Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.5s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := eh \cdot \sin t\\ t_3 := \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right)\\ t_4 := \cos t\_3 \cdot t\_1 - t\_2 \cdot \sin t\_3\\ t_5 := \frac{\tan t}{ew} \cdot eh\\ t_6 := \left|\frac{t\_2 \cdot t\_5 + t\_1}{\frac{-1}{\cos \tan^{-1} t\_5}}\right|\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{+280}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_4 \leq -4 \cdot 10^{+221}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-277}:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t, -eh, \frac{1}{\sqrt{1 + {t\_5}^{2}}} \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew))
        (t_2 (* eh (sin t)))
        (t_3 (atan (/ (* (tan t) eh) (- ew))))
        (t_4 (- (* (cos t_3) t_1) (* t_2 (sin t_3))))
        (t_5 (* (/ (tan t) ew) eh))
        (t_6 (fabs (/ (+ (* t_2 t_5) t_1) (/ -1.0 (cos (atan t_5)))))))
   (if (<= t_4 -2e+280)
     t_6
     (if (<= t_4 -4e+221)
       (fabs
        (*
         (sin
          (atan (* (/ (fma -0.3333333333333333 (* (* eh t) t) (- eh)) ew) t)))
         (* (- (sin t)) eh)))
       (if (<= t_4 -2e-277)
         t_6
         (fma
          (* (sin (atan (* (/ (- eh) ew) (tan t)))) (sin t))
          (- eh)
          (* (/ 1.0 (sqrt (+ 1.0 (pow t_5 2.0)))) t_1)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = eh * sin(t);
	double t_3 = atan(((tan(t) * eh) / -ew));
	double t_4 = (cos(t_3) * t_1) - (t_2 * sin(t_3));
	double t_5 = (tan(t) / ew) * eh;
	double t_6 = fabs((((t_2 * t_5) + t_1) / (-1.0 / cos(atan(t_5)))));
	double tmp;
	if (t_4 <= -2e+280) {
		tmp = t_6;
	} else if (t_4 <= -4e+221) {
		tmp = fabs((sin(atan(((fma(-0.3333333333333333, ((eh * t) * t), -eh) / ew) * t))) * (-sin(t) * eh)));
	} else if (t_4 <= -2e-277) {
		tmp = t_6;
	} else {
		tmp = fma((sin(atan(((-eh / ew) * tan(t)))) * sin(t)), -eh, ((1.0 / sqrt((1.0 + pow(t_5, 2.0)))) * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = Float64(eh * sin(t))
	t_3 = atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))
	t_4 = Float64(Float64(cos(t_3) * t_1) - Float64(t_2 * sin(t_3)))
	t_5 = Float64(Float64(tan(t) / ew) * eh)
	t_6 = abs(Float64(Float64(Float64(t_2 * t_5) + t_1) / Float64(-1.0 / cos(atan(t_5)))))
	tmp = 0.0
	if (t_4 <= -2e+280)
		tmp = t_6;
	elseif (t_4 <= -4e+221)
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(eh * t) * t), Float64(-eh)) / ew) * t))) * Float64(Float64(-sin(t)) * eh)));
	elseif (t_4 <= -2e-277)
		tmp = t_6;
	else
		tmp = fma(Float64(sin(atan(Float64(Float64(Float64(-eh) / ew) * tan(t)))) * sin(t)), Float64(-eh), Float64(Float64(1.0 / sqrt(Float64(1.0 + (t_5 ^ 2.0)))) * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[t$95$3], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$2 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$6 = N[Abs[N[(N[(N[(t$95$2 * t$95$5), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[t$95$5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -2e+280], t$95$6, If[LessEqual[t$95$4, -4e+221], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] * t), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, -2e-277], t$95$6, N[(N[(N[Sin[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision] * (-eh) + N[(N[(1.0 / N[Sqrt[N[(1.0 + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -2.0000000000000001e280 or -4.0000000000000002e221 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -1.99999999999999994e-277

   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 rewrites 67.2%

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

   if -2.0000000000000001e280 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew))))) < -4.0000000000000002e221

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 77.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 78.0%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 78.0%

