Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 24.0s

Alternatives: 12

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 3: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.2 \cdot 10^{+25} \lor \neg \left(ew \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(-\cos t, \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), eh \cdot \frac{\sin t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t - ew \cdot \frac{\cos t \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{eh}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.2e+25) (not (<= ew 1.5e-16)))
   (fabs
    (*
     ew
     (fma
      (- (cos t))
      (cos (atan (* eh (/ (tan t) (- ew)))))
      (* eh (/ (sin t) ew)))))
   (fabs
    (*
     eh
     (-
      (sin t)
      (* ew (/ (* (cos t) (cos (atan (* (tan t) (/ eh (- ew)))))) eh)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.2e+25) || !(ew <= 1.5e-16)) {
		tmp = fabs((ew * fma(-cos(t), cos(atan((eh * (tan(t) / -ew)))), (eh * (sin(t) / ew)))));
	} else {
		tmp = fabs((eh * (sin(t) - (ew * ((cos(t) * cos(atan((tan(t) * (eh / -ew))))) / eh)))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.2e+25) || !(ew <= 1.5e-16))
		tmp = abs(Float64(ew * fma(Float64(-cos(t)), cos(atan(Float64(eh * Float64(tan(t) / Float64(-ew))))), Float64(eh * Float64(sin(t) / ew)))));
	else
		tmp = abs(Float64(eh * Float64(sin(t) - Float64(ew * Float64(Float64(cos(t) * cos(atan(Float64(tan(t) * Float64(eh / Float64(-ew)))))) / eh)))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.2e+25], N[Not[LessEqual[ew, 1.5e-16]], $MachinePrecision]], N[Abs[N[(ew * N[((-N[Cos[t], $MachinePrecision]) * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(eh * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] - N[(ew * N[(N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.19999999999999998e25 or 1.49999999999999997e-16 < ew

