Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.2s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.6

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\frac{t\_1 \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{if}\;ew \leq -1.85 \cdot 10^{+22}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t, ew, t\_1 \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2
         (fabs
          (/
           (+ (* t_1 (* eh (/ (tan t) ew))) (* (cos t) ew))
           (/ 1.0 (pow (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)) -0.5))))))
   (if (<= ew -1.85e+22)
     t_2
     (if (<= ew 1.8e-10)
       (fabs
        (fma
         (* (cos (atan (/ (* eh t) ew))) (cos t))
         ew
         (* t_1 (sin (atan (* (/ t ew) eh))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs((((t_1 * (eh * (tan(t) / ew))) + (cos(t) * ew)) / (1.0 / pow((1.0 + pow((ew / (eh * tan(t))), -2.0)), -0.5))));
	double tmp;
	if (ew <= -1.85e+22) {
		tmp = t_2;
	} else if (ew <= 1.8e-10) {
		tmp = fabs(fma((cos(atan(((eh * t) / ew))) * cos(t)), ew, (t_1 * sin(atan(((t / ew) * eh))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = abs(Float64(Float64(Float64(t_1 * Float64(eh * Float64(tan(t) / ew))) + Float64(cos(t) * ew)) / Float64(1.0 / (Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0)) ^ -0.5))))
	tmp = 0.0
	if (ew <= -1.85e+22)
		tmp = t_2;
	elseif (ew <= 1.8e-10)
		tmp = abs(fma(Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t)), ew, Float64(t_1 * sin(atan(Float64(Float64(t / ew) * eh))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(N[(t$95$1 * N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.85e+22], t$95$2, If[LessEqual[ew, 1.8e-10], N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew + N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.8499999999999999e22 or 1.8e-10 < 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 86.4%

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

   6. Applied rewrites 95.8%

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

   if -1.8499999999999999e22 < ew < 1.8e-10

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 95.8

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

   10. Applied rewrites 95.8%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.85 \cdot 10^{+22}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{elif}\;ew \leq 1.8 \cdot 10^{-10}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t, ew, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t, ew, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{if}\;eh \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 9.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}}}{\left|\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right|}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (fma
           (* (cos (atan (/ (* eh t) ew))) (cos t))
           ew
           (* (* (sin t) eh) (sin (atan (* (/ t ew) eh))))))))
   (if (<= eh -1.55e+60)
     t_1
     (if (<= eh 9.6e-20)
       (/
        1.0
        (/
         (sqrt (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)))
         (fabs (+ (* (* (* eh (/ (tan t) ew)) eh) (sin t)) (* (cos t) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(fma((cos(atan(((eh * t) / ew))) * cos(t)), ew, ((sin(t) * eh) * sin(atan(((t / ew) * eh))))));
	double tmp;
	if (eh <= -1.55e+60) {
		tmp = t_1;
	} else if (eh <= 9.6e-20) {
		tmp = 1.0 / (sqrt((1.0 + pow((ew / (eh * tan(t))), -2.0))) / fabs(((((eh * (tan(t) / ew)) * eh) * sin(t)) + (cos(t) * ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(fma(Float64(cos(atan(Float64(Float64(eh * t) / ew))) * cos(t)), ew, Float64(Float64(sin(t) * eh) * sin(atan(Float64(Float64(t / ew) * eh))))))
	tmp = 0.0
	if (eh <= -1.55e+60)
		tmp = t_1;
	elseif (eh <= 9.6e-20)
		tmp = Float64(1.0 / Float64(sqrt(Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0))) / abs(Float64(Float64(Float64(Float64(eh * Float64(tan(t) / ew)) * eh) * sin(t)) + Float64(cos(t) * ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Cos[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * ew + N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.55e+60], t$95$1, If[LessEqual[eh, 9.6e-20], N[(1.0 / N[(N[Sqrt[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.55e60 or 9.59999999999999971e-20 < eh

