Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.7s

Alternatives: 10

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{eh \cdot \tan t}{-ew}\\ \left|\mathsf{fma}\left(eh \cdot \sin t, -\sin \tan^{-1} t\_1, \frac{ew \cdot \cos t}{\sqrt{1 + {t\_1}^{2}}}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (* eh (tan t)) (- ew))))
   (fabs
    (fma
     (* eh (sin t))
     (- (sin (atan t_1)))
     (/ (* ew (cos t)) (sqrt (+ 1.0 (pow t_1 2.0))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh * tan(t)) / -ew;
	return fabs(fma((eh * sin(t)), -sin(atan(t_1)), ((ew * cos(t)) / sqrt((1.0 + pow(t_1, 2.0))))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh * tan(t)) / Float64(-ew))
	return abs(fma(Float64(eh * sin(t)), Float64(-sin(atan(t_1))), Float64(Float64(ew * cos(t)) / sqrt(Float64(1.0 + (t_1 ^ 2.0))))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]}, N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * (-N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]) + N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.9%

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

Alternative 2: 98.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in eh around 0

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

   1. mul-1-neg N/A

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

   2. associate-*r* N/A

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

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

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

   4. mul-1-neg N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

7. Simplified 97.7%

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

8. Final simplification 97.7%

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

Alternative 3: 74.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;eh \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 8 \cdot 10^{+66}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, \left(eh \cdot \frac{0.5}{ew}\right) \cdot \left(t\_1 \cdot \tan t\right)\right)\right|\\ \mathbf{elif}\;eh \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh \cdot \left(-t\right), \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right), ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))) (t_2 (fabs t_1)))
   (if (<= eh -3.6e+62)
     t_2
     (if (<= eh 8e+66)
       (fabs (fma (cos t) ew (* (* eh (/ 0.5 ew)) (* t_1 (tan t)))))
       (if (<= eh 6e+148)
         (fabs (fma (* eh (- t)) (sin (atan (/ (* eh (tan t)) (- ew)))) ew))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double t_2 = fabs(t_1);
	double tmp;
	if (eh <= -3.6e+62) {
		tmp = t_2;
	} else if (eh <= 8e+66) {
		tmp = fabs(fma(cos(t), ew, ((eh * (0.5 / ew)) * (t_1 * tan(t)))));
	} else if (eh <= 6e+148) {
		tmp = fabs(fma((eh * -t), sin(atan(((eh * tan(t)) / -ew))), ew));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	t_2 = abs(t_1)
	tmp = 0.0
	if (eh <= -3.6e+62)
		tmp = t_2;
	elseif (eh <= 8e+66)
		tmp = abs(fma(cos(t), ew, Float64(Float64(eh * Float64(0.5 / ew)) * Float64(t_1 * tan(t)))));
	elseif (eh <= 6e+148)
		tmp = abs(fma(Float64(eh * Float64(-t)), sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))), ew));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[eh, -3.6e+62], t$95$2, If[LessEqual[eh, 8e+66], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(eh * N[(0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 6e+148], N[Abs[N[(N[(eh * (-t)), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + ew), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -3.6e62 or 6.00000000000000029e148 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 77.7

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

   7. Simplified 77.7%

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

   if -3.6e62 < eh < 7.99999999999999956e66

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 91.7%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 86.0

          \[\leadsto \left|\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \color{blue}{\cos t}\right)\right| \]

   7. Simplified 86.0%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-/l* N/A

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

      15. tan-quot N/A

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

      16. *-lowering-*.f64 N/A

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

      17. sin-lowering-sin.f64 N/A

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

      18. tan-lowering-tan.f64 86.0

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

   9. Applied egg-rr 86.0%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t, ew, \frac{0.5}{ew} \cdot \left(\left(eh \cdot eh\right) \cdot \left(\sin t \cdot \tan t\right)\right)\right)}\right| \]
   10. Step-by-step derivation