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 98.4%

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

      2. associate-*r/ 98.4%

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

      3. associate-*r/ 98.3%

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

      4. *-commutative 98.3%

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

      5. associate-/l* 98.3%

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

      6. add-sqr-sqrt 57.1%

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

      7. sqrt-unprod 80.1%

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

      8. sqr-neg 80.1%

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

      9. sqrt-unprod 40.8%

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

      10. add-sqr-sqrt 97.9%

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

      11. hypot-1-def 99.3%

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

      12. associate-*r/ 99.3%

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

      13. *-commutative 99.3%

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

      14. associate-/l* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*l* 99.4%

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

      3. associate-*r/ 99.3%

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

      4. associate-*l/ 99.3%

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

      5. *-commutative 99.3%

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

   8. Simplified 99.3%

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

   9. Taylor expanded in eh around inf 77.1%

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

       1. mul-1-neg 77.1%

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

       2. unsub-neg 77.1%

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

       3. associate-/l* 76.9%

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

   11. Simplified 76.9%

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

   12. Taylor expanded in ew around inf 99.0%

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

       1. associate-*r* 99.0%

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

       2. fma-define 99.0%

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

       3. mul-1-neg 99.0%

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

       4. mul-1-neg 99.0%

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

       5. associate-/l* 99.0%

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

       6. distribute-rgt-neg-in 99.0%

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

       7. associate-/l* 99.0%

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

   14. Simplified 99.0%

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

   if -1.19999999999999998e25 < ew < 1.49999999999999997e-16

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

      1. sin-atan 62.5%

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

      2. associate-*r/ 62.5%

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

      3. associate-*r/ 62.4%

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

      4. *-commutative 62.4%

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

      5. associate-/l* 59.1%

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

      6. add-sqr-sqrt 32.9%

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

      7. sqrt-unprod 52.4%

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

      8. sqr-neg 52.4%

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

      9. sqrt-unprod 26.0%

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

      10. add-sqr-sqrt 58.1%

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

      11. hypot-1-def 76.9%

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

      12. associate-*r/ 76.9%

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

      13. *-commutative 76.9%

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

      14. associate-/l* 76.8%

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

   6. Applied egg-rr 76.8%

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

      1. *-commutative 76.8%

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

      2. associate-*l* 76.7%

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

      3. associate-*r/ 76.8%

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

      4. associate-*l/ 76.9%

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

      5. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in eh around inf 98.2%

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

       1. mul-1-neg 98.2%

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

       2. unsub-neg 98.2%

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

       3. associate-/l* 98.2%

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

   11. Simplified 98.2%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.2 \cdot 10^{+25} \lor \neg \left(ew \leq 1.5 \cdot 10^{-16}\right):\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(-\cos t, \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{-ew}\right), eh \cdot \frac{\sin t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t - ew \cdot \frac{\cos t \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{eh}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.15 \cdot 10^{+76} \lor \neg \left(ew \leq 2.4 \cdot 10^{+145}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t - ew \cdot \frac{\cos t \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{eh}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.15e+76) (not (<= ew 2.4e+145)))
   (fabs (* ew (cos t)))
   (fabs
    (*
     eh
     (-
      (sin t)
      (* ew (/ (* (cos t) (cos (atan (* (tan t) (/ eh (- ew)))))) eh)))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.15e+76) || !(ew <= 2.4e+145)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((eh * (sin(t) - (ew * ((cos(t) * cos(atan((tan(t) * (eh / -ew))))) / eh)))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.15d+76)) .or. (.not. (ew <= 2.4d+145))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((eh * (sin(t) - (ew * ((cos(t) * cos(atan((tan(t) * (eh / -ew))))) / eh)))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.15e+76) || !(ew <= 2.4e+145)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((eh * (Math.sin(t) - (ew * ((Math.cos(t) * Math.cos(Math.atan((Math.tan(t) * (eh / -ew))))) / eh)))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.15e+76) or not (ew <= 2.4e+145):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((eh * (math.sin(t) - (ew * ((math.cos(t) * math.cos(math.atan((math.tan(t) * (eh / -ew))))) / eh)))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.15e+76) || !(ew <= 2.4e+145))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(eh * Float64(sin(t) - Float64(ew * Float64(Float64(cos(t) * cos(atan(Float64(tan(t) * Float64(eh / Float64(-ew)))))) / eh)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.15e+76) || ~((ew <= 2.4e+145)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((eh * (sin(t) - (ew * ((cos(t) * cos(atan((tan(t) * (eh / -ew))))) / eh)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.15e+76], N[Not[LessEqual[ew, 2.4e+145]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] - N[(ew * N[(N[(N[Cos[t], $MachinePrecision] * N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / (-ew)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.15000000000000001e76 or 2.39999999999999992e145 < ew

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.5%

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

      2. pow3 99.5%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in ew around inf 96.0%