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 92.3

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

   10. Applied rewrites 92.3%

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

   if -1.55e60 < eh < 9.59999999999999971e-20

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 83.9%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. remove-double-div N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   8. Applied rewrites 97.4%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.55 \cdot 10^{+60}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t, ew, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{elif}\;eh \leq 9.6 \cdot 10^{-20}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}}}{\left|\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot \cos t, ew, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cos t \cdot ew\\ t_3 := \frac{1}{\frac{\sqrt{1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}}}{\left|\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t + t\_2\right|}}\\ t_4 := \left|\sin \tan^{-1} \left(\frac{\sin t}{t\_2} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;eh \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right) \cdot t\_1}\right|}\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 4 \cdot 10^{+192}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* (cos t) ew))
        (t_3
         (/
          1.0
          (/
           (sqrt (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)))
           (fabs (+ (* (* (* eh (/ (tan t) ew)) eh) (sin t)) t_2)))))
        (t_4
         (fabs
          (* (sin (atan (* (/ (sin t) t_2) (- eh)))) (* (- eh) (sin t))))))
   (if (<= eh -3.3e+93)
     (/ 1.0 (fabs (/ 1.0 (* (sin (atan (/ (/ t_1 ew) (cos t)))) t_1))))
     (if (<= eh 4.8e+70)
       t_3
       (if (<= eh 1.95e+139) t_4 (if (<= eh 4e+192) t_3 t_4))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cos(t) * ew;
	double t_3 = 1.0 / (sqrt((1.0 + pow((ew / (eh * tan(t))), -2.0))) / fabs(((((eh * (tan(t) / ew)) * eh) * sin(t)) + t_2)));
	double t_4 = fabs((sin(atan(((sin(t) / t_2) * -eh))) * (-eh * sin(t))));
	double tmp;
	if (eh <= -3.3e+93) {
		tmp = 1.0 / fabs((1.0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)));
	} else if (eh <= 4.8e+70) {
		tmp = t_3;
	} else if (eh <= 1.95e+139) {
		tmp = t_4;
	} else if (eh <= 4e+192) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	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 = sin(t) * eh
    t_2 = cos(t) * ew
    t_3 = 1.0d0 / (sqrt((1.0d0 + ((ew / (eh * tan(t))) ** (-2.0d0)))) / abs(((((eh * (tan(t) / ew)) * eh) * sin(t)) + t_2)))
    t_4 = abs((sin(atan(((sin(t) / t_2) * -eh))) * (-eh * sin(t))))
    if (eh <= (-3.3d+93)) then
        tmp = 1.0d0 / abs((1.0d0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)))
    else if (eh <= 4.8d+70) then
        tmp = t_3
    else if (eh <= 1.95d+139) then
        tmp = t_4
    else if (eh <= 4d+192) then
        tmp = t_3
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double t_2 = Math.cos(t) * ew;
	double t_3 = 1.0 / (Math.sqrt((1.0 + Math.pow((ew / (eh * Math.tan(t))), -2.0))) / Math.abs(((((eh * (Math.tan(t) / ew)) * eh) * Math.sin(t)) + t_2)));
	double t_4 = Math.abs((Math.sin(Math.atan(((Math.sin(t) / t_2) * -eh))) * (-eh * Math.sin(t))));
	double tmp;
	if (eh <= -3.3e+93) {
		tmp = 1.0 / Math.abs((1.0 / (Math.sin(Math.atan(((t_1 / ew) / Math.cos(t)))) * t_1)));
	} else if (eh <= 4.8e+70) {
		tmp = t_3;
	} else if (eh <= 1.95e+139) {
		tmp = t_4;
	} else if (eh <= 4e+192) {
		tmp = t_3;
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.cos(t) * ew
	t_3 = 1.0 / (math.sqrt((1.0 + math.pow((ew / (eh * math.tan(t))), -2.0))) / math.fabs(((((eh * (math.tan(t) / ew)) * eh) * math.sin(t)) + t_2)))
	t_4 = math.fabs((math.sin(math.atan(((math.sin(t) / t_2) * -eh))) * (-eh * math.sin(t))))
	tmp = 0
	if eh <= -3.3e+93:
		tmp = 1.0 / math.fabs((1.0 / (math.sin(math.atan(((t_1 / ew) / math.cos(t)))) * t_1)))
	elif eh <= 4.8e+70:
		tmp = t_3
	elif eh <= 1.95e+139:
		tmp = t_4
	elif eh <= 4e+192:
		tmp = t_3
	else:
		tmp = t_4
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(cos(t) * ew)
	t_3 = Float64(1.0 / Float64(sqrt(Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0))) / abs(Float64(Float64(Float64(Float64(eh * Float64(tan(t) / ew)) * eh) * sin(t)) + t_2))))
	t_4 = abs(Float64(sin(atan(Float64(Float64(sin(t) / t_2) * Float64(-eh)))) * Float64(Float64(-eh) * sin(t))))
	tmp = 0.0
	if (eh <= -3.3e+93)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(Float64(t_1 / ew) / cos(t)))) * t_1))));
	elseif (eh <= 4.8e+70)
		tmp = t_3;
	elseif (eh <= 1.95e+139)
		tmp = t_4;
	elseif (eh <= 4e+192)
		tmp = t_3;
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = cos(t) * ew;
	t_3 = 1.0 / (sqrt((1.0 + ((ew / (eh * tan(t))) ^ -2.0))) / abs(((((eh * (tan(t) / ew)) * eh) * sin(t)) + t_2)));
	t_4 = abs((sin(atan(((sin(t) / t_2) * -eh))) * (-eh * sin(t))));
	tmp = 0.0;
	if (eh <= -3.3e+93)
		tmp = 1.0 / abs((1.0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)));
	elseif (eh <= 4.8e+70)
		tmp = t_3;
	elseif (eh <= 1.95e+139)
		tmp = t_4;
	elseif (eh <= 4e+192)
		tmp = t_3;
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[N[(N[(N[(N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / t$95$2), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.3e+93], N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, 4.8e+70], t$95$3, If[LessEqual[eh, 1.95e+139], t$95$4, If[LessEqual[eh, 4e+192], t$95$3, t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.30000000000000009e93

