Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 15.7s

Alternatives: 8

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 98.5

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

7. Applied rewrites 98.5%

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

8. Final simplification 98.5%

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

Alternative 3: 94.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\\ t_4 := \left|\mathsf{fma}\left(t\_2, \cos t\_3, t\_1 \cdot \sin t\_3\right)\right|\\ \mathbf{if}\;eh \leq -4.2 \cdot 10^{-64}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 6.4 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{t\_1 \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + t\_2}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* ew (cos t)))
        (t_3 (atan (* (/ t ew) eh)))
        (t_4 (fabs (fma t_2 (cos t_3) (* t_1 (sin t_3))))))
   (if (<= eh -4.2e-64)
     t_4
     (if (<= eh 6.4e-97)
       (fabs
        (/
         (+ (* t_1 (* eh (/ (tan t) ew))) t_2)
         (/ 1.0 (pow (+ 1.0 (pow (/ ew (* eh (tan t))) -2.0)) -0.5))))
       t_4))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = ew * cos(t);
	double t_3 = atan(((t / ew) * eh));
	double t_4 = fabs(fma(t_2, cos(t_3), (t_1 * sin(t_3))));
	double tmp;
	if (eh <= -4.2e-64) {
		tmp = t_4;
	} else if (eh <= 6.4e-97) {
		tmp = fabs((((t_1 * (eh * (tan(t) / ew))) + t_2) / (1.0 / pow((1.0 + pow((ew / (eh * tan(t))), -2.0)), -0.5))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(t / ew) * eh))
	t_4 = abs(fma(t_2, cos(t_3), Float64(t_1 * sin(t_3))))
	tmp = 0.0
	if (eh <= -4.2e-64)
		tmp = t_4;
	elseif (eh <= 6.4e-97)
		tmp = abs(Float64(Float64(Float64(t_1 * Float64(eh * Float64(tan(t) / ew))) + t_2) / Float64(1.0 / (Float64(1.0 + (Float64(ew / Float64(eh * tan(t))) ^ -2.0)) ^ -0.5))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision] + N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -4.2e-64], t$95$4, If[LessEqual[eh, 6.4e-97], N[Abs[N[(N[(N[(t$95$1 * N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 + N[Power[N[(ew / N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -4.20000000000000023e-64 or 6.39999999999999961e-97 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 98.8

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

   7. Applied rewrites 98.8%

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

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 93.2

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

   10. Applied rewrites 93.2%

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

   if -4.20000000000000023e-64 < eh < 6.39999999999999961e-97

   1. Initial program 99.8%

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

   4. Applied rewrites 89.8%

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

   6. Applied rewrites 98.8%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(\frac{ew}{\tan t \cdot eh}\right)}^{-2} + 1\right)}^{-0.5}}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.2 \cdot 10^{-64}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \cos t, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right), \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{elif}\;eh \leq 6.4 \cdot 10^{-97}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + ew \cdot \cos t}{\frac{1}{{\left(1 + {\left(\frac{ew}{eh \cdot \tan t}\right)}^{-2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \cos t, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right), \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := ew \cdot \cos t\\ t_3 := \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\\ t_4 := \left|\mathsf{fma}\left(t\_2, \cos t\_3, t\_1 \cdot \sin t\_3\right)\right|\\ \mathbf{if}\;eh \leq -1 \cdot 10^{-144}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{t\_1}{ew}, eh \cdot \tan t, t\_2\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* ew (cos t)))
        (t_3 (atan (* (/ t ew) eh)))
        (t_4 (fabs (fma t_2 (cos t_3) (* t_1 (sin t_3))))))
   (if (<= eh -1e-144)
     t_4
     (if (<= eh 2.5e-98)
       (*
        (fabs (fma (/ t_1 ew) (* eh (tan t)) t_2))
        (cos (atan (* eh (/ (tan t) ew)))))
       t_4))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = ew * cos(t);
	double t_3 = atan(((t / ew) * eh));
	double t_4 = fabs(fma(t_2, cos(t_3), (t_1 * sin(t_3))));
	double tmp;
	if (eh <= -1e-144) {
		tmp = t_4;
	} else if (eh <= 2.5e-98) {
		tmp = fabs(fma((t_1 / ew), (eh * tan(t)), t_2)) * cos(atan((eh * (tan(t) / ew))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(ew * cos(t))
	t_3 = atan(Float64(Float64(t / ew) * eh))
	t_4 = abs(fma(t_2, cos(t_3), Float64(t_1 * sin(t_3))))
	tmp = 0.0
	if (eh <= -1e-144)
		tmp = t_4;
	elseif (eh <= 2.5e-98)
		tmp = Float64(abs(fma(Float64(t_1 / ew), Float64(eh * tan(t)), t_2)) * cos(atan(Float64(eh * Float64(tan(t) / ew)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Abs[N[(t$95$2 * N[Cos[t$95$3], $MachinePrecision] + N[(t$95$1 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1e-144], t$95$4, If[LessEqual[eh, 2.5e-98], N[(N[Abs[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -9.9999999999999995e-145 or 2.50000000000000009e-98 < 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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 98.9