       1. accelerator-lowering-fma.f64 N/A

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

       2. cos-lowering-cos.f64 N/A

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

       3. associate-*l* N/A

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

       4. associate-*r* N/A

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

       5. associate-*r* N/A

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

       6. *-commutative N/A

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

       7. *-lowering-*.f64 N/A

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

       8. *-lowering-*.f64 N/A

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

       9. /-lowering-/.f64 N/A

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

       10. *-lowering-*.f64 N/A

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

       11. tan-lowering-tan.f64 N/A

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

       12. *-lowering-*.f64 N/A

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

       13. sin-lowering-sin.f64 86.0

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

   11. Applied egg-rr 86.0%

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

   if 7.99999999999999956e66 < eh < 6.00000000000000029e148

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-*r* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

   7. Simplified 75.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{elif}\;eh \leq 8 \cdot 10^{+66}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, \left(eh \cdot \frac{0.5}{ew}\right) \cdot \left(\left(eh \cdot \sin t\right) \cdot \tan t\right)\right)\right|\\ \mathbf{elif}\;eh \leq 6 \cdot 10^{+148}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh \cdot \left(-t\right), \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right), ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -1.6 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;\left|\mathsf{fma}\left(\left(eh \cdot 0.5\right) \cdot \left(\tan t \cdot \frac{\sin t}{ew}\right), eh, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -1.6e+64)
     t_1
     (if (<= eh 4.4e+148)
       (fabs (fma (* (* eh 0.5) (* (tan t) (/ (sin t) ew))) eh (* ew (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -1.6e+64) {
		tmp = t_1;
	} else if (eh <= 4.4e+148) {
		tmp = fabs(fma(((eh * 0.5) * (tan(t) * (sin(t) / ew))), eh, (ew * cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -1.6e+64)
		tmp = t_1;
	elseif (eh <= 4.4e+148)
		tmp = abs(fma(Float64(Float64(eh * 0.5) * Float64(tan(t) * Float64(sin(t) / ew))), eh, Float64(ew * cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.6e+64], t$95$1, If[LessEqual[eh, 4.4e+148], N[Abs[N[(N[(N[(eh * 0.5), $MachinePrecision] * N[(N[Tan[t], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * eh + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.60000000000000009e64 or 4.3999999999999998e148 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 77.7

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

   7. Simplified 77.7%

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

   if -1.60000000000000009e64 < eh < 4.3999999999999998e148

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 82.2

          \[\leadsto \left|\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \color{blue}{\cos t}\right)\right| \]

   7. Simplified 82.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-lowering-*.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. unpow2 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. tan-quot N/A

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

      12. *-lowering-*.f64 N/A

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

      13. tan-lowering-tan.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. sin-lowering-sin.f64 N/A

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

      16. *-lowering-*.f64 N/A

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

      17. cos-lowering-cos.f64 82.2

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

   9. Applied egg-rr 82.2%

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

Alternative 5: 74.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -1.8 \cdot 10^{+66}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5 \cdot 10^{+148}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t, ew, \frac{0.5}{ew} \cdot \left(\left(eh \cdot eh\right) \cdot \left(\sin t \cdot \tan t\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -1.8e+66)
     t_1
     (if (<= eh 5e+148)
       (fabs (fma (cos t) ew (* (/ 0.5 ew) (* (* eh eh) (* (sin t) (tan t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -1.8e+66) {
		tmp = t_1;
	} else if (eh <= 5e+148) {
		tmp = fabs(fma(cos(t), ew, ((0.5 / ew) * ((eh * eh) * (sin(t) * tan(t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -1.8e+66)
		tmp = t_1;
	elseif (eh <= 5e+148)
		tmp = abs(fma(cos(t), ew, Float64(Float64(0.5 / ew) * Float64(Float64(eh * eh) * Float64(sin(t) * tan(t))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.8e+66], t$95$1, If[LessEqual[eh, 5e+148], N[Abs[N[(N[Cos[t], $MachinePrecision] * ew + N[(N[(0.5 / ew), $MachinePrecision] * N[(N[(eh * eh), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.8e66 or 5.00000000000000024e148 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 77.7