   1. Initial program 99.8%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 74.4

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

   7. Applied rewrites 74.4%

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

   if -3.30000000000000009e93 < eh < 4.79999999999999974e70 or 1.95000000000000003e139 < eh < 4.00000000000000016e192

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 79.9%

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

      1. lift-pow.f64 N/A

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

      2. inv-pow N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-atan.f64 N/A

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

      5. cos-atan N/A

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

      6. remove-double-div N/A

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

      7. lower-sqrt.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

   8. Applied rewrites 95.5%

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

   if 4.79999999999999974e70 < eh < 1.95000000000000003e139 or 4.00000000000000016e192 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 14.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 13.5%

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

   9. Taylor expanded in ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   10. 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. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. mul-1-neg N/A

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

       14. associate-/l* N/A

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

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

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

   11. Applied rewrites 86.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.3 \cdot 10^{+93}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}}}{\left|\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right|}}\\ \mathbf{elif}\;eh \leq 1.95 \cdot 10^{+139}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 4 \cdot 10^{+192}:\\ \;\;\;\;\frac{1}{\frac{\sqrt{1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}}}{\left|\left(\left(eh \cdot \frac{\tan t}{ew}\right) \cdot eh\right) \cdot \sin t + \cos t \cdot ew\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cos t \cdot ew\\ \mathbf{if}\;eh \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right) \cdot t\_1}\right|}\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left(eh \cdot \tan t\right) \cdot eh, \frac{\sin t}{ew}, t\_2\right)\right|}{\sqrt{1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{t\_2} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)) (t_2 (* (cos t) ew)))
   (if (<= eh -8e+86)
     (/ 1.0 (fabs (/ 1.0 (* (sin (atan (/ (/ t_1 ew) (cos t)))) t_1))))
     (if (<= eh 4.8e+70)
       (/
        (fabs (fma (* (* eh (tan t)) eh) (/ (sin t) ew) t_2))
        (sqrt (+ 1.0 (pow (* eh (/ (tan t) ew)) 2.0))))
       (fabs
        (* (sin (atan (* (/ (sin t) t_2) (- eh)))) (* (- eh) (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cos(t) * ew;
	double tmp;
	if (eh <= -8e+86) {
		tmp = 1.0 / fabs((1.0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)));
	} else if (eh <= 4.8e+70) {
		tmp = fabs(fma(((eh * tan(t)) * eh), (sin(t) / ew), t_2)) / sqrt((1.0 + pow((eh * (tan(t) / ew)), 2.0)));
	} else {
		tmp = fabs((sin(atan(((sin(t) / t_2) * -eh))) * (-eh * sin(t))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(cos(t) * ew)
	tmp = 0.0
	if (eh <= -8e+86)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(Float64(t_1 / ew) / cos(t)))) * t_1))));
	elseif (eh <= 4.8e+70)
		tmp = Float64(abs(fma(Float64(Float64(eh * tan(t)) * eh), Float64(sin(t) / ew), t_2)) / sqrt(Float64(1.0 + (Float64(eh * Float64(tan(t) / ew)) ^ 2.0))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(sin(t) / t_2) * Float64(-eh)))) * Float64(Float64(-eh) * sin(t))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, If[LessEqual[eh, -8e+86], N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, 4.8e+70], N[(N[Abs[N[(N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / t$95$2), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -8.0000000000000001e86

