Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.6s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.8

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. atan-neg-rev N/A

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

   9. lower-atan.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. lift-neg.f64 N/A

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

   14. distribute-lft-neg-out N/A

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

   15. remove-double-neg N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 99.8

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 91.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := ew \cdot \cos t\\ t_3 := \frac{\tan t}{ew}\\ t_4 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_5 := t\_2 \cdot \cos t\_4 - t\_1 \cdot \sin t\_4\\ \mathbf{if}\;t\_5 \leq 2 \cdot 10^{-245} \lor \neg \left(t\_5 \leq 10^{-9}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - t\_2 \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(eh \cdot t\_3, \sin t \cdot eh, t\_2\right)}{\cosh \sinh^{-1} \left(t\_3 \cdot eh\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t)))
        (t_2 (* ew (cos t)))
        (t_3 (/ (tan t) ew))
        (t_4 (atan (/ (* eh (tan t)) (- ew))))
        (t_5 (- (* t_2 (cos t_4)) (* t_1 (sin t_4)))))
   (if (or (<= t_5 2e-245) (not (<= t_5 1e-9)))
     (fabs
      (-
       (* t_1 (sin (atan (/ (* (- t) eh) ew))))
       (* t_2 (cos (atan (/ (* (- eh) t) ew))))))
     (/ (fma (* eh t_3) (* (sin t) eh) t_2) (cosh (asinh (* t_3 eh)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = ew * cos(t);
	double t_3 = tan(t) / ew;
	double t_4 = atan(((eh * tan(t)) / -ew));
	double t_5 = (t_2 * cos(t_4)) - (t_1 * sin(t_4));
	double tmp;
	if ((t_5 <= 2e-245) || !(t_5 <= 1e-9)) {
		tmp = fabs(((t_1 * sin(atan(((-t * eh) / ew)))) - (t_2 * cos(atan(((-eh * t) / ew))))));
	} else {
		tmp = fma((eh * t_3), (sin(t) * eh), t_2) / cosh(asinh((t_3 * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = Float64(ew * cos(t))
	t_3 = Float64(tan(t) / ew)
	t_4 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	t_5 = Float64(Float64(t_2 * cos(t_4)) - Float64(t_1 * sin(t_4)))
	tmp = 0.0
	if ((t_5 <= 2e-245) || !(t_5 <= 1e-9))
		tmp = abs(Float64(Float64(t_1 * sin(atan(Float64(Float64(Float64(-t) * eh) / ew)))) - Float64(t_2 * cos(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	else
		tmp = Float64(fma(Float64(eh * t_3), Float64(sin(t) * eh), t_2) / cosh(asinh(Float64(t_3 * eh))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, Block[{t$95$4 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$2 * N[Cos[t$95$4], $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[Sin[t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$5, 2e-245], N[Not[LessEqual[t$95$5, 1e-9]], $MachinePrecision]], N[Abs[N[(N[(t$95$1 * N[Sin[N[ArcTan[N[(N[((-t) * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * N[Cos[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[(eh * t$95$3), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] + t$95$2), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$3 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1.9999999999999999e-245 or 1.00000000000000006e-9 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 93.4

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

   8. Applied rewrites 93.4%

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

   if 1.9999999999999999e-245 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < 1.00000000000000006e-9

   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 90.5%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l* N/A

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

      8. lift-*.f64 N/A

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

      9. lower-fma.f64 97.6

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 97.6

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 97.6

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 97.6

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

   6. Applied rewrites 97.6%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 2 \cdot 10^{-245} \lor \neg \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq 10^{-9}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(eh \cdot \frac{\tan t}{ew}, \sin t \cdot eh, ew \cdot \cos t\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_2 := \left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\sin \left(\left(-t\right) + \mathsf{PI}\left(\right)\right) \cdot eh\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew))))
        (t_2 (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))))
   (if (<= t_2 -1e+151)
     (* (sin (+ (- t) (PI))) eh)
     (if (<= t_2 -4e-278) (* (sin t) eh) (* (cos t) ew)))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.00000000000000002e151

   1. Initial program 99.9%

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.7%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.0%

       \[\leadsto \sin \left(\left(-t\right) + \mathsf{PI}\left(\right)\right) \cdot eh \]

   if -1.00000000000000002e151 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   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 1.9%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 21.2%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\sin \left(\left(-t\right) + \mathsf{PI}\left(\right)\right) \cdot eh\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ t_2 := \left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) + t\right) \cdot \left(-eh\right)\\ \mathbf{elif}\;t\_2 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew))))
        (t_2 (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))))
   (if (<= t_2 -1e+151)
     (* (sin (+ (PI) t)) (- eh))
     (if (<= t_2 -4e-278) (* (sin t) eh) (* (cos t) ew)))))

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.00000000000000002e151

   1. Initial program 99.9%

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

   4. Applied rewrites 0.2%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 1.7%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 7.7%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]
   11. Step-by-step derivation

