Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.8s

Alternatives: 11

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;ew \leq 1.8 \cdot 10^{-268}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \frac{\sin t}{ew}, eh \cdot \left(-ew\right), t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(t\_2, eh \cdot \sin t, t\_1\right)\right|}{\sqrt{{t\_2}^{2} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (* eh (/ (tan t) ew))))
   (if (<= ew 1.8e-268)
     (fabs
      (*
       ew
       (fma
        eh
        (/ (* (sin t) (sin (atan (/ (* eh t) (- ew))))) ew)
        (- (cos t)))))
     (if (<= ew 1.02e+42)
       (fabs
        (fma
         (* (sin (atan (/ (* eh (tan t)) (- ew)))) (/ (sin t) ew))
         (* eh (- ew))
         t_1))
       (/ (fabs (fma t_2 (* eh (sin t)) t_1)) (sqrt (+ (pow t_2 2.0) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = eh * (tan(t) / ew);
	double tmp;
	if (ew <= 1.8e-268) {
		tmp = fabs((ew * fma(eh, ((sin(t) * sin(atan(((eh * t) / -ew)))) / ew), -cos(t))));
	} else if (ew <= 1.02e+42) {
		tmp = fabs(fma((sin(atan(((eh * tan(t)) / -ew))) * (sin(t) / ew)), (eh * -ew), t_1));
	} else {
		tmp = fabs(fma(t_2, (eh * sin(t)), t_1)) / sqrt((pow(t_2, 2.0) + 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if (ew <= 1.8e-268)
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(Float64(eh * t) / Float64(-ew))))) / ew), Float64(-cos(t)))));
	elseif (ew <= 1.02e+42)
		tmp = abs(fma(Float64(sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(sin(t) / ew)), Float64(eh * Float64(-ew)), t_1));
	else
		tmp = Float64(abs(fma(t_2, Float64(eh * sin(t)), t_1)) / sqrt(Float64((t_2 ^ 2.0) + 1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, 1.8e-268], N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.02e+42], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[(eh * (-ew)), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(t$95$2 * N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < 1.8000000000000001e-268

   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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

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

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

   7. Simplified 94.4%

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

   8. Taylor expanded in t around 0

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

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

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

      2. *-lowering-*.f64 94.4

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

   10. Simplified 94.4%

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

   if 1.8000000000000001e-268 < ew < 1.01999999999999996e42

   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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

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

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

   7. Simplified 84.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   9. Applied egg-rr 96.0%

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

   if 1.01999999999999996e42 < ew

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. 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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 99.8

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

   8. Applied egg-rr 99.8%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 1.8 \cdot 10^{-268}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 1.02 \cdot 10^{+42}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \frac{\sin t}{ew}, eh \cdot \left(-ew\right), ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(eh \cdot \frac{\tan t}{ew}, eh \cdot \sin t, ew \cdot \cos t\right)\right|}{\sqrt{{\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;ew \leq 2.5 \cdot 10^{-268}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \frac{\sin t}{ew}, eh \cdot \left(-ew\right), t\_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(eh, \sin t \cdot t\_2, t\_1\right)\right|}{\sqrt{{t\_2}^{2} + 1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (* eh (/ (tan t) ew))))
   (if (<= ew 2.5e-268)
     (fabs
      (*
       ew
       (fma
        eh
        (/ (* (sin t) (sin (atan (/ (* eh t) (- ew))))) ew)
        (- (cos t)))))
     (if (<= ew 8e+41)
       (fabs
        (fma
         (* (sin (atan (/ (* eh (tan t)) (- ew)))) (/ (sin t) ew))
         (* eh (- ew))
         t_1))
       (/ (fabs (fma eh (* (sin t) t_2) t_1)) (sqrt (+ (pow t_2 2.0) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = eh * (tan(t) / ew);
	double tmp;
	if (ew <= 2.5e-268) {
		tmp = fabs((ew * fma(eh, ((sin(t) * sin(atan(((eh * t) / -ew)))) / ew), -cos(t))));
	} else if (ew <= 8e+41) {
		tmp = fabs(fma((sin(atan(((eh * tan(t)) / -ew))) * (sin(t) / ew)), (eh * -ew), t_1));
	} else {
		tmp = fabs(fma(eh, (sin(t) * t_2), t_1)) / sqrt((pow(t_2, 2.0) + 1.0));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if (ew <= 2.5e-268)
		tmp = abs(Float64(ew * fma(eh, Float64(Float64(sin(t) * sin(atan(Float64(Float64(eh * t) / Float64(-ew))))) / ew), Float64(-cos(t)))));
	elseif (ew <= 8e+41)
		tmp = abs(fma(Float64(sin(atan(Float64(Float64(eh * tan(t)) / Float64(-ew)))) * Float64(sin(t) / ew)), Float64(eh * Float64(-ew)), t_1));
	else
		tmp = Float64(abs(fma(eh, Float64(sin(t) * t_2), t_1)) / sqrt(Float64((t_2 ^ 2.0) + 1.0)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, 2.5e-268], N[Abs[N[(ew * N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + (-N[Cos[t], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8e+41], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * N[(eh * (-ew)), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(eh * N[(N[Sin[t], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$2, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < 2.5e-268

