Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 13.3s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 90.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \cos t \cdot ew\\ t_3 := \tan^{-1} t\_1\\ \mathbf{if}\;eh \leq -1.8 \cdot 10^{-45} \lor \neg \left(eh \leq 1.5 \cdot 10^{-158}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t\_3 \cdot \left(-\sin t\right), -eh, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot t\_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot t\_1 + t\_2}{{\cos t\_3}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh)) (t_2 (* (cos t) ew)) (t_3 (atan t_1)))
   (if (or (<= eh -1.8e-45) (not (<= eh 1.5e-158)))
     (fabs
      (fma
       (* (sin t_3) (- (sin t)))
       (- eh)
       (* (cos (atan (* (/ t ew) eh))) t_2)))
     (fabs (/ (+ (* (* (sin t) eh) t_1) t_2) (pow (cos t_3) -1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = cos(t) * ew;
	double t_3 = atan(t_1);
	double tmp;
	if ((eh <= -1.8e-45) || !(eh <= 1.5e-158)) {
		tmp = fabs(fma((sin(t_3) * -sin(t)), -eh, (cos(atan(((t / ew) * eh))) * t_2)));
	} else {
		tmp = fabs(((((sin(t) * eh) * t_1) + t_2) / pow(cos(t_3), -1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = Float64(cos(t) * ew)
	t_3 = atan(t_1)
	tmp = 0.0
	if ((eh <= -1.8e-45) || !(eh <= 1.5e-158))
		tmp = abs(fma(Float64(sin(t_3) * Float64(-sin(t))), Float64(-eh), Float64(cos(atan(Float64(Float64(t / ew) * eh))) * t_2)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * eh) * t_1) + t_2) / (cos(t_3) ^ -1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$1], $MachinePrecision]}, If[Or[LessEqual[eh, -1.8e-45], N[Not[LessEqual[eh, 1.5e-158]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t$95$3], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] * (-eh) + N[(N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / N[Power[N[Cos[t$95$3], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8e-45 or 1.5e-158 < 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 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 94.7

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

   7. Applied rewrites 94.7%

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

   if -1.8e-45 < eh < 1.5e-158

   1. Initial program 99.8%

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

   4. Applied rewrites 96.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{-45} \lor \neg \left(eh \leq 1.5 \cdot 10^{-158}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(-\sin t\right), -eh, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \cos t \cdot ew\\ t_3 := \tan^{-1} t\_1\\ \mathbf{if}\;eh \leq -1.8 \cdot 10^{-45} \lor \neg \left(eh \leq 1.7 \cdot 10^{-158}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\sin t\_3 \cdot \left(-\sin t\right), -eh, \frac{t\_2}{\sqrt{{\left(\frac{t}{ew} \cdot eh\right)}^{2} + 1}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot t\_1 + t\_2}{{\cos t\_3}^{-1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh)) (t_2 (* (cos t) ew)) (t_3 (atan t_1)))
   (if (or (<= eh -1.8e-45) (not (<= eh 1.7e-158)))
     (fabs
      (fma
       (* (sin t_3) (- (sin t)))
       (- eh)
       (/ t_2 (sqrt (+ (pow (* (/ t ew) eh) 2.0) 1.0)))))
     (fabs (/ (+ (* (* (sin t) eh) t_1) t_2) (pow (cos t_3) -1.0))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = cos(t) * ew;
	double t_3 = atan(t_1);
	double tmp;
	if ((eh <= -1.8e-45) || !(eh <= 1.7e-158)) {
		tmp = fabs(fma((sin(t_3) * -sin(t)), -eh, (t_2 / sqrt((pow(((t / ew) * eh), 2.0) + 1.0)))));
	} else {
		tmp = fabs(((((sin(t) * eh) * t_1) + t_2) / pow(cos(t_3), -1.0)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = Float64(cos(t) * ew)
	t_3 = atan(t_1)
	tmp = 0.0
	if ((eh <= -1.8e-45) || !(eh <= 1.7e-158))
		tmp = abs(fma(Float64(sin(t_3) * Float64(-sin(t))), Float64(-eh), Float64(t_2 / sqrt(Float64((Float64(Float64(t / ew) * eh) ^ 2.0) + 1.0)))));
	else
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * eh) * t_1) + t_2) / (cos(t_3) ^ -1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[t$95$1], $MachinePrecision]}, If[Or[LessEqual[eh, -1.8e-45], N[Not[LessEqual[eh, 1.7e-158]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t$95$3], $MachinePrecision] * (-N[Sin[t], $MachinePrecision])), $MachinePrecision] * (-eh) + N[(t$95$2 / N[Sqrt[N[(N[Power[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / N[Power[N[Cos[t$95$3], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8e-45 or 1.7e-158 < 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 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 94.7