   12. Applied rewrites 29.0%

       \[\leadsto \sin \left(-\left(\mathsf{PI}\left(\right) + t\right)\right) \cdot eh \]

   if -1.00000000000000002e151 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   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 1.9%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 21.2%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -1 \cdot 10^{+151}:\\ \;\;\;\;\sin \left(\mathsf{PI}\left(\right) + t\right) \cdot \left(-eh\right)\\ \mathbf{elif}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ t_2 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_2 - \left(eh \cdot \sin t\right) \cdot \sin t\_2 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(t\_1, \frac{t\_1}{2}, \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (PI))) (t_2 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_2)) (* (* eh (sin t)) (sin t_2)))
        -4e-278)
     (fabs (* (sin (fma t_1 (/ t_1 2.0) (atan (* (/ (tan t) ew) eh)))) ew))
     (* (cos t) ew))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 50.1%

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

   7. Applied rewrites 49.9%

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

   9. Applied rewrites 50.1%

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

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -4e-278)
     (fabs (* (cos (atan (* eh (/ (tan t) ew)))) ew))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -4e-278) {
		tmp = fabs((cos(atan((eh * (tan(t) / ew)))) * ew));
	} else {
		tmp = cos(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 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-4d-278)) then
        tmp = abs((cos(atan((eh * (tan(t) / ew)))) * ew))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -4e-278) {
		tmp = Math.abs((Math.cos(Math.atan((eh * (Math.tan(t) / ew)))) * ew));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -4e-278:
		tmp = math.fabs((math.cos(math.atan((eh * (math.tan(t) / ew)))) * ew))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -4e-278)
		tmp = abs(Float64(cos(atan(Float64(eh * Float64(tan(t) / ew)))) * ew));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -4e-278)
		tmp = abs((cos(atan((eh * (tan(t) / ew)))) * ew));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -4e-278], N[Abs[N[(N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 50.1%

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

   7. Applied rewrites 50.1%

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

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -4e-278)
     (fabs (* (sin (+ (/ (PI) 2.0) (atan (/ (* eh t) ew)))) ew))
     (* (cos t) ew))))

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 50.1%

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 48.6%

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

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\left|\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\\ \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1 \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- ew)))))
   (if (<=
        (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1)))
        -4e-278)
     (* (sin t) eh)
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / -ew));
	double tmp;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -4e-278) {
		tmp = sin(t) * eh;
	} else {
		tmp = cos(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 = atan(((eh * tan(t)) / -ew))
    if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= (-4d-278)) then
        tmp = sin(t) * eh
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / -ew));
	double tmp;
	if ((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))) <= -4e-278) {
		tmp = Math.sin(t) * eh;
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / -ew))
	tmp = 0
	if (((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))) <= -4e-278:
		tmp = math.sin(t) * eh
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))) <= -4e-278)
		tmp = Float64(sin(t) * eh);
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / -ew));
	tmp = 0.0;
	if ((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))) <= -4e-278)
		tmp = sin(t) * eh;
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[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], -4e-278], N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -3.99999999999999975e-278

   1. Initial program 99.9%

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

   4. Applied rewrites 1.2%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 10.6%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.9%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   if -3.99999999999999975e-278 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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 65.7%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 63.7

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

   7. Applied rewrites 63.7%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \leq -4 \cdot 10^{-278}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 10: 93.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -4.3 \cdot 10^{+77} \lor \neg \left(eh \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, t\_1 \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_1 \cdot eh\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (tan t) ew)))
   (if (or (<= eh -4.3e+77) (not (<= eh 2.6e+77)))
     (fabs
      (-
       (* (* eh (sin t)) (sin (atan (/ (* (- t) eh) ew))))
       (* (* ew (cos t)) (cos (atan (/ (* (- eh) t) ew))))))
     (fabs
      (/
       (fma (sin t) (* t_1 (* eh eh)) (* (cos t) ew))
       (cosh (asinh (* t_1 eh))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) / ew;
	double tmp;
	if ((eh <= -4.3e+77) || !(eh <= 2.6e+77)) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-t * eh) / ew)))) - ((ew * cos(t)) * cos(atan(((-eh * t) / ew))))));
	} else {
		tmp = fabs((fma(sin(t), (t_1 * (eh * eh)), (cos(t) * ew)) / cosh(asinh((t_1 * eh)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) / ew)
	tmp = 0.0
	if ((eh <= -4.3e+77) || !(eh <= 2.6e+77))
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-t) * eh) / ew)))) - Float64(Float64(ew * cos(t)) * cos(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	else
		tmp = abs(Float64(fma(sin(t), Float64(t_1 * Float64(eh * eh)), Float64(cos(t) * ew)) / cosh(asinh(Float64(t_1 * eh)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -4.3e+77], N[Not[LessEqual[eh, 2.6e+77]], $MachinePrecision]], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(t$95$1 * N[(eh * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$1 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.29999999999999991e77 or 2.6000000000000002e77 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.5

