Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.5s

Alternatives: 6

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) (/ eh ew))))
   (fabs
    (+
     (* ew (/ (cos t) (hypot 1.0 t_1)))
     (* eh (* (sin t) (sin (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * (eh / ew);
	return fabs(((ew * (cos(t) / hypot(1.0, t_1))) + (eh * (sin(t) * sin(atan(t_1))))));
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) * (eh / ew);
	return Math.abs(((ew * (Math.cos(t) / Math.hypot(1.0, t_1))) + (eh * (Math.sin(t) * Math.sin(Math.atan(t_1))))));
}

Python

def code(eh, ew, t):
	t_1 = math.tan(t) * (eh / ew)
	return math.fabs(((ew * (math.cos(t) / math.hypot(1.0, t_1))) + (eh * (math.sin(t) * math.sin(math.atan(t_1))))))

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * Float64(eh / ew))
	return abs(Float64(Float64(ew * Float64(cos(t) / hypot(1.0, t_1))) + Float64(eh * Float64(sin(t) * sin(atan(t_1))))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = tan(t) * (eh / ew);
	tmp = abs(((ew * (cos(t) / hypot(1.0, t_1))) + (eh * (sin(t) * sin(atan(t_1))))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(ew * N[(N[Cos[t], $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

4. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*r/ 99.8%

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

   4. associate-*r/ 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r/ 99.8%

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

   7. *-commutative 99.8%

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

   8. associate-*r/ 99.8%

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 98.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg 99.8%

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around inf 98.3%

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

6. Final simplification 98.3%

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

Alternative 3: 74.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;t \leq -3.75 \cdot 10^{-6} \lor \neg \left(t \leq 4.4 \cdot 10^{-18}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= t -9.5e+68)
   (fabs (* (cos t) ew))
   (if (or (<= t -3.75e-6) (not (<= t 4.4e-18)))
     (fabs (* eh (* (sin t) (sin (atan (/ (* t eh) ew))))))
     (fabs (+ ew (* eh (* t (sin (atan (/ (* (tan t) eh) ew))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -9.5e+68) {
		tmp = fabs((cos(t) * ew));
	} else if ((t <= -3.75e-6) || !(t <= 4.4e-18)) {
		tmp = fabs((eh * (sin(t) * sin(atan(((t * eh) / ew))))));
	} else {
		tmp = fabs((ew + (eh * (t * sin(atan(((tan(t) * eh) / ew)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-9.5d+68)) then
        tmp = abs((cos(t) * ew))
    else if ((t <= (-3.75d-6)) .or. (.not. (t <= 4.4d-18))) then
        tmp = abs((eh * (sin(t) * sin(atan(((t * eh) / ew))))))
    else
        tmp = abs((ew + (eh * (t * sin(atan(((tan(t) * eh) / ew)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (t <= -9.5e+68) {
		tmp = Math.abs((Math.cos(t) * ew));
	} else if ((t <= -3.75e-6) || !(t <= 4.4e-18)) {
		tmp = Math.abs((eh * (Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew))))));
	} else {
		tmp = Math.abs((ew + (eh * (t * Math.sin(Math.atan(((Math.tan(t) * eh) / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if t <= -9.5e+68:
		tmp = math.fabs((math.cos(t) * ew))
	elif (t <= -3.75e-6) or not (t <= 4.4e-18):
		tmp = math.fabs((eh * (math.sin(t) * math.sin(math.atan(((t * eh) / ew))))))
	else:
		tmp = math.fabs((ew + (eh * (t * math.sin(math.atan(((math.tan(t) * eh) / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (t <= -9.5e+68)
		tmp = abs(Float64(cos(t) * ew));
	elseif ((t <= -3.75e-6) || !(t <= 4.4e-18))
		tmp = abs(Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew))))));
	else
		tmp = abs(Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(tan(t) * eh) / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (t <= -9.5e+68)
		tmp = abs((cos(t) * ew));
	elseif ((t <= -3.75e-6) || ~((t <= 4.4e-18)))
		tmp = abs((eh * (sin(t) * sin(atan(((t * eh) / ew))))));
	else
		tmp = abs((ew + (eh * (t * sin(atan(((tan(t) * eh) / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[t, -9.5e+68], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], If[Or[LessEqual[t, -3.75e-6], N[Not[LessEqual[t, 4.4e-18]], $MachinePrecision]], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.50000000000000069e68

   1. Initial program 99.6%

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

      1. add-cbrt-cube 29.7%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

      2. pow3 29.8%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

   4. Applied egg-rr 29.8%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

   5. Taylor expanded in ew around inf 20.7%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\cos t}^{3}}}\right| \]
   6. Step-by-step derivation

      1. pow-prod-down 20.8%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \cos t\right)}^{3}}}\right| \]

      2. rem-cbrt-cube 67.6%

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

      3. *-commutative 67.6%

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

   7. Applied egg-rr 67.6%

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

   if -9.50000000000000069e68 < t < -3.7500000000000001e-6 or 4.3999999999999997e-18 < t