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

   if -1.15000000000000001e76 < ew < 2.39999999999999992e145

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 71.6%

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

      2. associate-*r/ 71.5%

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

      3. associate-*r/ 71.5%

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

      4. *-commutative 71.5%

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

      5. associate-/l* 69.0%

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

      6. add-sqr-sqrt 39.3%

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

      7. sqrt-unprod 60.2%

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

      8. sqr-neg 60.2%

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

      9. sqrt-unprod 29.5%

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

      10. add-sqr-sqrt 68.3%

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

      11. hypot-1-def 83.1%

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

      12. associate-*r/ 83.1%

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

      13. *-commutative 83.1%

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

      14. associate-/l* 83.0%

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

   6. Applied egg-rr 83.0%

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

      1. *-commutative 83.0%

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

      2. associate-*l* 83.0%

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

      3. associate-*r/ 83.1%

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

      4. associate-*l/ 83.1%

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

      5. *-commutative 83.1%

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

   8. Simplified 83.1%

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

   9. Taylor expanded in eh around inf 96.7%

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

       1. mul-1-neg 96.7%

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

       2. unsub-neg 96.7%

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

       3. associate-/l* 96.6%

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

   11. Simplified 96.6%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.15 \cdot 10^{+76} \lor \neg \left(ew \leq 2.4 \cdot 10^{+145}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(\sin t - ew \cdot \frac{\cos t \cdot \cos \tan^{-1} \left(\tan t \cdot \frac{eh}{-ew}\right)}{eh}\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -6 \cdot 10^{-157}:\\ \;\;\;\;\cos t \cdot \left(-ew\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\\ \mathbf{elif}\;ew \leq 5.5 \cdot 10^{-97}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -3.7e+62)
     t_1
     (if (<= ew -6e-157)
       (-
        (* (cos t) (- ew))
        (* eh (* (sin t) (sin (atan (* eh (/ (tan t) ew)))))))
       (if (<= ew 5.5e-97) (fabs (* eh (sin t))) t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -3.7e+62) {
		tmp = t_1;
	} else if (ew <= -6e-157) {
		tmp = (cos(t) * -ew) - (eh * (sin(t) * sin(atan((eh * (tan(t) / ew))))));
	} else if (ew <= 5.5e-97) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-3.7d+62)) then
        tmp = t_1
    else if (ew <= (-6d-157)) then
        tmp = (cos(t) * -ew) - (eh * (sin(t) * sin(atan((eh * (tan(t) / ew))))))
    else if (ew <= 5.5d-97) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -3.7e+62) {
		tmp = t_1;
	} else if (ew <= -6e-157) {
		tmp = (Math.cos(t) * -ew) - (eh * (Math.sin(t) * Math.sin(Math.atan((eh * (Math.tan(t) / ew))))));
	} else if (ew <= 5.5e-97) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -3.7e+62:
		tmp = t_1
	elif ew <= -6e-157:
		tmp = (math.cos(t) * -ew) - (eh * (math.sin(t) * math.sin(math.atan((eh * (math.tan(t) / ew))))))
	elif ew <= 5.5e-97:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -3.7e+62)
		tmp = t_1;
	elseif (ew <= -6e-157)
		tmp = Float64(Float64(cos(t) * Float64(-ew)) - Float64(eh * Float64(sin(t) * sin(atan(Float64(eh * Float64(tan(t) / ew)))))));
	elseif (ew <= 5.5e-97)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -3.7e+62)
		tmp = t_1;
	elseif (ew <= -6e-157)
		tmp = (cos(t) * -ew) - (eh * (sin(t) * sin(atan((eh * (tan(t) / ew))))));
	elseif (ew <= 5.5e-97)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.7e+62], t$95$1, If[LessEqual[ew, -6e-157], N[(N[(N[Cos[t], $MachinePrecision] * (-ew)), $MachinePrecision] - N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[ew, 5.5e-97], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.70000000000000014e62 or 5.49999999999999948e-97 < ew

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in ew around inf 87.5%

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

   if -3.70000000000000014e62 < ew < -6e-157

   1. Initial program 99.7%

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

   4. Applied egg-rr 82.4%

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

   5. Taylor expanded in eh around 0 82.4%

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

      1. mul-1-neg 82.4%

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

      2. *-commutative 82.4%

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

      3. distribute-lft-neg-in 82.4%

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

      4. mul-1-neg 82.4%

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

      5. associate-*r* 82.4%

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

      6. mul-1-neg 82.4%

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

      7. associate-*r/ 82.4%

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

   7. Simplified 82.4%

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

   if -6e-157 < ew < 5.49999999999999948e-97

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 56.1%

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

      2. associate-*r/ 56.1%

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

      3. associate-*r/ 56.0%

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

      4. *-commutative 56.0%

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

      5. associate-/l* 51.3%

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

      6. add-sqr-sqrt 30.1%

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

      7. sqrt-unprod 41.1%

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

      8. sqr-neg 41.1%

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

      9. sqrt-unprod 20.8%

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

      10. add-sqr-sqrt 50.1%

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

      11. hypot-1-def 67.4%

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

      12. associate-*r/ 67.4%

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

      13. *-commutative 67.4%

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

      14. associate-/l* 67.2%

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

   6. Applied egg-rr 67.2%

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

      1. *-commutative 67.2%

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

      2. associate-*l* 67.1%

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

      3. associate-*r/ 67.3%

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

      4. associate-*l/ 67.4%

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

      5. *-commutative 67.4%

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

   8. Simplified 67.4%

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

   9. Taylor expanded in ew around 0 79.1%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq -6 \cdot 10^{-157}:\\ \;\;\;\;\cos t \cdot \left(-ew\right) - eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right)\\ \mathbf{elif}\;ew \leq 5.5 \cdot 10^{-97}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.3 \cdot 10^{-123} \lor \neg \left(ew \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.3e-123) (not (<= ew 5e-99)))
   (fabs (* ew (cos t)))
   (fabs (* eh (sin t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.3e-123) || !(ew <= 5e-99)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((eh * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-1.3d-123)) .or. (.not. (ew <= 5d-99))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((eh * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.3e-123) || !(ew <= 5e-99)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((eh * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -1.3e-123) or not (ew <= 5e-99):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((eh * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.3e-123) || !(ew <= 5e-99))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(eh * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -1.3e-123) || ~((ew <= 5e-99)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((eh * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.3e-123], N[Not[LessEqual[ew, 5e-99]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.29999999999999998e-123 or 4.99999999999999969e-99 < ew