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

   7. Simplified 77.7%

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

   if -1.8e66 < eh < 5.00000000000000024e148

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 82.2

          \[\leadsto \left|\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \color{blue}{\cos t}\right)\right| \]

   7. Simplified 82.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]
   8. Step-by-step derivation

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. cos-lowering-cos.f64 N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. *-lowering-*.f64 N/A

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

      10. /-lowering-/.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. unpow2 N/A

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

      14. associate-/l* N/A

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

      15. tan-quot N/A

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

      16. *-lowering-*.f64 N/A

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

      17. sin-lowering-sin.f64 N/A

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

      18. tan-lowering-tan.f64 82.2

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

   9. Applied egg-rr 82.2%

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

Alternative 6: 74.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -3.3 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew}, ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -3.3e+64)
     t_1
     (if (<= eh 4.4e+148)
       (fabs (fma (* eh eh) (/ (* 0.5 (pow (sin t) 2.0)) ew) (* ew (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -3.3e+64) {
		tmp = t_1;
	} else if (eh <= 4.4e+148) {
		tmp = fabs(fma((eh * eh), ((0.5 * pow(sin(t), 2.0)) / ew), (ew * cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -3.3e+64)
		tmp = t_1;
	elseif (eh <= 4.4e+148)
		tmp = abs(fma(Float64(eh * eh), Float64(Float64(0.5 * (sin(t) ^ 2.0)) / ew), Float64(ew * cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.3e+64], t$95$1, If[LessEqual[eh, 4.4e+148], N[Abs[N[(N[(eh * eh), $MachinePrecision] * N[(N[(0.5 * N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.29999999999999988e64 or 4.3999999999999998e148 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 77.7

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

   7. Simplified 77.7%

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

   if -3.29999999999999988e64 < eh < 4.3999999999999998e148

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. sin-atan N/A

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

      5. associate-/l* N/A

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

      6. associate-/l* N/A

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

      7. associate-*r* N/A

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

   4. Applied egg-rr 90.8%

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

   5. Taylor expanded in eh around 0

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

      1. +-commutative N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. /-lowering-/.f64 N/A

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

      9. *-lowering-*.f64 N/A

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

      10. pow-lowering-pow.f64 N/A

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

      11. sin-lowering-sin.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. cos-lowering-cos.f64 N/A

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

      14. *-lowering-*.f64 N/A

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

      15. cos-lowering-cos.f64 82.2

          \[\leadsto \left|\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \color{blue}{\cos t}\right)\right| \]

   7. Simplified 82.2%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(eh \cdot eh, \frac{0.5 \cdot {\sin t}^{2}}{ew \cdot \cos t}, ew \cdot \cos t\right)}\right| \]

   8. Taylor expanded in t around 0

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

   10. Simplified 82.1%

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

Alternative 7: 74.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -1.12 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 4.4 \cdot 10^{+148}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -1.12e+61) t_1 (if (<= eh 4.4e+148) (fabs (* ew (cos t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -1.12e+61) {
		tmp = t_1;
	} else if (eh <= 4.4e+148) {
		tmp = fabs((ew * cos(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((eh * sin(t)))
    if (eh <= (-1.12d+61)) then
        tmp = t_1
    else if (eh <= 4.4d+148) then
        tmp = abs((ew * cos(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((eh * Math.sin(t)));
	double tmp;
	if (eh <= -1.12e+61) {
		tmp = t_1;
	} else if (eh <= 4.4e+148) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -1.12e+61:
		tmp = t_1
	elif eh <= 4.4e+148:
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -1.12e+61)
		tmp = t_1;
	elseif (eh <= 4.4e+148)
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -1.12e+61)
		tmp = t_1;
	elseif (eh <= 4.4e+148)
		tmp = abs((ew * cos(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.12e+61], t$95$1, If[LessEqual[eh, 4.4e+148], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.12e61 or 4.3999999999999998e148 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 47.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 77.7

