Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 10.1s

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

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* eh (/ (tan t) ew)))))
   (fabs (fma (* (cos t_1) ew) (cos t) (* (* (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_1) * ew), cos(t), ((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_1) * ew), cos(t), 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$95$1], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t], $MachinePrecision] + 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%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Final simplification 99.0%

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

Alternative 3: 94.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot ew, \cos t, t\_1 \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{if}\;eh \leq -2300000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;{\left(1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}\right)}^{-0.5} \cdot \left|\mathsf{fma}\left(\frac{t\_1}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2
         (fabs
          (fma
           (* (cos (atan (* (/ eh ew) t))) ew)
           (cos t)
           (* t_1 (sin (atan (* (/ t ew) eh))))))))
   (if (<= eh -2300000.0)
     t_2
     (if (<= eh 1.9e-23)
       (*
        (pow (+ 1.0 (pow (* eh (/ (tan t) ew)) 2.0)) -0.5)
        (fabs (fma (/ t_1 ew) (* eh (tan t)) (* (cos t) ew))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs(fma((cos(atan(((eh / ew) * t))) * ew), cos(t), (t_1 * sin(atan(((t / ew) * eh))))));
	double tmp;
	if (eh <= -2300000.0) {
		tmp = t_2;
	} else if (eh <= 1.9e-23) {
		tmp = pow((1.0 + pow((eh * (tan(t) / ew)), 2.0)), -0.5) * fabs(fma((t_1 / ew), (eh * tan(t)), (cos(t) * ew)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = abs(fma(Float64(cos(atan(Float64(Float64(eh / ew) * t))) * ew), cos(t), Float64(t_1 * sin(atan(Float64(Float64(t / ew) * eh))))))
	tmp = 0.0
	if (eh <= -2300000.0)
		tmp = t_2;
	elseif (eh <= 1.9e-23)
		tmp = Float64((Float64(1.0 + (Float64(eh * Float64(tan(t) / ew)) ^ 2.0)) ^ -0.5) * abs(fma(Float64(t_1 / ew), Float64(eh * tan(t)), Float64(cos(t) * ew))));
	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[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[t], $MachinePrecision] + N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2300000.0], t$95$2, If[LessEqual[eh, 1.9e-23], N[(N[Power[N[(1.0 + N[Power[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[Abs[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.3e6 or 1.90000000000000006e-23 < 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%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 99.0

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

   7. Applied rewrites 99.0%

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

   8. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 94.3

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

   10. Applied rewrites 94.3%

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

   if -2.3e6 < eh < 1.90000000000000006e-23

   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%

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

   6. Applied rewrites 85.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

      6. cos-atan N/A

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

      7. pow1/2 N/A

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

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

   8. Applied rewrites 96.6%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2300000:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot ew, \cos t, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{elif}\;eh \leq 1.9 \cdot 10^{-23}:\\ \;\;\;\;{\left(1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}\right)}^{-0.5} \cdot \left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, \cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot t\right) \cdot ew, \cos t, \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 91.7

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

10. Applied rewrites 91.7%

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

11. Final simplification 91.7%

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

Alternative 5: 90.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.0

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

7. Applied rewrites 99.0%

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

8. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 91.7

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

10. Applied rewrites 91.7%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-cos.f64 N/A

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

    4. lift-atan.f64 N/A

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

    5. cos-atan N/A

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

    6. un-div-inv N/A

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

    7. lower-/.f64 N/A

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

    8. lower-sqrt.f64 N/A

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

    9. +-commutative N/A

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

    10. lower-+.f64 N/A

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

12. Applied rewrites 91.6%

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

13. Final simplification 91.6%

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

Alternative 6: 74.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos t \cdot ew\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -2.1 \cdot 10^{-92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.3 \cdot 10^{+16}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (fabs (* (cos t) ew)) (cos (atan (* eh (/ (tan t) ew)))))))
   (if (<= ew -2.1e-92)
     t_1
     (if (<= ew 3.3e+16)
       (fabs
        (* (sin (atan (* (/ (sin t) ew) (/ eh (cos t))))) (* (- eh) (sin t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(t) * ew)) * cos(atan((eh * (tan(t) / ew))));
	double tmp;
	if (ew <= -2.1e-92) {
		tmp = t_1;
	} else if (ew <= 3.3e+16) {
		tmp = fabs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (-eh * sin(t))));
	} 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((cos(t) * ew)) * cos(atan((eh * (tan(t) / ew))))
    if (ew <= (-2.1d-92)) then
        tmp = t_1
    else if (ew <= 3.3d+16) then
        tmp = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (-eh * sin(t))))
    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.cos(t) * ew)) * Math.cos(Math.atan((eh * (Math.tan(t) / ew))));
	double tmp;
	if (ew <= -2.1e-92) {
		tmp = t_1;
	} else if (ew <= 3.3e+16) {
		tmp = Math.abs((Math.sin(Math.atan(((Math.sin(t) / ew) * (eh / Math.cos(t))))) * (-eh * Math.sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(t) * ew)) * math.cos(math.atan((eh * (math.tan(t) / ew))))
	tmp = 0
	if ew <= -2.1e-92:
		tmp = t_1
	elif ew <= 3.3e+16:
		tmp = math.fabs((math.sin(math.atan(((math.sin(t) / ew) * (eh / math.cos(t))))) * (-eh * math.sin(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(abs(Float64(cos(t) * ew)) * cos(atan(Float64(eh * Float64(tan(t) / ew)))))
	tmp = 0.0
	if (ew <= -2.1e-92)
		tmp = t_1;
	elseif (ew <= 3.3e+16)
		tmp = abs(Float64(sin(atan(Float64(Float64(sin(t) / ew) * Float64(eh / cos(t))))) * Float64(Float64(-eh) * sin(t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(t) * ew)) * cos(atan((eh * (tan(t) / ew))));
	tmp = 0.0;
	if (ew <= -2.1e-92)
		tmp = t_1;
	elseif (ew <= 3.3e+16)
		tmp = abs((sin(atan(((sin(t) / ew) * (eh / cos(t))))) * (-eh * sin(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -2.1e-92], t$95$1, If[LessEqual[ew, 3.3e+16], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision] * N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.1e-92 or 3.3e16 < ew

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 83.6%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 83.0

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

   9. Applied rewrites 83.0%

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

   if -2.1e-92 < ew < 3.3e16

   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 31.5%

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

   6. Applied rewrites 60.2%

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

   7. Taylor expanded in eh around inf

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

      1. unpow2 N/A

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

      2. rem-square-sqrt N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. neg-mul-1 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. lower-atan.f64 N/A

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

   9. Applied rewrites 75.0%

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

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.1 \cdot 10^{-92}:\\ \;\;\;\;\left|\cos t \cdot ew\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 3.3 \cdot 10^{+16}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\sin t}{ew} \cdot \frac{eh}{\cos t}\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos t \cdot ew\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

6. Applied rewrites 60.4%

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

7. Taylor expanded in eh around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-cos.f64 58.9

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

9. Applied rewrites 58.9%

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

10. Final simplification 58.9%

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

Alternative 8: 52.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(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 60.4%

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

5. Taylor expanded in eh around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-cos.f64 58.9

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

7. Applied rewrites 58.9%

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

8. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 51.1

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

10. Applied rewrites 51.1%

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

11. Final simplification 51.1%

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

Alternative 9: 41.7% 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 38.7%

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

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 37.6%

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

10. Applied rewrites 36.7%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 38.9%

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

Reproduce

herbie shell --seed 2024332 
(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)))))))