Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.8s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan((eh / (ew * tan(t))));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / (ew * tan(t))))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-tan.f64 N/A

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

   2. associate-/l/ N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.8

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

4. Applied egg-rr 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-tan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-atan.f64 N/A

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

   5. lift-cos.f64 99.8

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. associate-/r* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-/.f64 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 2: 93.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|t\_2 \cdot \sin t\_3 + t\_1 \cdot \cos t\_3\right| \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\left|t\_1 + t\_2 \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 + t\_2 \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (* eh (cos t)))
        (t_3 (atan (/ (/ eh ew) (tan t)))))
   (if (<= (fabs (+ (* t_2 (sin t_3)) (* t_1 (cos t_3)))) 5e+47)
     (fabs
      (+
       t_1
       (*
        t_2
        (sin
         (atan
          (/ (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew)) t))))))
     (fabs (+ t_1 (* t_2 (sin (atan (/ eh (* ew t))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = eh * cos(t);
	double t_3 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs(((t_2 * sin(t_3)) + (t_1 * cos(t_3)))) <= 5e+47) {
		tmp = fabs((t_1 + (t_2 * sin(atan((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))))));
	} else {
		tmp = fabs((t_1 + (t_2 * sin(atan((eh / (ew * t)))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(eh * cos(t))
	t_3 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(t_2 * sin(t_3)) + Float64(t_1 * cos(t_3)))) <= 5e+47)
		tmp = abs(Float64(t_1 + Float64(t_2 * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = abs(Float64(t_1 + Float64(t_2 * sin(atan(Float64(eh / Float64(ew * t)))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(t$95$2 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 5e+47], N[Abs[N[(t$95$1 + N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 + N[(t$95$2 * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 5.00000000000000022e47

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.3

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

   5. Simplified 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 81.7

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

   7. Applied egg-rr 81.7%

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

   8. Taylor expanded in eh around 0

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

   10. Simplified 95.8%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 95.7

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

   13. Simplified 95.7%

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

   if 5.00000000000000022e47 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.3

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

   5. Simplified 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 90.0

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

   7. Applied egg-rr 90.0%

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

   8. Taylor expanded in eh around 0

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

   10. Simplified 99.3%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 93.3

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

   13. Simplified 93.3%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 5 \cdot 10^{+47}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.9

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

5. Simplified 98.9%

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

Alternative 4: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. sqr-neg N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. sqr-neg N/A