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-*r/ N/A

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

      5. lower-*.f64 N/A

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

      6. associate-*r/ N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 93.2

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

   10. Applied rewrites 93.2%

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

   if -9.9999999999999995e-145 < eh < 2.50000000000000009e-98

   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%

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

   6. Applied rewrites 92.5%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1 \cdot 10^{-144}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \cos t, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right), \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \mathbf{elif}\;eh \leq 2.5 \cdot 10^{-98}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, ew \cdot \cos t\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(ew \cdot \cos t, \cos \tan^{-1} \left(\frac{t}{ew} \cdot eh\right), \left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \left|\mathsf{fma}\left(\frac{t\_1}{ew}, eh \cdot \tan t, ew \cdot \cos t\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -1.66 \cdot 10^{-217}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2
         (*
          (fabs (fma (/ t_1 ew) (* eh (tan t)) (* ew (cos t))))
          (cos (atan (* eh (/ (tan t) ew)))))))
   (if (<= ew -1.66e-217)
     t_2
     (if (<= ew 1.35e-71) (fabs (* t_1 (sin (atan (* (/ t ew) eh))))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = fabs(fma((t_1 / ew), (eh * tan(t)), (ew * cos(t)))) * cos(atan((eh * (tan(t) / ew))));
	double tmp;
	if (ew <= -1.66e-217) {
		tmp = t_2;
	} else if (ew <= 1.35e-71) {
		tmp = fabs((t_1 * sin(atan(((t / ew) * eh)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(abs(fma(Float64(t_1 / ew), Float64(eh * tan(t)), Float64(ew * cos(t)))) * cos(atan(Float64(eh * Float64(tan(t) / ew)))))
	tmp = 0.0
	if (ew <= -1.66e-217)
		tmp = t_2;
	elseif (ew <= 1.35e-71)
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(t / ew) * eh)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[N[(N[(t$95$1 / ew), $MachinePrecision] * N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] + N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.66e-217], t$95$2, If[LessEqual[ew, 1.35e-71], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.66e-217 or 1.3500000000000001e-71 < 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%

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

   6. Applied rewrites 80.4%

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

   if -1.66e-217 < ew < 1.3500000000000001e-71

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 81.6

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

   7. Applied rewrites 81.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 81.7%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.66 \cdot 10^{-217}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, ew \cdot \cos t\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 1.35 \cdot 10^{-71}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{\sin t \cdot eh}{ew}, eh \cdot \tan t, ew \cdot \cos t\right)\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(-ew\right) \cdot \cos t\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{if}\;ew \leq -1.66 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (fabs (* (- ew) (cos t))) (cos (atan (* eh (/ (tan t) ew)))))))
   (if (<= ew -1.66e-217)
     t_1
     (if (<= ew 2.2e-65)
       (fabs (* (* (sin t) eh) (sin (atan (* (/ t ew) eh)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((-ew * cos(t))) * cos(atan((eh * (tan(t) / ew))));
	double tmp;
	if (ew <= -1.66e-217) {
		tmp = t_1;
	} else if (ew <= 2.2e-65) {
		tmp = fabs(((sin(t) * eh) * sin(atan(((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((-ew * cos(t))) * cos(atan((eh * (tan(t) / ew))))
    if (ew <= (-1.66d-217)) then
        tmp = t_1
    else if (ew <= 2.2d-65) then
        tmp = abs(((sin(t) * eh) * sin(atan(((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((-ew * Math.cos(t))) * Math.cos(Math.atan((eh * (Math.tan(t) / ew))));
	double tmp;
	if (ew <= -1.66e-217) {
		tmp = t_1;
	} else if (ew <= 2.2e-65) {
		tmp = Math.abs(((Math.sin(t) * eh) * Math.sin(Math.atan(((t / ew) * eh)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((-ew * math.cos(t))) * math.cos(math.atan((eh * (math.tan(t) / ew))))
	tmp = 0
	if ew <= -1.66e-217:
		tmp = t_1
	elif ew <= 2.2e-65:
		tmp = math.fabs(((math.sin(t) * eh) * math.sin(math.atan(((t / ew) * eh)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(abs(Float64(Float64(-ew) * cos(t))) * cos(atan(Float64(eh * Float64(tan(t) / ew)))))
	tmp = 0.0
	if (ew <= -1.66e-217)
		tmp = t_1;
	elseif (ew <= 2.2e-65)
		tmp = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(Float64(t / ew) * eh)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((-ew * cos(t))) * cos(atan((eh * (tan(t) / ew))));
	tmp = 0.0;
	if (ew <= -1.66e-217)
		tmp = t_1;
	elseif (ew <= 2.2e-65)
		tmp = abs(((sin(t) * eh) * sin(atan(((t / ew) * eh)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Abs[N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -1.66e-217], t$95$1, If[LessEqual[ew, 2.2e-65], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.66e-217 or 2.20000000000000021e-65 < 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 80.4%