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if -8.0000000000000001e86 < eh < 4.79999999999999974e70

   1. Initial program 99.7%

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 80.6%

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

   8. Applied rewrites 92.0%

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

   if 4.79999999999999974e70 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 21.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 19.9%

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

   9. Taylor expanded in ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   10. 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. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. mul-1-neg N/A

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

       14. associate-/l* N/A

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

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

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

   11. Applied rewrites 77.5%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq 4.8 \cdot 10^{+70}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(\left(eh \cdot \tan t\right) \cdot eh, \frac{\sin t}{ew}, \cos t \cdot ew\right)\right|}{\sqrt{1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ \mathbf{if}\;eh \leq -6 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{t\_1}{ew}}{\cos t}\right) \cdot t\_1}\right|}\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh)))
   (if (<= eh -6e+85)
     (/ 1.0 (fabs (/ 1.0 (* (sin (atan (/ (/ t_1 ew) (cos t)))) t_1))))
     (if (<= eh 3.7e+63)
       (fabs
        (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* eh (/ (tan t) ew)))))))
       (fabs
        (*
         (sin (atan (* (/ (sin t) (* (cos t) ew)) (- eh))))
         (* (- eh) (sin t))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double tmp;
	if (eh <= -6e+85) {
		tmp = 1.0 / fabs((1.0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)));
	} else if (eh <= 3.7e+63) {
		tmp = fabs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	} else {
		tmp = fabs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = sin(t) * eh
    if (eh <= (-6d+85)) then
        tmp = 1.0d0 / abs((1.0d0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)))
    else if (eh <= 3.7d+63) then
        tmp = abs(((-ew * cos(t)) / ((-1.0d0) / cos(atan((eh * (tan(t) / ew)))))))
    else
        tmp = abs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.sin(t) * eh;
	double tmp;
	if (eh <= -6e+85) {
		tmp = 1.0 / Math.abs((1.0 / (Math.sin(Math.atan(((t_1 / ew) / Math.cos(t)))) * t_1)));
	} else if (eh <= 3.7e+63) {
		tmp = Math.abs(((-ew * Math.cos(t)) / (-1.0 / Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((Math.sin(t) / (Math.cos(t) * ew)) * -eh))) * (-eh * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	tmp = 0
	if eh <= -6e+85:
		tmp = 1.0 / math.fabs((1.0 / (math.sin(math.atan(((t_1 / ew) / math.cos(t)))) * t_1)))
	elif eh <= 3.7e+63:
		tmp = math.fabs(((-ew * math.cos(t)) / (-1.0 / math.cos(math.atan((eh * (math.tan(t) / ew)))))))
	else:
		tmp = math.fabs((math.sin(math.atan(((math.sin(t) / (math.cos(t) * ew)) * -eh))) * (-eh * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	tmp = 0.0
	if (eh <= -6e+85)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(Float64(t_1 / ew) / cos(t)))) * t_1))));
	elseif (eh <= 3.7e+63)
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(-1.0 / cos(atan(Float64(eh * Float64(tan(t) / ew)))))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(sin(t) / Float64(cos(t) * ew)) * Float64(-eh)))) * Float64(Float64(-eh) * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	tmp = 0.0;
	if (eh <= -6e+85)
		tmp = 1.0 / abs((1.0 / (sin(atan(((t_1 / ew) / cos(t)))) * t_1)));
	elseif (eh <= 3.7e+63)
		tmp = abs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	else
		tmp = abs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -6e+85], N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(N[(t$95$1 / ew), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[eh, 3.7e+63], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.0000000000000001e85