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

   if -1.99999999999999994e-277 < (-.f64 (*.f64 (*.f64 ew (cos.f64 t)) (cos.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) ew)))) (*.f64 (*.f64 eh (sin.f64 t)) (sin.f64 (atan.f64 (/.f64 (*.f64 (neg.f64 eh) (tan.f64 t)) 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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

   8. Applied rewrites 99.8%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \cdot \left(\cos t \cdot ew\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \leq -2 \cdot 10^{+280}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{elif}\;\cos \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \cdot \left(\cos t \cdot ew\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \leq -4 \cdot 10^{+221}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;\cos \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \cdot \left(\cos t \cdot ew\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \leq -2 \cdot 10^{-277}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot \tan t\right) \cdot \sin t, -eh, \frac{1}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}} \cdot \left(\cos t \cdot ew\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\tan t}{ew} \cdot eh\\ t_2 := \left(eh \cdot \sin t\right) \cdot t\_1 + \cos t \cdot ew\\ t_3 := \left|\frac{t\_2}{\frac{-1}{\cos \tan^{-1} t\_1}}\right|\\ \mathbf{if}\;ew \leq -3.8 \cdot 10^{-64}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 3.55 \cdot 10^{-171}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-t}{ew}\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{t\_2}{\sqrt{1 + {t\_1}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (/ (tan t) ew) eh))
        (t_2 (+ (* (* eh (sin t)) t_1) (* (cos t) ew)))
        (t_3 (fabs (/ t_2 (/ -1.0 (cos (atan t_1)))))))
   (if (<= ew -3.8e-64)
     t_3
     (if (<= ew 3.55e-171)
       (fabs
        (* (sin (atan (* (/ eh (cos t)) (/ (- t) ew)))) (* (- (sin t)) eh)))
       (if (<= ew 2.6e-90) (/ t_2 (sqrt (+ 1.0 (pow t_1 2.0)))) t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (tan(t) / ew) * eh;
	double t_2 = ((eh * sin(t)) * t_1) + (cos(t) * ew);
	double t_3 = fabs((t_2 / (-1.0 / cos(atan(t_1)))));
	double tmp;
	if (ew <= -3.8e-64) {
		tmp = t_3;
	} else if (ew <= 3.55e-171) {
		tmp = fabs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)));
	} else if (ew <= 2.6e-90) {
		tmp = t_2 / sqrt((1.0 + pow(t_1, 2.0)));
	} else {
		tmp = t_3;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (tan(t) / ew) * eh
    t_2 = ((eh * sin(t)) * t_1) + (cos(t) * ew)
    t_3 = abs((t_2 / ((-1.0d0) / cos(atan(t_1)))))
    if (ew <= (-3.8d-64)) then
        tmp = t_3
    else if (ew <= 3.55d-171) then
        tmp = abs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)))
    else if (ew <= 2.6d-90) then
        tmp = t_2 / sqrt((1.0d0 + (t_1 ** 2.0d0)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (Math.tan(t) / ew) * eh;
	double t_2 = ((eh * Math.sin(t)) * t_1) + (Math.cos(t) * ew);
	double t_3 = Math.abs((t_2 / (-1.0 / Math.cos(Math.atan(t_1)))));
	double tmp;
	if (ew <= -3.8e-64) {
		tmp = t_3;
	} else if (ew <= 3.55e-171) {
		tmp = Math.abs((Math.sin(Math.atan(((eh / Math.cos(t)) * (-t / ew)))) * (-Math.sin(t) * eh)));
	} else if (ew <= 2.6e-90) {
		tmp = t_2 / Math.sqrt((1.0 + Math.pow(t_1, 2.0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (math.tan(t) / ew) * eh
	t_2 = ((eh * math.sin(t)) * t_1) + (math.cos(t) * ew)
	t_3 = math.fabs((t_2 / (-1.0 / math.cos(math.atan(t_1)))))
	tmp = 0
	if ew <= -3.8e-64:
		tmp = t_3
	elif ew <= 3.55e-171:
		tmp = math.fabs((math.sin(math.atan(((eh / math.cos(t)) * (-t / ew)))) * (-math.sin(t) * eh)))
	elif ew <= 2.6e-90:
		tmp = t_2 / math.sqrt((1.0 + math.pow(t_1, 2.0)))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(tan(t) / ew) * eh)
	t_2 = Float64(Float64(Float64(eh * sin(t)) * t_1) + Float64(cos(t) * ew))
	t_3 = abs(Float64(t_2 / Float64(-1.0 / cos(atan(t_1)))))
	tmp = 0.0
	if (ew <= -3.8e-64)
		tmp = t_3;
	elseif (ew <= 3.55e-171)
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(Float64(-t) / ew)))) * Float64(Float64(-sin(t)) * eh)));
	elseif (ew <= 2.6e-90)
		tmp = Float64(t_2 / sqrt(Float64(1.0 + (t_1 ^ 2.0))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (tan(t) / ew) * eh;
	t_2 = ((eh * sin(t)) * t_1) + (cos(t) * ew);
	t_3 = abs((t_2 / (-1.0 / cos(atan(t_1)))));
	tmp = 0.0;
	if (ew <= -3.8e-64)
		tmp = t_3;
	elseif (ew <= 3.55e-171)
		tmp = abs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)));
	elseif (ew <= 2.6e-90)
		tmp = t_2 / sqrt((1.0 + (t_1 ^ 2.0)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$2 / N[(-1.0 / N[Cos[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -3.8e-64], t$95$3, If[LessEqual[ew, 3.55e-171], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[((-t) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.6e-90], N[(t$95$2 / N[Sqrt[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.8000000000000002e-64 or 2.6e-90 < 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