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

   7. Applied rewrites 94.7%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 94.5

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

   9. Applied rewrites 94.5%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. un-div-inv N/A

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

       5. lower-/.f64 94.5

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

   11. Applied rewrites 94.5%

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

   if -1.8e-45 < eh < 1.7e-158

   1. Initial program 99.8%

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

   4. Applied rewrites 96.6%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.8 \cdot 10^{-45} \lor \neg \left(eh \leq 1.7 \cdot 10^{-158}\right):\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right) \cdot \left(-\sin t\right), -eh, \frac{\cos t \cdot ew}{\sqrt{{\left(\frac{t}{ew} \cdot eh\right)}^{2} + 1}}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ t_2 := \frac{\tan t}{ew} \cdot eh\\ \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot t\_2 + \cos t \cdot ew}{{\cos \tan^{-1} t\_2}^{-1}}\right|\\ \mathbf{elif}\;eh \leq 2.6 \cdot 10^{+263}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t}{ew} \cdot eh\right) - \mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (- eh) (sin t))
           (sin
            (atan
             (*
              (fma -0.3333333333333333 (/ (* (* t t) eh) ew) (/ (- eh) ew))
              t))))))
        (t_2 (* (/ (tan t) ew) eh)))
   (if (<= eh -3.4e+53)
     t_1
     (if (<= eh 8.5e+17)
       (fabs
        (/
         (+ (* (* (sin t) eh) t_2) (* (cos t) ew))
         (pow (cos (atan t_2)) -1.0)))
       (if (<= eh 2.6e+263)
         (fabs
          (-
           (* (* eh (sin t)) (sin (atan (* (/ (- t) ew) eh))))
           (*
            (fma (* -0.5 ew) (* t t) ew)
            (cos (atan (/ (* eh (tan t)) (- ew)))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((-eh * sin(t)) * sin(atan((fma(-0.3333333333333333, (((t * t) * eh) / ew), (-eh / ew)) * t)))));
	double t_2 = (tan(t) / ew) * eh;
	double tmp;
	if (eh <= -3.4e+53) {
		tmp = t_1;
	} else if (eh <= 8.5e+17) {
		tmp = fabs(((((sin(t) * eh) * t_2) + (cos(t) * ew)) / pow(cos(atan(t_2)), -1.0)));
	} else if (eh <= 2.6e+263) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-t / ew) * eh)))) - (fma((-0.5 * ew), (t * t), ew) * cos(atan(((eh * tan(t)) / -ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(-eh) * sin(t)) * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(Float64(t * t) * eh) / ew), Float64(Float64(-eh) / ew)) * t)))))
	t_2 = Float64(Float64(tan(t) / ew) * eh)
	tmp = 0.0
	if (eh <= -3.4e+53)
		tmp = t_1;
	elseif (eh <= 8.5e+17)
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * eh) * t_2) + Float64(cos(t) * ew)) / (cos(atan(t_2)) ^ -1.0)));
	elseif (eh <= 2.6e+263)
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-t) / ew) * eh)))) - Float64(fma(Float64(-0.5 * ew), Float64(t * t), ew) * cos(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] / ew), $MachinePrecision] + N[((-eh) / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[eh, -3.4e+53], t$95$1, If[LessEqual[eh, 8.5e+17], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 2.6e+263], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.39999999999999998e53 or 2.6000000000000002e263 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 80.6%

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

   if -3.39999999999999998e53 < eh < 8.5e17

   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%

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

   if 8.5e17 < eh < 2.6000000000000002e263

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.3

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

   8. Applied rewrites 77.3%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}^{-1}}\right|\\ \mathbf{elif}\;eh \leq 2.6 \cdot 10^{+263}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t}{ew} \cdot eh\right) - \mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. associate-*l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. *-commutative N/A