   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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

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

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

   7. Simplified 94.4%

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

   8. Taylor expanded in t around 0

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

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

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

      2. *-lowering-*.f64 94.4

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

   10. Simplified 94.4%

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

   if 2.5e-268 < ew < 8.00000000000000005e41

   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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in ew around -inf

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

      1. associate-*r* N/A

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

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

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

      3. mul-1-neg N/A

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

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

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

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

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

   7. Simplified 84.6%

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

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

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

   9. Applied egg-rr 96.0%

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

   if 8.00000000000000005e41 < ew

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. 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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.9%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq 2.5 \cdot 10^{-268}:\\ \;\;\;\;\left|ew \cdot \mathsf{fma}\left(eh, \frac{\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{-ew}\right)}{ew}, -\cos t\right)\right|\\ \mathbf{elif}\;ew \leq 8 \cdot 10^{+41}:\\ \;\;\;\;\left|\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{eh \cdot \tan t}{-ew}\right) \cdot \frac{\sin t}{ew}, eh \cdot \left(-ew\right), ew \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\mathsf{fma}\left(eh, \sin t \cdot \left(eh \cdot \frac{\tan t}{ew}\right), ew \cdot \cos t\right)\right|}{\sqrt{{\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[(eh * N[(1.0 / N[(ew / N[(N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-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. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

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

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

7. Simplified 92.6%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   7. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

   12. neg-lowering-neg.f64 92.6

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

9. Applied egg-rr 92.6%

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

10. Final simplification 92.6%

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

Alternative 5: 90.8% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

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

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

7. Simplified 92.6%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

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

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

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

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

   6. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

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

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

   12. sin-lowering-sin.f64 92.6

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

9. Applied egg-rr 92.6%

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

10. Final simplification 92.6%

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

Alternative 6: 90.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

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

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

7. Simplified 92.6%

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

   1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

   7. distribute-frac-neg2 N/A

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

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

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

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

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

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

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

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

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

   12. neg-lowering-neg.f64 92.6

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

9. Applied egg-rr 92.6%

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

10. Taylor expanded in t around 0

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

    1. mul-1-neg N/A

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

    2. associate-*l/ N/A

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

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

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

    4. mul-1-neg N/A

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

    5. associate-*r/ N/A

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

    6. associate-*l/ N/A

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

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

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

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

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

    9. mul-1-neg N/A

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

    10. neg-lowering-neg.f64 92.4

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

12. Simplified 92.4%

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

13. Final simplification 92.4%

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

Alternative 7: 90.6% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in ew around -inf