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

   14. lower-*.f64 86.5

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

7. Applied egg-rr 86.5%

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

8. Taylor expanded in eh around 0

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

10. Simplified 97.8%

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

    1. lift-cos.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-tan.f64 N/A

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

    4. lift-/.f64 N/A

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

    5. lift-/.f64 N/A

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

    6. lift-atan.f64 N/A

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

    7. lift-sin.f64 N/A

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

    8. *-commutative N/A

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

    9. lift-*.f64 N/A

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

    10. associate-*r* N/A

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

    11. lower-*.f64 N/A

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

12. Applied egg-rr 97.8%

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

13. Final simplification 97.8%

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

Alternative 5: 98.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \sin t\\ t_2 := eh \cdot \cos t\\ t_3 := \left|t\_1 \cdot \frac{1}{\sqrt{\mathsf{fma}\left(eh, \frac{eh}{ew \cdot \left(ew \cdot \left(t \cdot t\right)\right)}, 1\right)}} + t\_2 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} \cdot \mathsf{fma}\left(t, t \cdot -0.3333333333333333, 1\right)}{t}\right)\right|\\ \mathbf{if}\;eh \leq -2.12 \cdot 10^{+14}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 10:\\ \;\;\;\;\left|t\_1 + t\_2 \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (sin t)))
        (t_2 (* eh (cos t)))
        (t_3
         (fabs
          (+
           (* t_1 (/ 1.0 (sqrt (fma eh (/ eh (* ew (* ew (* t t)))) 1.0))))
           (*
            t_2
            (sin
             (atan
              (/ (* (/ eh ew) (fma t (* t -0.3333333333333333) 1.0)) t))))))))
   (if (<= eh -2.12e+14)
     t_3
     (if (<= eh 10.0)
       (fabs
        (+
         t_1
         (*
          t_2
          (sin
           (atan
            (/
             (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew))
             t))))))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * sin(t);
	double t_2 = eh * cos(t);
	double t_3 = fabs(((t_1 * (1.0 / sqrt(fma(eh, (eh / (ew * (ew * (t * t)))), 1.0)))) + (t_2 * sin(atan((((eh / ew) * fma(t, (t * -0.3333333333333333), 1.0)) / t))))));
	double tmp;
	if (eh <= -2.12e+14) {
		tmp = t_3;
	} else if (eh <= 10.0) {
		tmp = fabs((t_1 + (t_2 * sin(atan((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * sin(t))
	t_2 = Float64(eh * cos(t))
	t_3 = abs(Float64(Float64(t_1 * Float64(1.0 / sqrt(fma(eh, Float64(eh / Float64(ew * Float64(ew * Float64(t * t)))), 1.0)))) + Float64(t_2 * sin(atan(Float64(Float64(Float64(eh / ew) * fma(t, Float64(t * -0.3333333333333333), 1.0)) / t))))))
	tmp = 0.0
	if (eh <= -2.12e+14)
		tmp = t_3;
	elseif (eh <= 10.0)
		tmp = abs(Float64(t_1 + Float64(t_2 * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(t$95$1 * N[(1.0 / N[Sqrt[N[(eh * N[(eh / N[(ew * N[(ew * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(N[(eh / ew), $MachinePrecision] * N[(t * N[(t * -0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.12e+14], t$95$3, If[LessEqual[eh, 10.0], N[Abs[N[(t$95$1 + N[(t$95$2 * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.12e14 or 10 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh \cdot {t}^{2}}{ew} \cdot \frac{-1}{3}} + \frac{eh}{ew}}{t}\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\color{blue}{{t}^{2} \cdot eh}}{ew} \cdot \frac{-1}{3} + \frac{eh}{ew}}{t}\right)\right| \]

      3. associate-/l* N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left({t}^{2} \cdot \frac{eh}{ew}\right)} \cdot \frac{-1}{3} + \frac{eh}{ew}}{t}\right)\right| \]

      4. associate-*r* N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\color{blue}{{t}^{2} \cdot \left(\frac{eh}{ew} \cdot \frac{-1}{3}\right)} + \frac{eh}{ew}}{t}\right)\right| \]

      5. metadata-eval N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{{t}^{2} \cdot \left(\frac{eh}{ew} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}\right) + \frac{eh}{ew}}{t}\right)\right| \]

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

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{{t}^{2} \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{eh}{ew} \cdot \frac{1}{3}\right)\right)} + \frac{eh}{ew}}{t}\right)\right| \]

      7. *-commutative N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{{t}^{2} \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{1}{3} \cdot \frac{eh}{ew}}\right)\right) + \frac{eh}{ew}}{t}\right)\right| \]

      8. lower-/.f64 N/A

         \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{{t}^{2} \cdot \left(\mathsf{neg}\left(\frac{1}{3} \cdot \frac{eh}{ew}\right)\right) + \frac{eh}{ew}}{t}\right)}\right| \]

   5. Simplified 99.4%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t, t \cdot \frac{-1}{3}, 1\right) \cdot \frac{eh}{ew}}{t}\right)\right| \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.9

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

   8. Simplified 98.9%

      \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot t}\right)} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(t, t \cdot -0.3333333333333333, 1\right) \cdot \frac{eh}{ew}}{t}\right)\right| \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. times-frac N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-sqrt.f64 N/A