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.6%

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

      2. pow3 99.6%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in ew around inf 81.4%

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

   if -1.29999999999999998e-123 < ew < 4.99999999999999969e-99

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 55.8%

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

      2. associate-*r/ 55.8%

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

      3. associate-*r/ 55.7%

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

      4. *-commutative 55.7%

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

      5. associate-/l* 51.3%

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

      6. add-sqr-sqrt 31.2%

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

      7. sqrt-unprod 41.6%

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

      8. sqr-neg 41.6%

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

      9. sqrt-unprod 19.7%

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

      10. add-sqr-sqrt 50.1%

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

      11. hypot-1-def 66.5%

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

      12. associate-*r/ 66.5%

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

      13. *-commutative 66.5%

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

      14. associate-/l* 66.2%

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

   6. Applied egg-rr 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l* 66.1%

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

      3. associate-*r/ 66.4%

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

      4. associate-*l/ 66.5%

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

      5. *-commutative 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in ew around 0 79.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.3 \cdot 10^{-123} \lor \neg \left(ew \leq 5 \cdot 10^{-99}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 7.2 \cdot 10^{+37}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -4.8e+24)
   (* (cos t) (- ew))
   (if (<= ew 7.2e+37) (fabs (* eh (sin t))) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.8e+24) {
		tmp = cos(t) * -ew;
	} else if (ew <= 7.2e+37) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-4.8d+24)) then
        tmp = cos(t) * -ew
    else if (ew <= 7.2d+37) then
        tmp = abs((eh * sin(t)))
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -4.8e+24) {
		tmp = Math.cos(t) * -ew;
	} else if (ew <= 7.2e+37) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -4.8e+24:
		tmp = math.cos(t) * -ew
	elif ew <= 7.2e+37:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -4.8e+24)
		tmp = Float64(cos(t) * Float64(-ew));
	elseif (ew <= 7.2e+37)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -4.8e+24)
		tmp = cos(t) * -ew;
	elseif (ew <= 7.2e+37)
		tmp = abs((eh * sin(t)));
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -4.8e+24], N[(N[Cos[t], $MachinePrecision] * (-ew)), $MachinePrecision], If[LessEqual[ew, 7.2e+37], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -4.8000000000000001e24

   1. Initial program 99.8%

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

   4. Applied egg-rr 69.5%

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

   5. Taylor expanded in eh around 0 62.5%

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

      1. associate-*r* 62.5%

         \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

      2. neg-mul-1 62.5%

         \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

   7. Simplified 62.5%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -4.8000000000000001e24 < ew < 7.19999999999999995e37