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 95.2

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

   8. Applied rewrites 95.2%

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

   if -4.29999999999999991e77 < eh < 2.6000000000000002e77

   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 95.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.3 \cdot 10^{+77} \lor \neg \left(eh \leq 2.6 \cdot 10^{+77}\right):\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\tan t}{ew} \cdot \left(eh \cdot eh\right), \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 75.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ t_2 := \cos t \cdot ew\\ t_3 := \left|t\_2 \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{t\_2}\right)\right|\\ \mathbf{if}\;t \leq -1.35 \cdot 10^{+165}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -60000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+35}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \left(t \cdot t\right) \cdot ew, -0.5 \cdot ew\right), t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+136}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (- (sin t)) eh)
           (sin
            (atan
             (*
              (fma (* 0.3333333333333333 (/ eh ew)) (* t t) (/ eh ew))
              (- t)))))))
        (t_2 (* (cos t) ew))
        (t_3 (fabs (* t_2 (cos (atan (/ (* (- eh) (sin t)) t_2)))))))
   (if (<= t -1.35e+165)
     t_1
     (if (<= t -60000.0)
       t_3
       (if (<= t 1.15e+35)
         (fabs
          (-
           (* (* eh (sin t)) (sin (atan (/ (* (- t) eh) ew))))
           (*
            (fma
             (fma 0.041666666666666664 (* (* t t) ew) (* -0.5 ew))
             (* t t)
             ew)
            (cos (atan (/ (* (- eh) t) ew))))))
         (if (<= t 1.75e+136) t_3 t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((-sin(t) * eh) * sin(atan((fma((0.3333333333333333 * (eh / ew)), (t * t), (eh / ew)) * -t)))));
	double t_2 = cos(t) * ew;
	double t_3 = fabs((t_2 * cos(atan(((-eh * sin(t)) / t_2)))));
	double tmp;
	if (t <= -1.35e+165) {
		tmp = t_1;
	} else if (t <= -60000.0) {
		tmp = t_3;
	} else if (t <= 1.15e+35) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-t * eh) / ew)))) - (fma(fma(0.041666666666666664, ((t * t) * ew), (-0.5 * ew)), (t * t), ew) * cos(atan(((-eh * t) / ew))))));
	} else if (t <= 1.75e+136) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(-sin(t)) * eh) * sin(atan(Float64(fma(Float64(0.3333333333333333 * Float64(eh / ew)), Float64(t * t), Float64(eh / ew)) * Float64(-t))))))
	t_2 = Float64(cos(t) * ew)
	t_3 = abs(Float64(t_2 * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / t_2)))))
	tmp = 0.0
	if (t <= -1.35e+165)
		tmp = t_1;
	elseif (t <= -60000.0)
		tmp = t_3;
	elseif (t <= 1.15e+35)
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-t) * eh) / ew)))) - Float64(fma(fma(0.041666666666666664, Float64(Float64(t * t) * ew), Float64(-0.5 * ew)), Float64(t * t), ew) * cos(atan(Float64(Float64(Float64(-eh) * t) / ew))))));
	elseif (t <= 1.75e+136)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$2 * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -1.35e+165], t$95$1, If[LessEqual[t, -60000.0], t$95$3, If[LessEqual[t, 1.15e+35], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(0.041666666666666664 * N[(N[(t * t), $MachinePrecision] * ew), $MachinePrecision] + N[(-0.5 * ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 1.75e+136], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.35e165 or 1.75000000000000001e136 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 66.8%

      \[\leadsto \left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-1 \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)\right) \cdot t\right)\right| \]

   if -1.35e165 < t < -6e4 or 1.1499999999999999e35 < t < 1.75000000000000001e136

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 74.3

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

   8. Applied rewrites 74.3%

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

   9. Taylor expanded in eh around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 72.1

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

   11. Applied rewrites 72.1%

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

   if -6e4 < t < 1.1499999999999999e35

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 100.0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{\left(ew + {t}^{2} \cdot \left(\frac{-1}{2} \cdot ew + \frac{1}{24} \cdot \left(ew \cdot {t}^{2}\right)\right)\right)} \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \left|\color{blue}{\left({t}^{2} \cdot \left(\frac{-1}{2} \cdot ew + \frac{1}{24} \cdot \left(ew \cdot {t}^{2}\right)\right) + ew\right)} \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       2. *-commutative N/A

          \[\leadsto \left|\left(\color{blue}{\left(\frac{-1}{2} \cdot ew + \frac{1}{24} \cdot \left(ew \cdot {t}^{2}\right)\right) \cdot {t}^{2}} + ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       3. lower-fma.f64 N/A

          \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot ew + \frac{1}{24} \cdot \left(ew \cdot {t}^{2}\right), {t}^{2}, ew\right)} \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       4. +-commutative N/A

          \[\leadsto \left|\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot \left(ew \cdot {t}^{2}\right) + \frac{-1}{2} \cdot ew}, {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       5. lower-fma.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, ew \cdot {t}^{2}, \frac{-1}{2} \cdot ew\right)}, {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       6. *-commutative N/A