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

   8. lower-/.f64 N/A

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

   9. mul-1-neg N/A

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

   10. lower-neg.f64 99.7

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

5. Applied rewrites 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;eh \leq 2.6 \cdot 10^{+263}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t}{ew} \cdot eh\right) - \mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (- eh) (sin t))
           (sin
            (atan
             (*
              (fma -0.3333333333333333 (/ (* (* t t) eh) ew) (/ (- eh) ew))
              t)))))))
   (if (<= eh -3.4e+53)
     t_1
     (if (<= eh 8.5e+17)
       (fabs (* (cos t) ew))
       (if (<= eh 2.6e+263)
         (fabs
          (-
           (* (* eh (sin t)) (sin (atan (* (/ (- t) ew) eh))))
           (*
            (fma (* -0.5 ew) (* t t) ew)
            (cos (atan (/ (* eh (tan t)) (- ew)))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((-eh * sin(t)) * sin(atan((fma(-0.3333333333333333, (((t * t) * eh) / ew), (-eh / ew)) * t)))));
	double tmp;
	if (eh <= -3.4e+53) {
		tmp = t_1;
	} else if (eh <= 8.5e+17) {
		tmp = fabs((cos(t) * ew));
	} else if (eh <= 2.6e+263) {
		tmp = fabs((((eh * sin(t)) * sin(atan(((-t / ew) * eh)))) - (fma((-0.5 * ew), (t * t), ew) * cos(atan(((eh * tan(t)) / -ew))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(-eh) * sin(t)) * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(Float64(t * t) * eh) / ew), Float64(Float64(-eh) / ew)) * t)))))
	tmp = 0.0
	if (eh <= -3.4e+53)
		tmp = t_1;
	elseif (eh <= 8.5e+17)
		tmp = abs(Float64(cos(t) * ew));
	elseif (eh <= 2.6e+263)
		tmp = abs(Float64(Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(Float64(-t) / ew) * eh)))) - Float64(fma(Float64(-0.5 * ew), Float64(t * t), ew) * cos(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] / ew), $MachinePrecision] + N[((-eh) / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.4e+53], t$95$1, If[LessEqual[eh, 8.5e+17], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 2.6e+263], N[Abs[N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[((-t) / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.5 * ew), $MachinePrecision] * N[(t * t), $MachinePrecision] + ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.39999999999999998e53 or 2.6000000000000002e263 < eh

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 80.6%

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

   if -3.39999999999999998e53 < eh < 8.5e17

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 88.0

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 85.1

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

   12. Applied rewrites 85.1%

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

   if 8.5e17 < eh < 2.6000000000000002e263

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*l/ N/A

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

      7. *-commutative N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 77.3

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

   8. Applied rewrites 77.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;eh \leq 2.6 \cdot 10^{+263}:\\ \;\;\;\;\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{-t}{ew} \cdot eh\right) - \mathsf{fma}\left(-0.5 \cdot ew, t \cdot t, ew\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53} \lor \neg \left(eh \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -3.4e+53) (not (<= eh 1.15e+14)))
   (fabs
    (*
     (* (- eh) (sin t))
     (sin
      (atan
       (* (fma -0.3333333333333333 (/ (* (* t t) eh) ew) (/ (- eh) ew)) t)))))
   (fabs (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -3.4e+53) || !(eh <= 1.15e+14)) {
		tmp = fabs(((-eh * sin(t)) * sin(atan((fma(-0.3333333333333333, (((t * t) * eh) / ew), (-eh / ew)) * t)))));
	} else {
		tmp = fabs((cos(t) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -3.4e+53) || !(eh <= 1.15e+14))
		tmp = abs(Float64(Float64(Float64(-eh) * sin(t)) * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(Float64(t * t) * eh) / ew), Float64(Float64(-eh) / ew)) * t)))));
	else
		tmp = abs(Float64(cos(t) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -3.4e+53], N[Not[LessEqual[eh, 1.15e+14]], $MachinePrecision]], N[Abs[N[(N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] / ew), $MachinePrecision] + N[((-eh) / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.39999999999999998e53 or 1.15e14 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.1%

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

   if -3.39999999999999998e53 < eh < 1.15e14

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 88.2

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

   7. Applied rewrites 88.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 88.0

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

   9. Applied rewrites 88.0%

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

   10. Taylor expanded in eh around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-cos.f64 85.6

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

   12. Applied rewrites 85.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{+53} \lor \neg \left(eh \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\left|\left(\left(-eh\right) \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\mathsf{fma}\left(-0.3333333333333333, \frac{\left(t \cdot t\right) \cdot eh}{ew}, \frac{-eh}{ew}\right) \cdot t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.7% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 91.7

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

7. Applied rewrites 91.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 91.5

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

9. Applied rewrites 91.5%

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

10. Taylor expanded in eh around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-cos.f64 62.1

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

12. Applied rewrites 62.1%

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

Alternative 9: 41.7% accurate, 107.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

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 42.6%

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

7. Applied rewrites 42.3%

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

8. Taylor expanded in eh around 0

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

10. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2024327 
(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)))))))