   4. Applied rewrites 83.6%

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

   if -3.8000000000000002e-64 < ew < 3.54999999999999999e-171

   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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.0%

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

   if 3.54999999999999999e-171 < ew < 2.6e-90

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 91.0%

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

   8. Applied rewrites 86.4%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.8 \cdot 10^{-64}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{elif}\;ew \leq 3.55 \cdot 10^{-171}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-t}{ew}\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \frac{\tan t}{ew} \cdot eh\\ t_3 := \left|t\_1 \cdot \cos \tan^{-1} t\_2\right|\\ \mathbf{if}\;ew \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 3.55 \cdot 10^{-171}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-t}{ew}\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(eh \cdot \sin t\right) \cdot t\_2 + t\_1}{\sqrt{1 + {t\_2}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew))
        (t_2 (* (/ (tan t) ew) eh))
        (t_3 (fabs (* t_1 (cos (atan t_2))))))
   (if (<= ew -2.9e-58)
     t_3
     (if (<= ew 3.55e-171)
       (fabs
        (* (sin (atan (* (/ eh (cos t)) (/ (- t) ew)))) (* (- (sin t)) eh)))
       (if (<= ew 2.6e-90)
         (/ (+ (* (* eh (sin t)) t_2) t_1) (sqrt (+ 1.0 (pow t_2 2.0))))
         t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = (tan(t) / ew) * eh;
	double t_3 = fabs((t_1 * cos(atan(t_2))));
	double tmp;
	if (ew <= -2.9e-58) {
		tmp = t_3;
	} else if (ew <= 3.55e-171) {
		tmp = fabs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)));
	} else if (ew <= 2.6e-90) {
		tmp = (((eh * sin(t)) * t_2) + t_1) / sqrt((1.0 + pow(t_2, 2.0)));
	} else {
		tmp = t_3;
	}
	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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = cos(t) * ew
    t_2 = (tan(t) / ew) * eh
    t_3 = abs((t_1 * cos(atan(t_2))))
    if (ew <= (-2.9d-58)) then
        tmp = t_3
    else if (ew <= 3.55d-171) then
        tmp = abs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)))
    else if (ew <= 2.6d-90) then
        tmp = (((eh * sin(t)) * t_2) + t_1) / sqrt((1.0d0 + (t_2 ** 2.0d0)))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * ew;
	double t_2 = (Math.tan(t) / ew) * eh;
	double t_3 = Math.abs((t_1 * Math.cos(Math.atan(t_2))));
	double tmp;
	if (ew <= -2.9e-58) {
		tmp = t_3;
	} else if (ew <= 3.55e-171) {
		tmp = Math.abs((Math.sin(Math.atan(((eh / Math.cos(t)) * (-t / ew)))) * (-Math.sin(t) * eh)));
	} else if (ew <= 2.6e-90) {
		tmp = (((eh * Math.sin(t)) * t_2) + t_1) / Math.sqrt((1.0 + Math.pow(t_2, 2.0)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = (math.tan(t) / ew) * eh
	t_3 = math.fabs((t_1 * math.cos(math.atan(t_2))))
	tmp = 0
	if ew <= -2.9e-58:
		tmp = t_3
	elif ew <= 3.55e-171:
		tmp = math.fabs((math.sin(math.atan(((eh / math.cos(t)) * (-t / ew)))) * (-math.sin(t) * eh)))
	elif ew <= 2.6e-90:
		tmp = (((eh * math.sin(t)) * t_2) + t_1) / math.sqrt((1.0 + math.pow(t_2, 2.0)))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = Float64(Float64(tan(t) / ew) * eh)
	t_3 = abs(Float64(t_1 * cos(atan(t_2))))
	tmp = 0.0
	if (ew <= -2.9e-58)
		tmp = t_3;
	elseif (ew <= 3.55e-171)
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / cos(t)) * Float64(Float64(-t) / ew)))) * Float64(Float64(-sin(t)) * eh)));
	elseif (ew <= 2.6e-90)
		tmp = Float64(Float64(Float64(Float64(eh * sin(t)) * t_2) + t_1) / sqrt(Float64(1.0 + (t_2 ^ 2.0))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = (tan(t) / ew) * eh;
	t_3 = abs((t_1 * cos(atan(t_2))));
	tmp = 0.0;
	if (ew <= -2.9e-58)
		tmp = t_3;
	elseif (ew <= 3.55e-171)
		tmp = abs((sin(atan(((eh / cos(t)) * (-t / ew)))) * (-sin(t) * eh)));
	elseif (ew <= 2.6e-90)
		tmp = (((eh * sin(t)) * t_2) + t_1) / sqrt((1.0 + (t_2 ^ 2.0)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(t$95$1 * N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.9e-58], t$95$3, If[LessEqual[ew, 3.55e-171], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[((-t) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 2.6e-90], N[(N[(N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.8999999999999999e-58 or 2.6e-90 < 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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 83.2%