          \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{{t}^{2} \cdot ew}, \frac{-1}{2} \cdot ew\right), {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       7. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{{t}^{2} \cdot ew}, \frac{-1}{2} \cdot ew\right), {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       8. unpow2 N/A

          \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{\left(t \cdot t\right)} \cdot ew, \frac{-1}{2} \cdot ew\right), {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       9. lower-*.f64 N/A

          \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{\left(t \cdot t\right)} \cdot ew, \frac{-1}{2} \cdot ew\right), {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       10. lower-*.f64 N/A

           \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \left(t \cdot t\right) \cdot ew, \color{blue}{\frac{-1}{2} \cdot ew}\right), {t}^{2}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       11. unpow2 N/A

           \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \left(t \cdot t\right) \cdot ew, \frac{-1}{2} \cdot ew\right), \color{blue}{t \cdot t}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

       12. lower-*.f64 98.7

           \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \left(t \cdot t\right) \cdot ew, -0.5 \cdot ew\right), \color{blue}{t \cdot t}, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]

   11. Applied rewrites 98.7%

       \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \left(t \cdot t\right) \cdot ew, -0.5 \cdot ew\right), t \cdot t, ew\right)} \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right)\right| \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+165}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{elif}\;t \leq -60000:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+35}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-t\right) \cdot eh}{ew}\right) - \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, \left(t \cdot t\right) \cdot ew, -0.5 \cdot ew\right), t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{+136}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ t_2 := \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -7 \cdot 10^{+117}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(t\_2 \cdot \left(-eh\right)\right) \cdot t\_1\right) \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.4 \cdot 10^{+99}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\sin t, \frac{\left(eh \cdot eh\right) \cdot t}{ew}, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(t\_2 \cdot eh\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t\_1}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))) (t_2 (/ (tan t) ew)))
   (if (<= eh -7e+117)
     (fabs (* (* (tanh (asinh (* t_2 (- eh)))) t_1) eh))
     (if (<= eh 2.4e+99)
       (fabs
        (/
         (fma (sin t) (/ (* (* eh eh) t) ew) (* (cos t) ew))
         (cosh (asinh (* t_2 eh)))))
       (fabs (* (* t_1 eh) (sin (atan (* (/ t_1 ew) (/ eh (cos t)))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double t_2 = tan(t) / ew;
	double tmp;
	if (eh <= -7e+117) {
		tmp = fabs(((tanh(asinh((t_2 * -eh))) * t_1) * eh));
	} else if (eh <= 2.4e+99) {
		tmp = fabs((fma(sin(t), (((eh * eh) * t) / ew), (cos(t) * ew)) / cosh(asinh((t_2 * eh)))));
	} else {
		tmp = fabs(((t_1 * eh) * sin(atan(((t_1 / ew) * (eh / cos(t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	t_2 = Float64(tan(t) / ew)
	tmp = 0.0
	if (eh <= -7e+117)
		tmp = abs(Float64(Float64(tanh(asinh(Float64(t_2 * Float64(-eh)))) * t_1) * eh));
	elseif (eh <= 2.4e+99)
		tmp = abs(Float64(fma(sin(t), Float64(Float64(Float64(eh * eh) * t) / ew), Float64(cos(t) * ew)) / cosh(asinh(Float64(t_2 * eh)))));
	else
		tmp = abs(Float64(Float64(t_1 * eh) * sin(atan(Float64(Float64(t_1 / ew) * Float64(eh / cos(t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, Block[{t$95$2 = N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]}, If[LessEqual[eh, -7e+117], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(t$95$2 * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 2.4e+99], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * N[(N[(N[(eh * eh), $MachinePrecision] * t), $MachinePrecision] / ew), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[ArcSinh[N[(t$95$2 * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.99999999999999965e117

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 78.1%

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

   if -6.99999999999999965e117 < eh < 2.4000000000000001e99

   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 95.2%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\mathsf{fma}\left(\sin t, \color{blue}{\frac{{eh}^{2} \cdot t}{ew}}, \cos t \cdot ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right| \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 84.1

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

   7. Applied rewrites 84.1%

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

   if 2.4000000000000001e99 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 67.3%