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

   7. Simplified 77.7%

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

   if -1.12e61 < eh < 4.3999999999999998e148

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in eh around 0

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

      1. *-lowering-*.f64 N/A

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

      2. cos-lowering-cos.f64 82.1

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

   7. Simplified 82.1%

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

Alternative 8: 59.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -7.6 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.2 \cdot 10^{+60}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -7.6e+53) t_1 (if (<= eh 5.2e+60) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -7.6e+53) {
		tmp = t_1;
	} else if (eh <= 5.2e+60) {
		tmp = fabs(ew);
	} 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((eh * sin(t)))
    if (eh <= (-7.6d+53)) then
        tmp = t_1
    else if (eh <= 5.2d+60) then
        tmp = abs(ew)
    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((eh * Math.sin(t)));
	double tmp;
	if (eh <= -7.6e+53) {
		tmp = t_1;
	} else if (eh <= 5.2e+60) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -7.6e+53:
		tmp = t_1
	elif eh <= 5.2e+60:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -7.6e+53)
		tmp = t_1;
	elseif (eh <= 5.2e+60)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -7.6e+53)
		tmp = t_1;
	elseif (eh <= 5.2e+60)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -7.6e+53], t$95$1, If[LessEqual[eh, 5.2e+60], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.59999999999999995e53 or 5.20000000000000016e60 < eh

   1. Initial program 99.8%

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

      1. neg-fabs N/A

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

      2. fabs-lowering-fabs.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-neg-in N/A

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

   4. Applied egg-rr 54.3%

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

   5. Taylor expanded in eh around -inf

      \[\leadsto \left|\color{blue}{eh \cdot \sin t}\right| \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 72.1

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

   7. Simplified 72.1%

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

   if -7.59999999999999995e53 < eh < 5.20000000000000016e60

   1. Initial program 99.9%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

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

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

      4. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\color{blue}{ew}\right| \]
   6. Step-by-step derivation

   7. Simplified 59.0%

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

Alternative 9: 42.5% accurate, 287.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{ew}\right| \]
6. Step-by-step derivation

7. Simplified 41.2%

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

Alternative 10: 22.2% accurate, 862.0× 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 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
3. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. accelerator-lowering-fma.f64 N/A

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in t around 0

   \[\leadsto \left|\color{blue}{ew}\right| \]
6. Step-by-step derivation

7. Simplified 41.2%

   \[\leadsto \left|\color{blue}{ew}\right| \]
8. Step-by-step derivation

   1. neg-fabs N/A

      \[\leadsto \color{blue}{\left|\mathsf{neg}\left(ew\right)\right|} \]

   2. neg-sub0 N/A

      \[\leadsto \left|\color{blue}{0 - ew}\right| \]

   3. flip-- N/A

      \[\leadsto \left|\color{blue}{\frac{0 \cdot 0 - ew \cdot ew}{0 + ew}}\right| \]

   4. fabs-div N/A

      \[\leadsto \color{blue}{\frac{\left|0 \cdot 0 - ew \cdot ew\right|}{\left|0 + ew\right|}} \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left|0 \cdot 0 - ew \cdot ew\right|}{\left|0 + ew\right|}} \]

   6. fabs-lowering-fabs.f64 N/A

      \[\leadsto \frac{\color{blue}{\left|0 \cdot 0 - ew \cdot ew\right|}}{\left|0 + ew\right|} \]

   7. metadata-eval N/A

      \[\leadsto \frac{\left|\color{blue}{0} - ew \cdot ew\right|}{\left|0 + ew\right|} \]