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

   9. Applied rewrites 83.3%

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

   if -2.8999999999999999e-58 < ew < 3.54999999999999999e-171

   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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 76.4%

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

   if 3.54999999999999999e-171 < ew < 2.6e-90

   1. Initial program 100.0%

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 91.0%

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

   8. Applied rewrites 86.4%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 3.55 \cdot 10^{-171}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-t}{ew}\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 2.6 \cdot 10^{-90}:\\ \;\;\;\;\frac{\left(eh \cdot \sin t\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) + \cos t \cdot ew}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -2.6 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+131}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ (- eh) ew) t))) (* (- (sin t)) eh)))))
   (if (<= eh -2.6e+67)
     t_1
     (if (<= eh 5.5e+131)
       (fabs (* (* (cos t) ew) (cos (atan (* (/ (tan t) ew) eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	double tmp;
	if (eh <= -2.6e+67) {
		tmp = t_1;
	} else if (eh <= 5.5e+131) {
		tmp = fabs(((cos(t) * ew) * cos(atan(((tan(t) / ew) * eh)))));
	} 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((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)))
    if (eh <= (-2.6d+67)) then
        tmp = t_1
    else if (eh <= 5.5d+131) then
        tmp = abs(((cos(t) * ew) * cos(atan(((tan(t) / ew) * eh)))))
    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((Math.sin(Math.atan(((-eh / ew) * t))) * (-Math.sin(t) * eh)));
	double tmp;
	if (eh <= -2.6e+67) {
		tmp = t_1;
	} else if (eh <= 5.5e+131) {
		tmp = Math.abs(((Math.cos(t) * ew) * Math.cos(Math.atan(((Math.tan(t) / ew) * eh)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((-eh / ew) * t))) * (-math.sin(t) * eh)))
	tmp = 0
	if eh <= -2.6e+67:
		tmp = t_1
	elif eh <= 5.5e+131:
		tmp = math.fabs(((math.cos(t) * ew) * math.cos(math.atan(((math.tan(t) / ew) * eh)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(-eh) / ew) * t))) * Float64(Float64(-sin(t)) * eh)))
	tmp = 0.0
	if (eh <= -2.6e+67)
		tmp = t_1;
	elseif (eh <= 5.5e+131)
		tmp = abs(Float64(Float64(cos(t) * ew) * cos(atan(Float64(Float64(tan(t) / ew) * eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	tmp = 0.0;
	if (eh <= -2.6e+67)
		tmp = t_1;
	elseif (eh <= 5.5e+131)
		tmp = abs(((cos(t) * ew) * cos(atan(((tan(t) / ew) * eh)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.6e+67], t$95$1, If[LessEqual[eh, 5.5e+131], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision] * N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.6e67 or 5.49999999999999971e131 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.9%