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

Alternative 13: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{\cos t}\\ t_2 := \cos t \cdot ew\\ t_3 := -\sin t\\ t_4 := \frac{t\_3}{ew}\\ \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\left|t\_2 \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{t\_2}\right)\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;\left|\left(t\_3 \cdot eh\right) \cdot \sin \tan^{-1} \left(t\_4 \cdot t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(t\_1 \cdot t\_4\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (cos t)))
        (t_2 (* (cos t) ew))
        (t_3 (- (sin t)))
        (t_4 (/ t_3 ew)))
   (if (<= ew -4.3e-95)
     (fabs (* t_2 (cos (atan (/ (* (- eh) (sin t)) t_2)))))
     (if (<= ew 1.05e-140)
       (fabs (* (* t_3 eh) (sin (atan (* t_4 t_1)))))
       (fabs (* (* (cos (atan (* t_1 t_4))) (cos t)) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / cos(t);
	double t_2 = cos(t) * ew;
	double t_3 = -sin(t);
	double t_4 = t_3 / ew;
	double tmp;
	if (ew <= -4.3e-95) {
		tmp = fabs((t_2 * cos(atan(((-eh * sin(t)) / t_2)))));
	} else if (ew <= 1.05e-140) {
		tmp = fabs(((t_3 * eh) * sin(atan((t_4 * t_1)))));
	} else {
		tmp = fabs(((cos(atan((t_1 * t_4))) * cos(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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = eh / cos(t)
    t_2 = cos(t) * ew
    t_3 = -sin(t)
    t_4 = t_3 / ew
    if (ew <= (-4.3d-95)) then
        tmp = abs((t_2 * cos(atan(((-eh * sin(t)) / t_2)))))
    else if (ew <= 1.05d-140) then
        tmp = abs(((t_3 * eh) * sin(atan((t_4 * t_1)))))
    else
        tmp = abs(((cos(atan((t_1 * t_4))) * cos(t)) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh / Math.cos(t);
	double t_2 = Math.cos(t) * ew;
	double t_3 = -Math.sin(t);
	double t_4 = t_3 / ew;
	double tmp;
	if (ew <= -4.3e-95) {
		tmp = Math.abs((t_2 * Math.cos(Math.atan(((-eh * Math.sin(t)) / t_2)))));
	} else if (ew <= 1.05e-140) {
		tmp = Math.abs(((t_3 * eh) * Math.sin(Math.atan((t_4 * t_1)))));
	} else {
		tmp = Math.abs(((Math.cos(Math.atan((t_1 * t_4))) * Math.cos(t)) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh / math.cos(t)
	t_2 = math.cos(t) * ew
	t_3 = -math.sin(t)
	t_4 = t_3 / ew
	tmp = 0
	if ew <= -4.3e-95:
		tmp = math.fabs((t_2 * math.cos(math.atan(((-eh * math.sin(t)) / t_2)))))
	elif ew <= 1.05e-140:
		tmp = math.fabs(((t_3 * eh) * math.sin(math.atan((t_4 * t_1)))))
	else:
		tmp = math.fabs(((math.cos(math.atan((t_1 * t_4))) * math.cos(t)) * ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / cos(t))
	t_2 = Float64(cos(t) * ew)
	t_3 = Float64(-sin(t))
	t_4 = Float64(t_3 / ew)
	tmp = 0.0
	if (ew <= -4.3e-95)
		tmp = abs(Float64(t_2 * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / t_2)))));
	elseif (ew <= 1.05e-140)
		tmp = abs(Float64(Float64(t_3 * eh) * sin(atan(Float64(t_4 * t_1)))));
	else
		tmp = abs(Float64(Float64(cos(atan(Float64(t_1 * t_4))) * cos(t)) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh / cos(t);
	t_2 = cos(t) * ew;
	t_3 = -sin(t);
	t_4 = t_3 / ew;
	tmp = 0.0;
	if (ew <= -4.3e-95)
		tmp = abs((t_2 * cos(atan(((-eh * sin(t)) / t_2)))));
	elseif (ew <= 1.05e-140)
		tmp = abs(((t_3 * eh) * sin(atan((t_4 * t_1)))));
	else
		tmp = abs(((cos(atan((t_1 * t_4))) * cos(t)) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = (-N[Sin[t], $MachinePrecision])}, Block[{t$95$4 = N[(t$95$3 / ew), $MachinePrecision]}, If[LessEqual[ew, -4.3e-95], N[Abs[N[(t$95$2 * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.05e-140], N[Abs[N[(N[(t$95$3 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(t$95$4 * t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(t$95$1 * t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.29999999999999997e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.9

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in eh around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 80.3