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

   1. associate-*r* N/A

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

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

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

   3. mul-1-neg N/A

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

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

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

   5. +-commutative N/A

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

   6. associate-/l* N/A

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

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

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

7. Simplified 92.6%

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

8. Taylor expanded in t around 0

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

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

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

   2. *-lowering-*.f64 92.4

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

10. Simplified 92.4%

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

11. Final simplification 92.4%

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

Alternative 8: 73.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -2.4 \cdot 10^{-47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 0.0065:\\ \;\;\;\;\frac{-1}{\frac{-1}{\left|eh \cdot \sin t\right|}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -2.4e-47)
     t_1
     (if (<= ew 0.0065) (/ -1.0 (/ -1.0 (fabs (* eh (sin t))))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -2.4e-47) {
		tmp = t_1;
	} else if (ew <= 0.0065) {
		tmp = -1.0 / (-1.0 / fabs((eh * sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-2.4d-47)) then
        tmp = t_1
    else if (ew <= 0.0065d0) then
        tmp = (-1.0d0) / ((-1.0d0) / abs((eh * sin(t))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -2.4e-47) {
		tmp = t_1;
	} else if (ew <= 0.0065) {
		tmp = -1.0 / (-1.0 / Math.abs((eh * Math.sin(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -2.4e-47:
		tmp = t_1
	elif ew <= 0.0065:
		tmp = -1.0 / (-1.0 / math.fabs((eh * math.sin(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -2.4e-47)
		tmp = t_1;
	elseif (ew <= 0.0065)
		tmp = Float64(-1.0 / Float64(-1.0 / abs(Float64(eh * sin(t)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -2.4e-47)
		tmp = t_1;
	elseif (ew <= 0.0065)
		tmp = -1.0 / (-1.0 / abs((eh * sin(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.4e-47], t$95$1, If[LessEqual[ew, 0.0065], N[(-1.0 / N[(-1.0 / N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.3999999999999999e-47 or 0.0064999999999999997 < ew

   1. Initial program 99.8%

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

      1. 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. *-commutative N/A

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

   4. Applied egg-rr 99.8%

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

   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 83.9

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

   7. Simplified 83.9%

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

   if -2.3999999999999999e-47 < ew < 0.0064999999999999997

   1. Initial program 99.8%

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

   4. Applied egg-rr 57.4%

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

   5. Taylor expanded in ew around 0

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

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

         \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{eh \cdot \sin t}}\right|} \]

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

         \[\leadsto \frac{1}{\left|\frac{1}{\color{blue}{eh \cdot \sin t}}\right|} \]

      3. sin-lowering-sin.f64 71.3

         \[\leadsto \frac{1}{\left|\frac{1}{eh \cdot \color{blue}{\sin t}}\right|} \]

   7. Simplified 71.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.4 \cdot 10^{-47}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 0.0065:\\ \;\;\;\;\frac{-1}{\frac{-1}{\left|eh \cdot \sin t\right|}}\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $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. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

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 61.6

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

7. Simplified 61.6%

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

Alternative 10: 42.6% 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. *-commutative N/A

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

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

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

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

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

   6. associate-*r* N/A

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

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

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

4. Applied egg-rr 99.8%

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

5. Taylor expanded in t around 0

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

7. Simplified 42.4%

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

Alternative 11: 22.9% 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

4. Applied egg-rr 78.9%

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

5. Taylor expanded in t around 0

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

   1. /-lowering-/.f64 42.3

      \[\leadsto \frac{1}{\left|\color{blue}{\frac{1}{ew}}\right|} \]

7. Simplified 42.3%

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

   1. inv-pow N/A

      \[\leadsto \frac{1}{\left|\color{blue}{{ew}^{-1}}\right|} \]

   2. sqr-pow N/A

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

   3. fabs-sqr N/A

      \[\leadsto \frac{1}{\color{blue}{{ew}^{\left(\frac{-1}{2}\right)} \cdot {ew}^{\left(\frac{-1}{2}\right)}}} \]

   4. sqr-pow N/A

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

   5. inv-pow N/A

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

   6. clear-num N/A

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

   7. /-rgt-identity 25.8

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

9. Applied egg-rr 25.8%

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

Reproduce

herbie shell --seed 2024199 
(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)))))))