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

   if -2.6e67 < eh < 5.49999999999999971e131

   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 rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 81.0%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+67}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+131}:\\ \;\;\;\;\left|\left(\cos t \cdot ew\right) \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{if}\;eh \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\frac{1}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}} \cdot \left(\cos t \cdot ew\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ (- eh) ew) t))) (* (- (sin t)) eh)))))
   (if (<= eh -8.2e+64)
     t_1
     (if (<= eh 5.5e+131)
       (/
        1.0
        (fabs
         (/
          1.0
          (*
           (/ 1.0 (sqrt (+ 1.0 (pow (* (/ (tan t) ew) eh) 2.0))))
           (* (cos t) ew)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	double tmp;
	if (eh <= -8.2e+64) {
		tmp = t_1;
	} else if (eh <= 5.5e+131) {
		tmp = 1.0 / fabs((1.0 / ((1.0 / sqrt((1.0 + pow(((tan(t) / ew) * eh), 2.0)))) * (cos(t) * ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)))
    if (eh <= (-8.2d+64)) then
        tmp = t_1
    else if (eh <= 5.5d+131) then
        tmp = 1.0d0 / abs((1.0d0 / ((1.0d0 / sqrt((1.0d0 + (((tan(t) / ew) * eh) ** 2.0d0)))) * (cos(t) * ew))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((-eh / ew) * t))) * (-Math.sin(t) * eh)));
	double tmp;
	if (eh <= -8.2e+64) {
		tmp = t_1;
	} else if (eh <= 5.5e+131) {
		tmp = 1.0 / Math.abs((1.0 / ((1.0 / Math.sqrt((1.0 + Math.pow(((Math.tan(t) / ew) * eh), 2.0)))) * (Math.cos(t) * ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((-eh / ew) * t))) * (-math.sin(t) * eh)))
	tmp = 0
	if eh <= -8.2e+64:
		tmp = t_1
	elif eh <= 5.5e+131:
		tmp = 1.0 / math.fabs((1.0 / ((1.0 / math.sqrt((1.0 + math.pow(((math.tan(t) / ew) * eh), 2.0)))) * (math.cos(t) * ew))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(-eh) / ew) * t))) * Float64(Float64(-sin(t)) * eh)))
	tmp = 0.0
	if (eh <= -8.2e+64)
		tmp = t_1;
	elseif (eh <= 5.5e+131)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(Float64(1.0 / sqrt(Float64(1.0 + (Float64(Float64(tan(t) / ew) * eh) ^ 2.0)))) * Float64(cos(t) * ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	tmp = 0.0;
	if (eh <= -8.2e+64)
		tmp = t_1;
	elseif (eh <= 5.5e+131)
		tmp = 1.0 / abs((1.0 / ((1.0 / sqrt((1.0 + (((tan(t) / ew) * eh) ^ 2.0)))) * (cos(t) * ew))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -8.2e+64], t$95$1, If[LessEqual[eh, 5.5e+131], N[(1.0 / N[Abs[N[(1.0 / N[(N[(1.0 / N[Sqrt[N[(1.0 + N[Power[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -8.19999999999999956e64 or 5.49999999999999971e131 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 75.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.9%

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

   if -8.19999999999999956e64 < eh < 5.49999999999999971e131

   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 rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 80.8%