   1. Initial program 99.7%

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

      1. fabs-sub 99.7%

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

      2. sub-neg 99.7%

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

      3. +-commutative 99.7%

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

      4. associate-*l* 99.7%

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

      5. distribute-rgt-neg-in 99.7%

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

      6. fma-define 99.7%

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

   3. Simplified 99.7%

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

      1. sin-atan 64.1%

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

      2. associate-*r/ 64.0%

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

      3. associate-*r/ 64.0%

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

      4. *-commutative 64.0%

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

      5. associate-/l* 60.9%

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

      6. add-sqr-sqrt 33.2%

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

      7. sqrt-unprod 53.1%

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

      8. sqr-neg 53.1%

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

      9. sqrt-unprod 27.4%

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

      10. add-sqr-sqrt 59.9%

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

      11. hypot-1-def 78.3%

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

      12. associate-*r/ 78.3%

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

      13. *-commutative 78.3%

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

      14. associate-/l* 78.1%

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

   6. Applied egg-rr 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-*l* 78.1%

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

      3. associate-*r/ 78.2%

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

      4. associate-*l/ 78.3%

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

      5. *-commutative 78.3%

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

   8. Simplified 78.3%

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

   9. Taylor expanded in ew around 0 64.0%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]

   if 7.19999999999999995e37 < ew

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. add-cbrt-cube 99.7%

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

      2. pow3 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in t around 0 71.6%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -4.8 \cdot 10^{+24}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 7.2 \cdot 10^{+37}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.1% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.85 \cdot 10^{-128}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 4.1 \cdot 10^{-96}:\\ \;\;\;\;eh \cdot \left(-\sin t\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.85e-128)
   (* (cos t) (- ew))
   (if (<= ew 4.1e-96) (* eh (- (sin t))) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.85e-128) {
		tmp = cos(t) * -ew;
	} else if (ew <= 4.1e-96) {
		tmp = eh * -sin(t);
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.85d-128)) then
        tmp = cos(t) * -ew
    else if (ew <= 4.1d-96) then
        tmp = eh * -sin(t)
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.85e-128) {
		tmp = Math.cos(t) * -ew;
	} else if (ew <= 4.1e-96) {
		tmp = eh * -Math.sin(t);
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.85e-128:
		tmp = math.cos(t) * -ew
	elif ew <= 4.1e-96:
		tmp = eh * -math.sin(t)
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.85e-128)
		tmp = Float64(cos(t) * Float64(-ew));
	elseif (ew <= 4.1e-96)
		tmp = Float64(eh * Float64(-sin(t)));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.85e-128)
		tmp = cos(t) * -ew;
	elseif (ew <= 4.1e-96)
		tmp = eh * -sin(t);
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.85e-128], N[(N[Cos[t], $MachinePrecision] * (-ew)), $MachinePrecision], If[LessEqual[ew, 4.1e-96], N[(eh * (-N[Sin[t], $MachinePrecision])), $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.85e-128

   1. Initial program 99.7%

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

   4. Applied egg-rr 73.3%

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

   5. Taylor expanded in eh around 0 56.2%

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

      1. associate-*r* 56.2%

         \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

      2. neg-mul-1 56.2%

         \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

   7. Simplified 56.2%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -1.85e-128 < ew < 4.10000000000000024e-96

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 55.2%

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

      2. associate-*r/ 55.1%

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

      3. associate-*r/ 55.1%

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

      4. *-commutative 55.1%

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

      5. associate-/l* 50.6%

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

      6. add-sqr-sqrt 30.2%

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

      7. sqrt-unprod 40.7%

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

      8. sqr-neg 40.7%

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

      9. sqrt-unprod 19.9%

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

      10. add-sqr-sqrt 49.3%

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

      11. hypot-1-def 66.0%

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

      12. associate-*r/ 66.0%

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

      13. *-commutative 66.0%

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

      14. associate-/l* 65.7%

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

   6. Applied egg-rr 65.7%

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

      1. *-commutative 65.7%

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

      2. associate-*l* 65.6%

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

      3. associate-*r/ 65.9%

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

      4. associate-*l/ 66.0%

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

      5. *-commutative 66.0%

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

   8. Simplified 66.0%

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

      1. *-un-lft-identity 66.0%

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

      2. distribute-lft-neg-out 66.0%

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

      3. atan-neg 66.0%

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

      4. *-commutative 66.0%

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

   10. Applied egg-rr 66.0%

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

       1. *-lft-identity 66.0%

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

       2. rem-square-sqrt 36.1%

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

   12. Simplified 26.3%

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

   13. Taylor expanded in eh around -inf 49.2%

       \[\leadsto \color{blue}{-1 \cdot \left(eh \cdot \sin t\right)} \]
   14. Step-by-step derivation

       1. mul-1-neg 49.2%

          \[\leadsto \color{blue}{-eh \cdot \sin t} \]

       2. *-commutative 49.2%

          \[\leadsto -\color{blue}{\sin t \cdot eh} \]

       3. distribute-rgt-neg-in 49.2%

          \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   15. Simplified 49.2%

       \[\leadsto \color{blue}{\sin t \cdot \left(-eh\right)} \]

   if 4.10000000000000024e-96 < ew

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. add-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in t around 0 60.6%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.85 \cdot 10^{-128}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 4.1 \cdot 10^{-96}:\\ \;\;\;\;eh \cdot \left(-\sin t\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-14} \lor \neg \left(t \leq 7.8 \cdot 10^{-22}\right):\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -2e-14) (not (<= t 7.8e-22))) (* eh (sin t)) (fabs ew)))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2e-14) || !(t <= 7.8e-22)) {
		tmp = eh * sin(t);
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2d-14)) .or. (.not. (t <= 7.8d-22))) then
        tmp = eh * sin(t)
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -2e-14) || !(t <= 7.8e-22)) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (t <= -2e-14) or not (t <= 7.8e-22):
		tmp = eh * math.sin(t)
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -2e-14) || !(t <= 7.8e-22))
		tmp = Float64(eh * sin(t));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((t <= -2e-14) || ~((t <= 7.8e-22)))
		tmp = eh * sin(t);
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -2e-14], N[Not[LessEqual[t, 7.8e-22]], $MachinePrecision]], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[Abs[ew], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2e-14 or 7.79999999999999996e-22 < t