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if -6.0000000000000001e85 < eh < 3.69999999999999968e63

   1. Initial program 99.8%

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 79.7

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

   7. Applied rewrites 79.7%

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

   if 3.69999999999999968e63 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 22.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 21.1%

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

   9. Taylor expanded in ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   10. 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. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. mul-1-neg N/A

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

       14. associate-/l* N/A

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

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

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

   11. Applied rewrites 76.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6 \cdot 10^{+85}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{\sin t \cdot eh}{ew}}{\cos t}\right) \cdot \left(\sin t \cdot eh\right)}\right|}\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;eh \leq -6 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (sin (atan (* (/ (sin t) (* (cos t) ew)) (- eh))))
           (* (- eh) (sin t))))))
   (if (<= eh -6e+85)
     t_1
     (if (<= eh 3.7e+63)
       (fabs
        (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* eh (/ (tan t) ew)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))));
	double tmp;
	if (eh <= -6e+85) {
		tmp = t_1;
	} else if (eh <= 3.7e+63) {
		tmp = fabs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))))
    if (eh <= (-6d+85)) then
        tmp = t_1
    else if (eh <= 3.7d+63) then
        tmp = abs(((-ew * cos(t)) / ((-1.0d0) / cos(atan((eh * (tan(t) / ew)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((Math.sin(t) / (Math.cos(t) * ew)) * -eh))) * (-eh * Math.sin(t))));
	double tmp;
	if (eh <= -6e+85) {
		tmp = t_1;
	} else if (eh <= 3.7e+63) {
		tmp = Math.abs(((-ew * Math.cos(t)) / (-1.0 / Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((math.sin(t) / (math.cos(t) * ew)) * -eh))) * (-eh * math.sin(t))))
	tmp = 0
	if eh <= -6e+85:
		tmp = t_1
	elif eh <= 3.7e+63:
		tmp = math.fabs(((-ew * math.cos(t)) / (-1.0 / math.cos(math.atan((eh * (math.tan(t) / ew)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(sin(t) / Float64(cos(t) * ew)) * Float64(-eh)))) * Float64(Float64(-eh) * sin(t))))
	tmp = 0.0
	if (eh <= -6e+85)
		tmp = t_1;
	elseif (eh <= 3.7e+63)
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(-1.0 / cos(atan(Float64(eh * Float64(tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((sin(t) / (cos(t) * ew)) * -eh))) * (-eh * sin(t))));
	tmp = 0.0;
	if (eh <= -6e+85)
		tmp = t_1;
	elseif (eh <= 3.7e+63)
		tmp = abs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] * (-eh)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -6e+85], t$95$1, If[LessEqual[eh, 3.7e+63], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6.0000000000000001e85 or 3.69999999999999968e63 < 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

      \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)}\right| \]
   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 20.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 19.6%

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

   9. Taylor expanded in ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   10. 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. mul-1-neg N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. lower-neg.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-sin.f64 N/A