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

      12. lift-/.f64 87.6

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

   10. Applied egg-rr 98.9%

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

   if -2.12e14 < eh < 10

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.6

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

   5. Simplified 98.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 85.3

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

   7. Applied egg-rr 85.3%

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

   8. Taylor expanded in eh around 0

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

   10. Simplified 96.8%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 96.7

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

   13. Simplified 96.7%

       \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.12 \cdot 10^{+14}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(eh, \frac{eh}{ew \cdot \left(ew \cdot \left(t \cdot t\right)\right)}, 1\right)}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} \cdot \mathsf{fma}\left(t, t \cdot -0.3333333333333333, 1\right)}{t}\right)\right|\\ \mathbf{elif}\;eh \leq 10:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{\mathsf{fma}\left(eh, \frac{eh}{ew \cdot \left(ew \cdot \left(t \cdot t\right)\right)}, 1\right)}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew} \cdot \mathsf{fma}\left(t, t \cdot -0.3333333333333333, 1\right)}{t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh}{ew \cdot t}\\ t_2 := ew \cdot \sin t\\ t_3 := eh \cdot \cos t\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{+164}:\\ \;\;\;\;\left|t\_2 + t\_3 \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2 \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot t\_1}{ew \cdot t}}} + t\_3 \cdot \sin \tan^{-1} t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ eh (* ew t))) (t_2 (* ew (sin t))) (t_3 (* eh (cos t))))
   (if (<= t -1.7e+164)
     (fabs
      (+
       t_2
       (*
        t_3
        (sin
         (atan
          (/ (fma -0.3333333333333333 (/ (* eh (* t t)) ew) (/ eh ew)) t))))))
     (fabs
      (+
       (* t_2 (/ 1.0 (sqrt (+ 1.0 (/ (* eh t_1) (* ew t))))))
       (* t_3 (sin (atan t_1))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh / (ew * t);
	double t_2 = ew * sin(t);
	double t_3 = eh * cos(t);
	double tmp;
	if (t <= -1.7e+164) {
		tmp = fabs((t_2 + (t_3 * sin(atan((fma(-0.3333333333333333, ((eh * (t * t)) / ew), (eh / ew)) / t))))));
	} else {
		tmp = fabs(((t_2 * (1.0 / sqrt((1.0 + ((eh * t_1) / (ew * t)))))) + (t_3 * sin(atan(t_1)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh / Float64(ew * t))
	t_2 = Float64(ew * sin(t))
	t_3 = Float64(eh * cos(t))
	tmp = 0.0
	if (t <= -1.7e+164)
		tmp = abs(Float64(t_2 + Float64(t_3 * sin(atan(Float64(fma(-0.3333333333333333, Float64(Float64(eh * Float64(t * t)) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = abs(Float64(Float64(t_2 * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(eh * t_1) / Float64(ew * t)))))) + Float64(t_3 * sin(atan(t_1)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.7e+164], N[Abs[N[(t$95$2 + N[(t$95$3 * N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(t$95$2 * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(eh * t$95$1), $MachinePrecision] / N[(ew * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.7000000000000001e164

   1. Initial program 99.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.6

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

   5. Simplified 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 85.8

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

   7. Applied egg-rr 85.8%

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

   8. Taylor expanded in eh around 0

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

   10. Simplified 99.6%

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

   11. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left|\left(ew \cdot \sin t\right) \cdot 1 + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}\right| \]

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 N/A

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

       7. lower-/.f64 92.8

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

   13. Simplified 92.8%

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

   if -1.7000000000000001e164 < t

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.8

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

   5. Simplified 98.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 86.6

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

   7. Applied egg-rr 86.6%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 83.2

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

   10. Simplified 83.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. times-frac N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*r/ N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 94.6