   1. Initial program 99.6%

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

      1. fabs-sub 99.6%

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

      2. sub-neg 99.6%

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

      3. +-commutative 99.6%

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

      4. associate-*l* 99.6%

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

      5. distribute-rgt-neg-in 99.6%

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

      6. fma-define 99.6%

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

   3. Simplified 99.6%

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

      1. sin-atan 72.4%

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

      2. associate-*r/ 72.4%

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

      3. associate-*r/ 72.3%

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

      4. *-commutative 72.3%

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

      5. associate-/l* 72.3%

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

      6. add-sqr-sqrt 42.1%

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

      7. sqrt-unprod 61.2%

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

      8. sqr-neg 61.2%

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

      9. sqrt-unprod 30.2%

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

      10. add-sqr-sqrt 71.9%

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

      11. hypot-1-def 88.1%

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

      12. associate-*r/ 88.1%

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

      13. *-commutative 88.1%

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

      14. associate-/l* 88.2%

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

   6. Applied egg-rr 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l* 88.1%

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

      3. associate-*r/ 88.0%

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

      4. associate-*l/ 88.0%

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

      5. *-commutative 88.0%

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

   8. Simplified 88.0%

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

      1. *-un-lft-identity 88.0%

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

      2. distribute-lft-neg-out 88.0%

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

      3. atan-neg 88.0%

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

      4. *-commutative 88.0%

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

   10. Applied egg-rr 88.0%

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

       1. *-lft-identity 88.0%

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

       2. rem-square-sqrt 38.2%

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

   12. Simplified 31.4%

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

   13. Taylor expanded in ew around 0 24.5%

       \[\leadsto \color{blue}{eh \cdot \sin t} \]

   if -2e-14 < t < 7.79999999999999996e-22

   1. Initial program 100.0%

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

      1. fabs-sub 100.0%

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

      2. sub-neg 100.0%

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

      3. +-commutative 100.0%

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

      4. associate-*l* 100.0%

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

      5. distribute-rgt-neg-in 100.0%

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

      6. fma-define 100.0%

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

   3. Simplified 100.0%

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

      1. add-cbrt-cube 100.0%

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

      2. pow3 100.0%

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

   6. Applied egg-rr 98.9%

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

   7. Taylor expanded in t around 0 80.0%

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

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2 \cdot 10^{-14} \lor \neg \left(t \leq 7.8 \cdot 10^{-22}\right):\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.0% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.45 \cdot 10^{-266}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 4.9 \cdot 10^{-95}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -1.45e-266)
   (* (cos t) (- ew))
   (if (<= ew 4.9e-95) (* eh (sin t)) (fabs ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.45e-266) {
		tmp = cos(t) * -ew;
	} else if (ew <= 4.9e-95) {
		tmp = eh * sin(t);
	} else {
		tmp = fabs(ew);
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-1.45d-266)) then
        tmp = cos(t) * -ew
    else if (ew <= 4.9d-95) then
        tmp = eh * sin(t)
    else
        tmp = abs(ew)
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -1.45e-266) {
		tmp = Math.cos(t) * -ew;
	} else if (ew <= 4.9e-95) {
		tmp = eh * Math.sin(t);
	} else {
		tmp = Math.abs(ew);
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -1.45e-266:
		tmp = math.cos(t) * -ew
	elif ew <= 4.9e-95:
		tmp = eh * math.sin(t)
	else:
		tmp = math.fabs(ew)
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -1.45e-266)
		tmp = Float64(cos(t) * Float64(-ew));
	elseif (ew <= 4.9e-95)
		tmp = Float64(eh * sin(t));
	else
		tmp = abs(ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -1.45e-266)
		tmp = cos(t) * -ew;
	elseif (ew <= 4.9e-95)
		tmp = eh * sin(t);
	else
		tmp = abs(ew);
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -1.45e-266], N[(N[Cos[t], $MachinePrecision] * (-ew)), $MachinePrecision], If[LessEqual[ew, 4.9e-95], N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision], N[Abs[ew], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.44999999999999998e-266