   1. Initial program 99.6%

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

      1. add-cbrt-cube 39.5%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

      2. pow3 39.5%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

   4. Applied egg-rr 39.5%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

   5. Taylor expanded in ew around 0 64.0%

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

   6. Taylor expanded in t around 0 64.2%

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

   if -3.7500000000000001e-6 < t < 4.3999999999999997e-18

   1. Initial program 100.0%

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

      1. add-cbrt-cube 39.8%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

      2. pow3 39.9%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

   4. Applied egg-rr 39.9%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

   5. Taylor expanded in t around 0 99.1%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+68}:\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{elif}\;t \leq -3.75 \cdot 10^{-6} \lor \neg \left(t \leq 4.4 \cdot 10^{-18}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -3.6 \cdot 10^{-53} \lor \neg \left(ew \leq 3.1 \cdot 10^{-103}\right):\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -3.6e-53) (not (<= ew 3.1e-103)))
   (fabs (* (cos t) ew))
   (fabs (* eh (* (sin t) (sin (atan (/ (* t eh) ew))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.6e-53) || !(ew <= 3.1e-103)) {
		tmp = fabs((cos(t) * ew));
	} else {
		tmp = fabs((eh * (sin(t) * sin(atan(((t * eh) / ew))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-3.6d-53)) .or. (.not. (ew <= 3.1d-103))) then
        tmp = abs((cos(t) * ew))
    else
        tmp = abs((eh * (sin(t) * sin(atan(((t * eh) / ew))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -3.6e-53) || !(ew <= 3.1e-103)) {
		tmp = Math.abs((Math.cos(t) * ew));
	} else {
		tmp = Math.abs((eh * (Math.sin(t) * Math.sin(Math.atan(((t * eh) / ew))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -3.6e-53) or not (ew <= 3.1e-103):
		tmp = math.fabs((math.cos(t) * ew))
	else:
		tmp = math.fabs((eh * (math.sin(t) * math.sin(math.atan(((t * eh) / ew))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -3.6e-53) || !(ew <= 3.1e-103))
		tmp = abs(Float64(cos(t) * ew));
	else
		tmp = abs(Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(t * eh) / ew))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -3.6e-53) || ~((ew <= 3.1e-103)))
		tmp = abs((cos(t) * ew));
	else
		tmp = abs((eh * (sin(t) * sin(atan(((t * eh) / ew))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -3.6e-53], N[Not[LessEqual[ew, 3.1e-103]], $MachinePrecision]], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -3.5999999999999999e-53 or 3.1000000000000001e-103 < ew

   1. Initial program 99.8%

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

      1. add-cbrt-cube 38.3%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

      2. pow3 38.3%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

   4. Applied egg-rr 38.3%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

   5. Taylor expanded in ew around inf 31.8%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\cos t}^{3}}}\right| \]
   6. Step-by-step derivation

      1. pow-prod-down 31.8%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \cos t\right)}^{3}}}\right| \]

      2. rem-cbrt-cube 80.6%

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

      3. *-commutative 80.6%

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

   7. Applied egg-rr 80.6%

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

   if -3.5999999999999999e-53 < ew < 3.1000000000000001e-103

   1. Initial program 99.8%

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

      1. add-cbrt-cube 37.6%

         \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

      2. pow3 37.6%

         \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

   4. Applied egg-rr 37.6%

      \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

   5. Taylor expanded in ew around 0 73.4%

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

   6. Taylor expanded in t around 0 73.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.6 \cdot 10^{-53} \lor \neg \left(ew \leq 3.1 \cdot 10^{-103}\right):\\ \;\;\;\;\left|\cos t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{t \cdot eh}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 61.8% accurate, 4.5× 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
3. Step-by-step derivation

   1. add-cbrt-cube 38.0%

      \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

   2. pow3 38.0%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

4. Applied egg-rr 38.0%

   \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

5. Taylor expanded in ew around inf 22.1%

   \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3} \cdot {\cos t}^{3}}}\right| \]
6. Step-by-step derivation

   1. pow-prod-down 22.1%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(ew \cdot \cos t\right)}^{3}}}\right| \]

   2. rem-cbrt-cube 60.8%

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

   3. *-commutative 60.8%

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

7. Applied egg-rr 60.8%

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

Alternative 6: 42.4% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. add-cbrt-cube 38.0%

      \[\leadsto \left|\color{blue}{\sqrt[3]{\left(\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)\right) \cdot \left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}}\right| \]

   2. pow3 38.0%

      \[\leadsto \left|\sqrt[3]{\color{blue}{{\left(\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}^{3}}}\right| \]

4. Applied egg-rr 38.0%

   \[\leadsto \left|\color{blue}{\sqrt[3]{{\left(\mathsf{fma}\left(ew, \frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\right)\right)}^{3}}}\right| \]

5. Taylor expanded in t around 0 17.5%

   \[\leadsto \left|\sqrt[3]{\color{blue}{{ew}^{3}}}\right| \]

6. Taylor expanded in ew around 0 43.7%

   \[\leadsto \color{blue}{\left|ew\right|} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024141 
(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)))))))