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

   12. Applied egg-rr 94.6%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{+164}:\\ \;\;\;\;\left|ew \cdot \sin t + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \frac{eh \cdot \left(t \cdot t\right)}{ew}, \frac{eh}{ew}\right)}{t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot \frac{eh}{ew \cdot t}}{ew \cdot t}}} + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+129}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -4.5e+129)
   (fabs
    (+
     (* (* eh (cos t)) (sin (atan (/ eh (* ew t)))))
     (* (* ew t) (/ 1.0 (sqrt (+ 1.0 (/ (* eh eh) (* (* ew t) (* ew t)))))))))
   (fabs (+ (* ew (sin t)) (* eh (sin (atan (/ eh (* ew (tan t))))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -4.5e+129) {
		tmp = fabs((((eh * cos(t)) * sin(atan((eh / (ew * t))))) + ((ew * t) * (1.0 / sqrt((1.0 + ((eh * eh) / ((ew * t) * (ew * t)))))))));
	} else {
		tmp = fabs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * tan(t))))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-4.5d+129)) then
        tmp = abs((((eh * cos(t)) * sin(atan((eh / (ew * t))))) + ((ew * t) * (1.0d0 / sqrt((1.0d0 + ((eh * eh) / ((ew * t) * (ew * t)))))))))
    else
        tmp = abs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * tan(t))))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -4.5e+129) {
		tmp = Math.abs((((eh * Math.cos(t)) * Math.sin(Math.atan((eh / (ew * t))))) + ((ew * t) * (1.0 / Math.sqrt((1.0 + ((eh * eh) / ((ew * t) * (ew * t)))))))));
	} else {
		tmp = Math.abs(((ew * Math.sin(t)) + (eh * Math.sin(Math.atan((eh / (ew * Math.tan(t))))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -4.5e+129:
		tmp = math.fabs((((eh * math.cos(t)) * math.sin(math.atan((eh / (ew * t))))) + ((ew * t) * (1.0 / math.sqrt((1.0 + ((eh * eh) / ((ew * t) * (ew * t)))))))))
	else:
		tmp = math.fabs(((ew * math.sin(t)) + (eh * math.sin(math.atan((eh / (ew * math.tan(t))))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -4.5e+129)
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(eh / Float64(ew * t))))) + Float64(Float64(ew * t) * Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(eh * eh) / Float64(Float64(ew * t) * Float64(ew * t)))))))));
	else
		tmp = abs(Float64(Float64(ew * sin(t)) + Float64(eh * sin(atan(Float64(eh / Float64(ew * tan(t))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -4.5e+129)
		tmp = abs((((eh * cos(t)) * sin(atan((eh / (ew * t))))) + ((ew * t) * (1.0 / sqrt((1.0 + ((eh * eh) / ((ew * t) * (ew * t)))))))));
	else
		tmp = abs(((ew * sin(t)) + (eh * sin(atan((eh / (ew * tan(t))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -4.5e+129], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(eh * eh), $MachinePrecision] / N[(N[(ew * t), $MachinePrecision] * N[(ew * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] + N[(eh * N[Sin[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.5000000000000001e129

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.8

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

   5. Simplified 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 88.1

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

   7. Applied egg-rr 88.1%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 88.1

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

   10. Simplified 88.1%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 85.4

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

   13. Simplified 85.4%

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

   if -4.5000000000000001e129 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.7

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

   5. Simplified 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 86.2

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

   7. Applied egg-rr 86.2%

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

   8. Taylor expanded in eh around 0

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

   10. Simplified 97.5%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-atan.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-tan.f64 82.3

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

   13. Simplified 82.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.5 \cdot 10^{+129}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \sin t + eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 88.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. sqr-neg N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. sqr-neg N/A