   8. --lowering--.f64 N/A

      \[\leadsto \frac{\left|\color{blue}{0 - ew \cdot ew}\right|}{\left|0 + ew\right|} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{\left|0 - \color{blue}{ew \cdot ew}\right|}{\left|0 + ew\right|} \]

   10. fabs-lowering-fabs.f64 N/A

       \[\leadsto \frac{\left|0 - ew \cdot ew\right|}{\color{blue}{\left|0 + ew\right|}} \]

   11. +-lowering-+.f64 25.2

       \[\leadsto \frac{\left|0 - ew \cdot ew\right|}{\left|\color{blue}{0 + ew}\right|} \]

9. Applied egg-rr 25.2%

   \[\leadsto \color{blue}{\frac{\left|0 - ew \cdot ew\right|}{\left|0 + ew\right|}} \]
10. Step-by-step derivation

    1. div-fabs N/A

       \[\leadsto \color{blue}{\left|\frac{0 - ew \cdot ew}{0 + ew}\right|} \]

    2. metadata-eval N/A

       \[\leadsto \left|\frac{\color{blue}{0 \cdot 0} - ew \cdot ew}{0 + ew}\right| \]

    3. flip-- N/A

       \[\leadsto \left|\color{blue}{0 - ew}\right| \]

    4. neg-sub0 N/A

       \[\leadsto \left|\color{blue}{\mathsf{neg}\left(ew\right)}\right| \]

    5. neg-fabs N/A

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

    6. +-lft-identity N/A

       \[\leadsto \left|\color{blue}{0 + ew}\right| \]

    7. flip3-+ N/A

       \[\leadsto \left|\color{blue}{\frac{{0}^{3} + {ew}^{3}}{0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)}}\right| \]

    8. fabs-div N/A

       \[\leadsto \color{blue}{\frac{\left|{0}^{3} + {ew}^{3}\right|}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|}} \]

    9. metadata-eval N/A

       \[\leadsto \frac{\left|\color{blue}{0} + {ew}^{3}\right|}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    10. +-lft-identity N/A

        \[\leadsto \frac{\left|\color{blue}{{ew}^{3}}\right|}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    11. sqr-pow N/A

        \[\leadsto \frac{\left|\color{blue}{{ew}^{\left(\frac{3}{2}\right)} \cdot {ew}^{\left(\frac{3}{2}\right)}}\right|}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    12. fabs-sqr N/A

        \[\leadsto \frac{\color{blue}{{ew}^{\left(\frac{3}{2}\right)} \cdot {ew}^{\left(\frac{3}{2}\right)}}}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    13. sqr-pow N/A

        \[\leadsto \frac{\color{blue}{{ew}^{3}}}{\left|0 \cdot 0 + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    14. metadata-eval N/A

        \[\leadsto \frac{{ew}^{3}}{\left|\color{blue}{0} + \left(ew \cdot ew - 0 \cdot ew\right)\right|} \]

    15. +-lft-identity N/A

        \[\leadsto \frac{{ew}^{3}}{\left|\color{blue}{ew \cdot ew - 0 \cdot ew}\right|} \]

    16. mul0-lft N/A

        \[\leadsto \frac{{ew}^{3}}{\left|ew \cdot ew - \color{blue}{0}\right|} \]

    17. --rgt-identity N/A

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

    18. fabs-sqr N/A

        \[\leadsto \frac{{ew}^{3}}{\color{blue}{ew \cdot ew}} \]

    19. pow2 N/A

        \[\leadsto \frac{{ew}^{3}}{\color{blue}{{ew}^{2}}} \]

    20. pow-div N/A

        \[\leadsto \color{blue}{{ew}^{\left(3 - 2\right)}} \]

    21. metadata-eval N/A

        \[\leadsto {ew}^{\color{blue}{1}} \]

    22. unpow1 20.6

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

11. Applied egg-rr 20.6%

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

Reproduce

herbie shell --seed 2024198 
(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)))))))