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

   11. Applied rewrites 80.3%

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

   if -4.29999999999999997e-95 < ew < 1.05000000000000009e-140

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 73.8%

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

   if 1.05000000000000009e-140 < ew

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 78.9%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{-\sin t}{ew} \cdot \frac{eh}{\cos t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-\sin t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := -\sin t\\ \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{t\_1}\right)\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot t\_2\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{t\_2}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)) (t_2 (- (sin t))))
   (if (<= ew -4.3e-95)
     (fabs (* t_1 (cos (atan (/ (* (- eh) (sin t)) t_1)))))
     (if (<= ew 1.05e-140)
       (fabs (* (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) t_2) eh))
       (fabs (* (* (cos (atan (* (/ eh (cos t)) (/ t_2 ew)))) (cos t)) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = -sin(t);
	double tmp;
	if (ew <= -4.3e-95) {
		tmp = fabs((t_1 * cos(atan(((-eh * sin(t)) / t_1)))));
	} else if (ew <= 1.05e-140) {
		tmp = fabs(((tanh(asinh(((tan(t) / ew) * -eh))) * t_2) * eh));
	} else {
		tmp = fabs(((cos(atan(((eh / cos(t)) * (t_2 / ew)))) * cos(t)) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = -math.sin(t)
	tmp = 0
	if ew <= -4.3e-95:
		tmp = math.fabs((t_1 * math.cos(math.atan(((-eh * math.sin(t)) / t_1)))))
	elif ew <= 1.05e-140:
		tmp = math.fabs(((math.tanh(math.asinh(((math.tan(t) / ew) * -eh))) * t_2) * eh))
	else:
		tmp = math.fabs(((math.cos(math.atan(((eh / math.cos(t)) * (t_2 / ew)))) * math.cos(t)) * ew))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = Float64(-sin(t))
	tmp = 0.0
	if (ew <= -4.3e-95)
		tmp = abs(Float64(t_1 * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / t_1)))));
	elseif (ew <= 1.05e-140)
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * t_2) * eh));
	else
		tmp = abs(Float64(Float64(cos(atan(Float64(Float64(eh / cos(t)) * Float64(t_2 / ew)))) * cos(t)) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = -sin(t);
	tmp = 0.0;
	if (ew <= -4.3e-95)
		tmp = abs((t_1 * cos(atan(((-eh * sin(t)) / t_1)))));
	elseif (ew <= 1.05e-140)
		tmp = abs(((tanh(asinh(((tan(t) / ew) * -eh))) * t_2) * eh));
	else
		tmp = abs(((cos(atan(((eh / cos(t)) * (t_2 / ew)))) * cos(t)) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = (-N[Sin[t], $MachinePrecision])}, If[LessEqual[ew, -4.3e-95], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.05e-140], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.29999999999999997e-95

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 89.9

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

   8. Applied rewrites 89.9%

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

   9. Taylor expanded in eh around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 80.3

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

   11. Applied rewrites 80.3%

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

   if -4.29999999999999997e-95 < ew < 1.05000000000000009e-140

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 73.8%

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

   if 1.05000000000000009e-140 < ew

   1. Initial program 99.9%

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

   3. Taylor expanded in ew around inf

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

   5. Applied rewrites 96.5%

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

   6. Taylor expanded in eh around 0

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

   8. Applied rewrites 78.9%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{elif}\;ew \leq 1.05 \cdot 10^{-140}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-\sin t}{ew}\right) \cdot \cos t\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95} \lor \neg \left(ew \leq 1.05 \cdot 10^{-140}\right):\\ \;\;\;\;\left|t\_1 \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{t\_1}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)))
   (if (or (<= ew -4.3e-95) (not (<= ew 1.05e-140)))
     (fabs (* t_1 (cos (atan (/ (* (- eh) (sin t)) t_1)))))
     (fabs (* (* (tanh (asinh (* (/ (tan t) ew) (- eh)))) (- (sin t))) eh)))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double tmp;
	if ((ew <= -4.3e-95) || !(ew <= 1.05e-140)) {
		tmp = fabs((t_1 * cos(atan(((-eh * sin(t)) / t_1)))));
	} else {
		tmp = fabs(((tanh(asinh(((tan(t) / ew) * -eh))) * -sin(t)) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	tmp = 0
	if (ew <= -4.3e-95) or not (ew <= 1.05e-140):
		tmp = math.fabs((t_1 * math.cos(math.atan(((-eh * math.sin(t)) / t_1)))))
	else:
		tmp = math.fabs(((math.tanh(math.asinh(((math.tan(t) / ew) * -eh))) * -math.sin(t)) * eh))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	tmp = 0.0
	if ((ew <= -4.3e-95) || !(ew <= 1.05e-140))
		tmp = abs(Float64(t_1 * cos(atan(Float64(Float64(Float64(-eh) * sin(t)) / t_1)))));
	else
		tmp = abs(Float64(Float64(tanh(asinh(Float64(Float64(tan(t) / ew) * Float64(-eh)))) * Float64(-sin(t))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	tmp = 0.0;
	if ((ew <= -4.3e-95) || ~((ew <= 1.05e-140)))
		tmp = abs((t_1 * cos(atan(((-eh * sin(t)) / t_1)))));
	else
		tmp = abs(((tanh(asinh(((tan(t) / ew) * -eh))) * -sin(t)) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[Or[LessEqual[ew, -4.3e-95], N[Not[LessEqual[ew, 1.05e-140]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Tanh[N[ArcSinh[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if ew < -4.29999999999999997e-95 or 1.05000000000000009e-140 < ew

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 87.4

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

   8. Applied rewrites 87.4%

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

   9. Taylor expanded in eh around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-cos.f64 N/A

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

       6. lower-cos.f64 N/A

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

       7. lower-atan.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lower-/.f64 N/A