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

   14. lower-*.f64 86.5

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

7. Applied egg-rr 86.5%

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

8. Taylor expanded in eh around 0

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

10. Simplified 97.8%

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

11. Taylor expanded in t around 0

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

    1. lower-/.f64 N/A

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

    2. lower-*.f64 90.9

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

13. Simplified 90.9%

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

14. Final simplification 90.9%

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

Alternative 9: 62.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}}\\ t_2 := \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right)\\ t_3 := \left|\left(ew \cdot \sin t\right) \cdot t\_1 + t\_2 \cdot \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), -0.5, eh\right)\right|\\ \mathbf{if}\;ew \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot t\_2 + \left(ew \cdot t\right) \cdot t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (sqrt (+ 1.0 (/ (* eh eh) (* (* ew t) (* ew t)))))))
        (t_2 (sin (atan (/ eh (* ew t)))))
        (t_3
         (fabs
          (+ (* (* ew (sin t)) t_1) (* t_2 (fma (* eh (* t t)) -0.5 eh))))))
   (if (<= ew -1.9e-7)
     t_3
     (if (<= ew 9.2e-29)
       (fabs (+ (* (* eh (cos t)) t_2) (* (* ew t) t_1)))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / sqrt((1.0 + ((eh * eh) / ((ew * t) * (ew * t)))));
	double t_2 = sin(atan((eh / (ew * t))));
	double t_3 = fabs((((ew * sin(t)) * t_1) + (t_2 * fma((eh * (t * t)), -0.5, eh))));
	double tmp;
	if (ew <= -1.9e-7) {
		tmp = t_3;
	} else if (ew <= 9.2e-29) {
		tmp = fabs((((eh * cos(t)) * t_2) + ((ew * t) * t_1)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / sqrt(Float64(1.0 + Float64(Float64(eh * eh) / Float64(Float64(ew * t) * Float64(ew * t))))))
	t_2 = sin(atan(Float64(eh / Float64(ew * t))))
	t_3 = abs(Float64(Float64(Float64(ew * sin(t)) * t_1) + Float64(t_2 * fma(Float64(eh * Float64(t * t)), -0.5, eh))))
	tmp = 0.0
	if (ew <= -1.9e-7)
		tmp = t_3;
	elseif (ew <= 9.2e-29)
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * t_2) + Float64(Float64(ew * t) * t_1)));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Sqrt[N[(1.0 + N[(N[(eh * eh), $MachinePrecision] / N[(N[(ew * t), $MachinePrecision] * N[(ew * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$2 * N[(N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision] * -0.5 + eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.9e-7], t$95$3, If[LessEqual[ew, 9.2e-29], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.90000000000000007e-7 or 9.19999999999999965e-29 < ew

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 99.3

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

   5. Simplified 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 88.2

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

   7. Applied egg-rr 88.2%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 85.9

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

   10. Simplified 85.9%

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

   11. Taylor expanded in t around 0

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

       1. associate-*r* N/A

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

       2. +-commutative N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 65.6

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

   13. Simplified 65.6%

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

   if -1.90000000000000007e-7 < ew < 9.19999999999999965e-29

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 98.3

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

   5. Simplified 98.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. cos-atan N/A

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

      4. lower-/.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. frac-times N/A

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

      10. sqr-neg N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. sqr-neg N/A

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

      14. lower-*.f64 84.3

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

   7. Applied egg-rr 84.3%

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

   8. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 76.6

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

   10. Simplified 76.6%

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

   11. Taylor expanded in t around 0

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

       1. lower-*.f64 70.1

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

   13. Simplified 70.1%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.9 \cdot 10^{-7}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}} + \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), -0.5, eh\right)\right|\\ \mathbf{elif}\;ew \leq 9.2 \cdot 10^{-29}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) + \left(ew \cdot t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(ew \cdot \sin t\right) \cdot \frac{1}{\sqrt{1 + \frac{eh \cdot eh}{\left(ew \cdot t\right) \cdot \left(ew \cdot t\right)}}} + \sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \mathsf{fma}\left(eh \cdot \left(t \cdot t\right), -0.5, eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan((eh / (ew * t))))) + ((ew * t) * (1.0 / sqrt((1.0 + ((eh * eh) / ((ew * t) * (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((((eh * cos(t)) * sin(atan((eh / (ew * t))))) + ((ew * t) * (1.0d0 / sqrt((1.0d0 + ((eh * eh) / ((ew * t) * (ew * t)))))))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 98.9

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

5. Simplified 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. frac-times N/A

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

   10. sqr-neg N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. sqr-neg N/A

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

   14. lower-*.f64 86.5

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

7. Applied egg-rr 86.5%

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

8. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 81.8

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

10. Simplified 81.8%

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

11. Taylor expanded in t around 0

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

    1. lower-*.f64 61.8

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

13. Simplified 61.8%

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

14. Final simplification 61.8%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))