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

   9. Applied rewrites 80.7%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -8.2 \cdot 10^{+64}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 5.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\frac{1}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}} \cdot \left(\cos t \cdot ew\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-\sin t\right) \cdot eh\\ \mathbf{if}\;eh \leq -5.4 \cdot 10^{+64}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot t\_1\right|\\ \mathbf{elif}\;eh \leq 1.42 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot t\_1\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- (sin t)) eh)))
   (if (<= eh -5.4e+64)
     (fabs (* (sin (atan (* (/ (- eh) ew) t))) t_1))
     (if (<= eh 1.42e+97)
       (/ 1.0 (fabs (/ 1.0 (* (cos (atan (* (/ t ew) eh))) (* (cos t) ew)))))
       (fabs
        (*
         (sin
          (atan (* (/ (fma -0.3333333333333333 (* (* eh t) t) (- eh)) ew) t)))
         t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t) * eh;
	double tmp;
	if (eh <= -5.4e+64) {
		tmp = fabs((sin(atan(((-eh / ew) * t))) * t_1));
	} else if (eh <= 1.42e+97) {
		tmp = 1.0 / fabs((1.0 / (cos(atan(((t / ew) * eh))) * (cos(t) * ew))));
	} else {
		tmp = fabs((sin(atan(((fma(-0.3333333333333333, ((eh * t) * t), -eh) / ew) * t))) * t_1));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-sin(t)) * eh)
	tmp = 0.0
	if (eh <= -5.4e+64)
		tmp = abs(Float64(sin(atan(Float64(Float64(Float64(-eh) / ew) * t))) * t_1));
	elseif (eh <= 1.42e+97)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(cos(atan(Float64(Float64(t / ew) * eh))) * Float64(cos(t) * ew)))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(eh * t) * t), Float64(-eh)) / ew) * t))) * t_1));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]}, If[LessEqual[eh, -5.4e+64], N[Abs[N[(N[Sin[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.42e+97], N[(1.0 / N[Abs[N[(1.0 / N[(N[Cos[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] * t), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -5.3999999999999999e64

   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. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.4%

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

   if -5.3999999999999999e64 < eh < 1.41999999999999991e97

   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 rewrites 99.6%

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

   5. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

   7. Applied rewrites 81.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 72.7%

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

   if 1.41999999999999991e97 < eh

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.0%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 75.0%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.4 \cdot 10^{+64}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;eh \leq 1.42 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right) \cdot \left(\cos t \cdot ew\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (sin
            (atan
             (* (/ (fma -0.3333333333333333 (* (* eh t) t) (- eh)) ew) t)))
           (* (- (sin t)) eh)))))
   (if (<= t -7.8e-22) t_1 (if (<= t 4.8e-83) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((fma(-0.3333333333333333, ((eh * t) * t), -eh) / ew) * t))) * (-sin(t) * eh)));
	double tmp;
	if (t <= -7.8e-22) {
		tmp = t_1;
	} else if (t <= 4.8e-83) {
		tmp = fabs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(eh * t) * t), Float64(-eh)) / ew) * t))) * Float64(Float64(-sin(t)) * eh)))
	tmp = 0.0
	if (t <= -7.8e-22)
		tmp = t_1;
	elseif (t <= 4.8e-83)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] * t), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.8e-22], t$95$1, If[LessEqual[t, 4.8e-83], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.79999999999999996e-22 or 4.8000000000000002e-83 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.6%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 53.6%

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

   if -7.79999999999999996e-22 < t < 4.8000000000000002e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 78.8%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 79.1%

