(Float64(t_1 * sin(t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = atan(((t_1 / ew) / cos(t)));
	tmp = 0.0;
	if ((ew <= -1.65e-50) || ~((ew <= 1.7e-161)))
		tmp = abs(((cos(t) * ew) * cos(t_2)));
	else
		tmp = abs((t_1 * sin(t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[ew, -1.65e-50], N[Not[LessEqual[ew, 1.7e-161]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.6499999999999999e-50 or 1.69999999999999991e-161 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 84.4

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

   7. Applied rewrites 84.4%

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

   if -1.6499999999999999e-50 < ew < 1.69999999999999991e-161

   1. Initial program 99.7%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 71.2

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

   7. Applied rewrites 71.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.65 \cdot 10^{-50} \lor \neg \left(ew \leq 1.7 \cdot 10^{-161}\right):\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-atan.f64 N/A

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

   8. associate-/r* N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 67.4

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

7. Applied rewrites 67.4%

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

Alternative 12: 62.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((-ew * cos(t)) / (-1.0 / 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(((-ew * cos(t)) / ((-1.0d0) / cos(atan(((tan(t) / ew) * eh))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / 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 69.7%

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

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 67.4

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

7. Applied rewrites 67.4%

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

8. Final simplification 67.4%

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

Alternative 13: 51.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((cos(t) * ew) - (((eh * t) * (eh * t)) / -ew)) / (-1.0 / cos(atan(((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) - (((eh * t) * (eh * t)) / -ew)) / ((-1.0d0) / cos(atan(((t / ew) * eh))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] - N[(N[(N[(eh * t), $MachinePrecision] * N[(eh * t), $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[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 69.7%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 62.7

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

7. Applied rewrites 62.7%

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

8. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-neg.f64 60.6

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

10. Applied rewrites 60.6%

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

11. Final simplification 60.6%

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

Alternative 14: 42.6% 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 48.9%

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

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 48.1%

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

10. Applied rewrites 47.4%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 49.0%

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

Reproduce

herbie shell --seed 2024322 
(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)))))))