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

       10. associate-*r* N/A

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

       11. lower-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lower-neg.f64 N/A

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

       14. lower-sin.f64 N/A

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

       15. *-commutative N/A

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

       16. lower-*.f64 N/A

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

       17. lower-cos.f64 79.6

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

   11. Applied rewrites 79.6%

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

   if -4.29999999999999997e-95 < ew < 1.05000000000000009e-140

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 73.8%

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

   7. Applied rewrites 73.8%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.3 \cdot 10^{-95} \lor \neg \left(ew \leq 1.05 \cdot 10^{-140}\right):\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \sin t}{\cos t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\tanh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot \left(-eh\right)\right) \cdot \left(-\sin t\right)\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 62.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ t_3 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t \leq -6.2 \cdot 10^{-32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-92}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right), t \cdot t, ew\right)}{\cosh \sinh^{-1} t\_1}\right|\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-9}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(t\_3, \frac{t\_3}{2}, \tan^{-1} t\_1\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2
         (fabs
          (*
           (* (- (sin t)) eh)
           (sin
            (atan
             (*
              (fma (* 0.3333333333333333 (/ eh ew)) (* t t) (/ eh ew))
              (- t)))))))
        (t_3 (sqrt (PI))))
   (if (<= t -6.2e-32)
     t_2
     (if (<= t -2.8e-92)
       (fabs
        (/ (fma (fma -0.5 ew (/ (* eh eh) ew)) (* t t) ew) (cosh (asinh t_1))))
       (if (<= t 1.62e-9)
         (fabs (* (sin (fma t_3 (/ t_3 2.0) (atan t_1))) ew))
         t_2)))))

Derivation

1. Split input into 3 regimes

2. if t < -6.20000000000000021e-32 or 1.61999999999999999e-9 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 55.2%

      \[\leadsto \left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-1 \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)\right) \cdot t\right)\right| \]

   if -6.20000000000000021e-32 < t < -2.8e-92

   1. Initial program 99.9%

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 97.9

         \[\leadsto \left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right), \color{blue}{t \cdot t}, ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right| \]

   7. Applied rewrites 97.9%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right), t \cdot t, ew\right)}}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right| \]

   if -2.8e-92 < t < 1.61999999999999999e-9

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   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 80.9%

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.9%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-32}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-92}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right), t \cdot t, ew\right)}{\cosh \sinh^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}\right|\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{-9}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right) \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{\mathsf{PI}\left(\right)}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-39} \lor \neg \left(t \leq 1.62 \cdot 10^{-9}\right):\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(t\_1, \frac{t\_1}{2}, \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (sqrt (PI))))
   (if (or (<= t -9.2e-39) (not (<= t 1.62e-9)))
     (fabs
      (*
       (* (- (sin t)) eh)
       (sin
        (atan
         (* (fma (* 0.3333333333333333 (/ eh ew)) (* t t) (/ eh ew)) (- t))))))
     (fabs (* (sin (fma t_1 (/ t_1 2.0) (atan (* (/ (tan t) ew) eh)))) ew)))))

Derivation

1. Split input into 2 regimes

2. if t < -9.20000000000000033e-39 or 1.61999999999999999e-9 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. *-commutative N/A

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

      14. times-frac N/A

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

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

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

   5. Applied rewrites 54.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 55.2%

      \[\leadsto \left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-1 \cdot \mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right)\right) \cdot t\right)\right| \]

   if -9.20000000000000033e-39 < t < 1.61999999999999999e-9

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   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 80.3%

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 80.3%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.2 \cdot 10^{-39} \lor \neg \left(t \leq 1.62 \cdot 10^{-9}\right):\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(0.3333333333333333 \cdot \frac{eh}{ew}, t \cdot t, \frac{eh}{ew}\right) \cdot \left(-t\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \left(\mathsf{fma}\left(\sqrt{\mathsf{PI}\left(\right)}, \frac{\sqrt{\mathsf{PI}\left(\right)}}{2}, \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right)\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 34.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq 9.5 \cdot 10^{-141}:\\ \;\;\;\;\sin t \cdot eh\\ \mathbf{else}:\\ \;\;\;\;1 \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew 9.5e-141) (* (sin t) eh) (* 1.0 ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 9.5e-141) {
		tmp = sin(t) * eh;
	} else {
		tmp = 1.0 * 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) :: tmp
    if (ew <= 9.5d-141) then
        tmp = sin(t) * eh
    else
        tmp = 1.0d0 * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= 9.5e-141) {
		tmp = Math.sin(t) * eh;
	} else {
		tmp = 1.0 * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= 9.5e-141:
		tmp = math.sin(t) * eh
	else:
		tmp = 1.0 * ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= 9.5e-141)
		tmp = Float64(sin(t) * eh);
	else
		tmp = Float64(1.0 * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= 9.5e-141)
		tmp = sin(t) * eh;
	else
		tmp = 1.0 * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, 9.5e-141], N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision], N[(1.0 * ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < 9.49999999999999996e-141

   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 22.6%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 13.3%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 21.7%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   if 9.49999999999999996e-141 < ew