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

       12. lower-atan.f64 N/A

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

       13. mul-1-neg N/A

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

       14. associate-/l* N/A

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

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

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

   11. Applied rewrites 76.1%

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

   if -6.0000000000000001e85 < eh < 3.69999999999999968e63

   1. Initial program 99.8%

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

   4. Applied rewrites 81.5%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 79.7

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

   7. Applied rewrites 79.7%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6 \cdot 10^{+85}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 3.7 \cdot 10^{+63}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{\cos t \cdot ew} \cdot \left(-eh\right)\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{if}\;t \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{\frac{\left(eh \cdot t\right) \cdot \left(eh \cdot t\right)}{-ew} - \cos t \cdot ew}{\frac{-1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* eh (/ (tan t) ew)))))))))
   (if (<= t -3.5e+19)
     t_1
     (if (<= t 4.7e-17)
       (fabs
        (/
         (- (/ (* (* eh t) (* eh t)) (- ew)) (* (cos t) ew))
         (/ -1.0 (pow (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)) -0.5))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	double tmp;
	if (t <= -3.5e+19) {
		tmp = t_1;
	} else if (t <= 4.7e-17) {
		tmp = fabs((((((eh * t) * (eh * t)) / -ew) - (cos(t) * ew)) / (-1.0 / pow((1.0 + pow((ew / (eh * tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs(((-ew * cos(t)) / ((-1.0d0) / cos(atan((eh * (tan(t) / ew)))))))
    if (t <= (-3.5d+19)) then
        tmp = t_1
    else if (t <= 4.7d-17) then
        tmp = abs((((((eh * t) * (eh * t)) / -ew) - (cos(t) * ew)) / ((-1.0d0) / ((1.0d0 + ((ew / (eh * tan(t))) ** (-2.0d0))) ** (-0.5d0)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs(((-ew * Math.cos(t)) / (-1.0 / Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))));
	double tmp;
	if (t <= -3.5e+19) {
		tmp = t_1;
	} else if (t <= 4.7e-17) {
		tmp = Math.abs((((((eh * t) * (eh * t)) / -ew) - (Math.cos(t) * ew)) / (-1.0 / Math.pow((1.0 + Math.pow((ew / (eh * Math.tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((-ew * math.cos(t)) / (-1.0 / math.cos(math.atan((eh * (math.tan(t) / ew)))))))
	tmp = 0
	if t <= -3.5e+19:
		tmp = t_1
	elif t <= 4.7e-17:
		tmp = math.fabs((((((eh * t) * (eh * t)) / -ew) - (math.cos(t) * ew)) / (-1.0 / math.pow((1.0 + math.pow((ew / (eh * math.tan(t))), -2.0)), -0.5))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(-1.0 / cos(atan(Float64(eh * Float64(tan(t) / ew)))))))
	tmp = 0.0
	if (t <= -3.5e+19)
		tmp = t_1;
	elseif (t <= 4.7e-17)
		tmp = abs(Float64(Float64(Float64(Float64(Float64(eh * t) * Float64(eh * t)) / Float64(-ew)) - Float64(cos(t) * ew)) / Float64(-1.0 / (Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	tmp = 0.0;
	if (t <= -3.5e+19)
		tmp = t_1;
	elseif (t <= 4.7e-17)
		tmp = abs((((((eh * t) * (eh * t)) / -ew) - (cos(t) * ew)) / (-1.0 / ((1.0 + ((ew / (eh * tan(t))) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -3.5e+19], t$95$1, If[LessEqual[t, 4.7e-17], N[Abs[N[(N[(N[(N[(N[(eh * t), $MachinePrecision] * N[(eh * t), $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision] - N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Power[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.5e19 or 4.7e-17 < 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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 47.5

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

   7. Applied rewrites 47.5%

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

   if -3.5e19 < t < 4.7e-17

   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

   4. Applied rewrites 75.5%

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

   5. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unswap-sqr N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-neg.f64 73.8

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

   7. Applied rewrites 73.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. inv-pow N/A

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

      5. sqrt-pow2 N/A

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

      6. lower-pow.f64 N/A

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

   9. Applied rewrites 82.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+19}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{elif}\;t \leq 4.7 \cdot 10^{-17}:\\ \;\;\;\;\left|\frac{\frac{\left(eh \cdot t\right) \cdot \left(eh \cdot t\right)}{-ew} - \cos t \cdot ew}{\frac{-1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 62.3%

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

5. Taylor expanded in ew around inf

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 60.1

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

7. Applied rewrites 60.1%

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

8. Final simplification 60.1%

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

Alternative 11: 50.2% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 62.3%

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

5. Taylor expanded in t around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unswap-sqr N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-neg.f64 48.8

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

7. Applied rewrites 48.8%

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

8. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-*r/ N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 48.8

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

10. Applied rewrites 48.8%

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

11. Final simplification 48.8%

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

Alternative 12: 42.2% accurate, 61.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 41.6%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 40.2%

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

10. Applied rewrites 39.3%

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

11. Taylor expanded in ew around inf

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

13. Applied rewrites 41.7%

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

Reproduce

herbie shell --seed 2024259 
(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)))))))