       \[\leadsto \left|\frac{ew}{1}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 60.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{if}\;t \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ (- eh) ew) t))) (* (- (sin t)) eh)))))
   (if (<= t -7.8e-22) t_1 (if (<= t 4.8e-83) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	double tmp;
	if (t <= -7.8e-22) {
		tmp = t_1;
	} else if (t <= 4.8e-83) {
		tmp = fabs((ew / 1.0));
	} 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((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)))
    if (t <= (-7.8d-22)) then
        tmp = t_1
    else if (t <= 4.8d-83) then
        tmp = abs((ew / 1.0d0))
    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((Math.sin(Math.atan(((-eh / ew) * t))) * (-Math.sin(t) * eh)));
	double tmp;
	if (t <= -7.8e-22) {
		tmp = t_1;
	} else if (t <= 4.8e-83) {
		tmp = Math.abs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((-eh / ew) * t))) * (-math.sin(t) * eh)))
	tmp = 0
	if t <= -7.8e-22:
		tmp = t_1
	elif t <= 4.8e-83:
		tmp = math.fabs((ew / 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(Float64(-eh) / ew) * t))) * Float64(Float64(-sin(t)) * eh)))
	tmp = 0.0
	if (t <= -7.8e-22)
		tmp = t_1;
	elseif (t <= 4.8e-83)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((-eh / ew) * t))) * (-sin(t) * eh)));
	tmp = 0.0;
	if (t <= -7.8e-22)
		tmp = t_1;
	elseif (t <= 4.8e-83)
		tmp = abs((ew / 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[((-eh) / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -7.8e-22], t$95$1, If[LessEqual[t, 4.8e-83], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -7.79999999999999996e-22 or 4.8000000000000002e-83 < t

   1. Initial program 99.7%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.5%

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

   if -7.79999999999999996e-22 < t < 4.8000000000000002e-83

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.1%

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

   10. Applied rewrites 78.8%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 79.1%

       \[\leadsto \left|\frac{ew}{1}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-22}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{-eh}{ew} \cdot t\right) \cdot \left(\left(-\sin t\right) \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\left(-t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right)\right|\\ \mathbf{if}\;eh \leq -3.4 \cdot 10^{+137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (*
           (* (- t) eh)
           (sin
            (atan
             (* (/ (fma -0.3333333333333333 (* (* eh t) t) (- eh)) ew) t)))))))
   (if (<= eh -3.4e+137) t_1 (if (<= eh 3.5e+98) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((-t * eh) * sin(atan(((fma(-0.3333333333333333, ((eh * t) * t), -eh) / ew) * t)))));
	double tmp;
	if (eh <= -3.4e+137) {
		tmp = t_1;
	} else if (eh <= 3.5e+98) {
		tmp = fabs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(Float64(-t) * eh) * sin(atan(Float64(Float64(fma(-0.3333333333333333, Float64(Float64(eh * t) * t), Float64(-eh)) / ew) * t)))))
	tmp = 0.0
	if (eh <= -3.4e+137)
		tmp = t_1;
	elseif (eh <= 3.5e+98)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[((-t) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * N[(N[(eh * t), $MachinePrecision] * t), $MachinePrecision] + (-eh)), $MachinePrecision] / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.4e+137], t$95$1, If[LessEqual[eh, 3.5e+98], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.39999999999999986e137 or 3.5e98 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-atan.f64 N/A

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

      12. mul-1-neg N/A

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

      13. distribute-neg-frac2 N/A

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

      14. *-commutative N/A

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

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

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

      16. mul-1-neg N/A

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

   5. Applied rewrites 77.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 77.4%

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

   12. Taylor expanded in t around 0

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

   14. Applied rewrites 38.2%

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

   if -3.39999999999999986e137 < eh < 3.5e98

   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. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.7%

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

   10. Applied rewrites 52.1%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 54.0%

       \[\leadsto \left|\frac{ew}{1}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -3.4 \cdot 10^{+137}:\\ \;\;\;\;\left|\left(\left(-t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right)\right|\\ \mathbf{elif}\;eh \leq 3.5 \cdot 10^{+98}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(-t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333, \left(eh \cdot t\right) \cdot t, -eh\right)}{ew} \cdot t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 41.9% accurate, 61.6× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew}{1}\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (/ ew 1.0)))

C

double code(double eh, double ew, double t) {
	return fabs((ew / 1.0));
}

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 / 1.0d0))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew / 1.0));
}

Python

def code(eh, ew, t):
	return math.fabs((ew / 1.0))

Julia

function code(eh, ew, t)
	return abs(Float64(ew / 1.0))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew / 1.0));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew / 1.0), $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. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 43.5%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 42.5%

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

10. Applied rewrites 41.7%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 43.6%

    \[\leadsto \left|\frac{ew}{1}\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(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)))))))