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

   5. Taylor expanded in t around 0

      \[\leadsto \left|\frac{\color{blue}{{t}^{2} \cdot \left(\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} - \frac{1}{24} \cdot ew\right)\right) - \frac{-1}{2} \cdot ew\right) - ew}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \left|\frac{\color{blue}{{t}^{2} \cdot \left(\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} - \frac{1}{24} \cdot ew\right)\right) - \frac{-1}{2} \cdot ew\right) + \left(\mathsf{neg}\left(ew\right)\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{\color{blue}{\left(\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} - \frac{1}{24} \cdot ew\right)\right) - \frac{-1}{2} \cdot ew\right) \cdot {t}^{2}} + \left(\mathsf{neg}\left(ew\right)\right)}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\left(-1 \cdot \frac{{eh}^{2}}{ew} + {t}^{2} \cdot \left(\frac{-1}{6} \cdot \frac{{eh}^{2}}{ew} - \frac{1}{24} \cdot ew\right)\right) - \frac{-1}{2} \cdot ew, {t}^{2}, \mathsf{neg}\left(ew\right)\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

   7. Applied rewrites 39.9%

      \[\leadsto \left|\frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.041666666666666664, ew, \frac{eh \cdot eh}{ew} \cdot -0.16666666666666666\right) \cdot t, t, -\mathsf{fma}\left(-0.5, ew, \frac{eh \cdot eh}{ew}\right)\right), t \cdot t, -ew\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]

   9. Applied rewrites 39.9%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(t \cdot t, \mathsf{fma}\left(0.5, ew, \mathsf{fma}\left(eh, \frac{-eh}{ew}, \left(\mathsf{fma}\left(-0.041666666666666664, ew, -0.16666666666666666 \cdot \frac{eh \cdot eh}{ew}\right) \cdot t\right) \cdot t\right)\right), -ew\right)\right| \cdot \cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)} \]

   10. Taylor expanded in ew around inf

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

       1. associate-*r* N/A

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

       2. mul-1-neg N/A

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

       3. lower-*.f64 N/A

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

       4. lower-neg.f64 N/A

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

       5. lower-cos.f64 79.2

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

   12. Applied rewrites 79.2%

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

   if -1.66e-217 < ew < 2.20000000000000021e-65

   1. Initial program 99.7%

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 81.5

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 81.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.66 \cdot 10^{-217}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \mathbf{elif}\;ew \leq 2.2 \cdot 10^{-65}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(-ew\right) \cdot \cos t\right| \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{if}\;t \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (* (sin t) eh) (sin (atan (* (/ t ew) eh)))))))
   (if (<= t -4.9e-14) t_1 (if (<= t 1.5e-8) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * eh) * sin(atan(((t / ew) * eh)))));
	double tmp;
	if (t <= -4.9e-14) {
		tmp = t_1;
	} else if (t <= 1.5e-8) {
		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(t) * eh) * sin(atan(((t / ew) * eh)))))
    if (t <= (-4.9d-14)) then
        tmp = t_1
    else if (t <= 1.5d-8) 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(t) * eh) * Math.sin(Math.atan(((t / ew) * eh)))));
	double tmp;
	if (t <= -4.9e-14) {
		tmp = t_1;
	} else if (t <= 1.5e-8) {
		tmp = Math.abs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs(((math.sin(t) * eh) * math.sin(math.atan(((t / ew) * eh)))))
	tmp = 0
	if t <= -4.9e-14:
		tmp = t_1
	elif t <= 1.5e-8:
		tmp = math.fabs((ew / 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * eh) * sin(atan(Float64(Float64(t / ew) * eh)))))
	tmp = 0.0
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 1.5e-8)
		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(t) * eh) * sin(atan(((t / ew) * eh)))));
	tmp = 0.0;
	if (t <= -4.9e-14)
		tmp = t_1;
	elseif (t <= 1.5e-8)
		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[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(t / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -4.9e-14], t$95$1, If[LessEqual[t, 1.5e-8], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.89999999999999995e-14 or 1.49999999999999987e-8 < t

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-sin.f64 52.8

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

   7. Applied rewrites 52.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 53.2%

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

   if -4.89999999999999995e-14 < t < 1.49999999999999987e-8

   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 73.1%

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

   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 73.1%

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

   10. Applied rewrites 72.9%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 73.3%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.9 \cdot 10^{-14}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 1.5 \cdot 10^{-8}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{t}{ew} \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.1% 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 40.6%

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

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 39.4%

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

10. Applied rewrites 38.7%

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

11. Taylor expanded in ew around inf

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

13. Applied rewrites 40.8%

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

Reproduce

herbie shell --seed 2024284 
(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)))))))