   1. Initial program 99.9%

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

   4. Applied rewrites 58.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 39.1

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

   7. Applied rewrites 39.1%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 49.6%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 56.3%

       \[\leadsto 1 \cdot ew \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 25.0% accurate, 19.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.02 \cdot 10^{-303}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, 0.008333333333333333, -0.16666666666666666 \cdot eh\right), t \cdot t, eh\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.02e-303)
   (*
    (fma
     (fma (* (* t t) eh) 0.008333333333333333 (* -0.16666666666666666 eh))
     (* t t)
     eh)
    t)
   (* 1.0 ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.02e-303) {
		tmp = fma(fma(((t * t) * eh), 0.008333333333333333, (-0.16666666666666666 * eh)), (t * t), eh) * t;
	} else {
		tmp = 1.0 * ew;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.02e-303)
		tmp = Float64(fma(fma(Float64(Float64(t * t) * eh), 0.008333333333333333, Float64(-0.16666666666666666 * eh)), Float64(t * t), eh) * t);
	else
		tmp = Float64(1.0 * ew);
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.02e-303], N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * 0.008333333333333333 + N[(-0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] * t), $MachinePrecision], N[(1.0 * ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.01999999999999996e-303

   1. Initial program 99.9%

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

   4. Applied rewrites 18.3%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 10.8%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.5%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   11. Taylor expanded in t around 0

       \[\leadsto t \cdot \left(eh + \color{blue}{{t}^{2} \cdot \left(\frac{-1}{6} \cdot eh + \frac{1}{120} \cdot \left(eh \cdot {t}^{2}\right)\right)}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 9.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, 0.008333333333333333, -0.16666666666666666 \cdot eh\right), t \cdot t, eh\right) \cdot t \]

   if -1.01999999999999996e-303 < 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 52.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 33.4

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

   7. Applied rewrites 33.4%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 41.9%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 47.1%

       \[\leadsto 1 \cdot ew \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 25.1% accurate, 71.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.02 \cdot 10^{-303}:\\ \;\;\;\;eh \cdot t\\ \mathbf{else}:\\ \;\;\;\;1 \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.02e-303) (* eh t) (* 1.0 ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.02e-303) {
		tmp = eh * t;
	} else {
		tmp = 1.0 * 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) :: tmp
    if (ew <= (-1.02d-303)) then
        tmp = eh * t
    else
        tmp = 1.0d0 * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.02e-303) {
		tmp = eh * t;
	} else {
		tmp = 1.0 * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.02e-303:
		tmp = eh * t
	else:
		tmp = 1.0 * ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.02e-303)
		tmp = Float64(eh * t);
	else
		tmp = Float64(1.0 * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.02e-303)
		tmp = eh * t;
	else
		tmp = 1.0 * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.02e-303], N[(eh * t), $MachinePrecision], N[(1.0 * ew), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.01999999999999996e-303

   1. Initial program 99.9%

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

   4. Applied rewrites 18.3%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r/ N/A

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

      2. associate-*r* N/A

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

      3. times-frac N/A

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

      4. lower-*.f64 N/A

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

   7. Applied rewrites 10.8%

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

   8. Taylor expanded in eh around inf

      \[\leadsto eh \cdot \color{blue}{\sin t} \]
   9. Step-by-step derivation

   10. Applied rewrites 16.5%

       \[\leadsto \sin t \cdot \color{blue}{eh} \]

   11. Taylor expanded in t around 0

       \[\leadsto eh \cdot t \]
   12. Step-by-step derivation

   13. Applied rewrites 9.7%

       \[\leadsto eh \cdot t \]

   if -1.01999999999999996e-303 < 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 52.4%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 33.4

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

   7. Applied rewrites 33.4%

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

   8. Taylor expanded in ew around inf

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

   10. Applied rewrites 41.9%

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

   11. Taylor expanded in t around 0

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

   13. Applied rewrites 47.1%

       \[\leadsto 1 \cdot ew \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 9.9% accurate, 143.7× speedup

Math

\[\begin{array}{l} \\ eh \cdot t \end{array} \]

FPCore

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

C

double code(double eh, double ew, double t) {
	return eh * 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 = eh * t
end function

Java

public static double code(double eh, double ew, double t) {
	return eh * t;
}

Python

def code(eh, ew, t):
	return eh * t

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[(eh * t), $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 35.0%

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

5. Taylor expanded in ew around 0

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

   1. associate-*r/ N/A

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

   2. associate-*r* N/A

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

   3. times-frac N/A

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

   4. lower-*.f64 N/A

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

7. Applied rewrites 10.8%

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

8. Taylor expanded in eh around inf

   \[\leadsto eh \cdot \color{blue}{\sin t} \]
9. Step-by-step derivation

10. Applied rewrites 18.6%

    \[\leadsto \sin t \cdot \color{blue}{eh} \]

11. Taylor expanded in t around 0

    \[\leadsto eh \cdot t \]
12. Step-by-step derivation

13. Applied rewrites 9.5%

    \[\leadsto eh \cdot t \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024343 
(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)))))))