   1. Initial program 99.7%

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

   4. Applied egg-rr 74.3%

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

   5. Taylor expanded in eh around 0 50.9%

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

      1. associate-*r* 50.9%

         \[\leadsto \color{blue}{\left(-1 \cdot ew\right) \cdot \cos t} \]

      2. neg-mul-1 50.9%

         \[\leadsto \color{blue}{\left(-ew\right)} \cdot \cos t \]

   7. Simplified 50.9%

      \[\leadsto \color{blue}{\left(-ew\right) \cdot \cos t} \]

   if -1.44999999999999998e-266 < ew < 4.9e-95

   1. Initial program 99.8%

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

      1. fabs-sub 99.8%

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

      2. sub-neg 99.8%

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

      3. +-commutative 99.8%

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

      4. associate-*l* 99.8%

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

      5. distribute-rgt-neg-in 99.8%

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

      6. fma-define 99.8%

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

   3. Simplified 99.8%

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

      1. sin-atan 59.8%

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

      2. associate-*r/ 59.8%

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

      3. associate-*r/ 59.7%

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

      4. *-commutative 59.7%

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

      5. associate-/l* 55.6%

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

      6. add-sqr-sqrt 32.8%

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

      7. sqrt-unprod 45.4%

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

      8. sqr-neg 45.4%

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

      9. sqrt-unprod 22.2%

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

      10. add-sqr-sqrt 53.9%

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

      11. hypot-1-def 69.6%

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

      12. associate-*r/ 69.6%

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

      13. *-commutative 69.6%

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

      14. associate-/l* 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. *-commutative 69.5%

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

      2. associate-*l* 69.3%

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

      3. associate-*r/ 69.4%

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

      4. associate-*l/ 69.5%

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

      5. *-commutative 69.5%

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

   8. Simplified 69.5%

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

      1. *-un-lft-identity 69.5%

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

      2. distribute-lft-neg-out 69.5%

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

      3. atan-neg 69.5%

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

      4. *-commutative 69.5%

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

   10. Applied egg-rr 69.5%

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

       1. *-lft-identity 69.5%

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

       2. rem-square-sqrt 29.4%

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

   12. Simplified 19.0%

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

   13. Taylor expanded in ew around 0 40.6%

       \[\leadsto \color{blue}{eh \cdot \sin t} \]

   if 4.9e-95 < ew

   1. Initial program 99.9%

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

      1. fabs-sub 99.9%

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

      2. sub-neg 99.9%

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

      3. +-commutative 99.9%

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

      4. associate-*l* 99.9%

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

      5. distribute-rgt-neg-in 99.9%

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

      6. fma-define 99.9%

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

   3. Simplified 99.9%

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

      1. add-cbrt-cube 99.7%

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

      2. pow3 99.7%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in t around 0 61.6%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.45 \cdot 10^{-266}:\\ \;\;\;\;\cos t \cdot \left(-ew\right)\\ \mathbf{elif}\;ew \leq 4.9 \cdot 10^{-95}:\\ \;\;\;\;eh \cdot \sin t\\ \mathbf{else}:\\ \;\;\;\;\left|ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.4% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. fabs-sub 99.8%

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

   2. sub-neg 99.8%

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

   3. +-commutative 99.8%

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

   4. associate-*l* 99.8%

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

   5. distribute-rgt-neg-in 99.8%

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

   6. fma-define 99.8%

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

3. Simplified 99.8%

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

   1. add-cbrt-cube 99.6%

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

   2. pow3 99.6%

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

6. Applied egg-rr 98.9%

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

7. Taylor expanded in t around 0 45.1%

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

Alternative 12: 21.7% accurate, 460.5× speedup

Math

\[\begin{array}{l} \\ -ew \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double eh, double ew, double t) {
	return -ew;
}

Python

def code(eh, ew, t):
	return -ew

Julia

function code(eh, ew, t)
	return Float64(-ew)
end

MATLAB

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

Wolfram

code[eh_, ew_, t_] := (-ew)

Derivation

1. Initial program 99.8%

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

4. Applied egg-rr 48.2%

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

5. Taylor expanded in t around 0 21.3%

   \[\leadsto \color{blue}{-1 \cdot ew} \]
6. Step-by-step derivation

   1. neg-mul-1 21.3%

      \[\leadsto \color{blue}{-ew} \]

7. Simplified 21.3%

   \[\leadsto \color{blue}{-ew} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))