Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 16.2s

Alternatives: 10

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

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

FPCore

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

C

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

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

Alternative 2: 51.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot ew\\ t_2 := \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right)\\ \mathbf{if}\;\cos t\_2 \cdot t\_1 - \sin t\_2 \cdot \left(\sin t \cdot eh\right) \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) ew)) (t_2 (atan (/ (* (tan t) eh) (- ew)))))
   (if (<= (- (* (cos t_2) t_1) (* (sin t_2) (* (sin t) eh))) -1e-281)
     (fabs (/ ew 1.0))
     t_1)))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * ew;
	double t_2 = atan(((tan(t) * eh) / -ew));
	double tmp;
	if (((cos(t_2) * t_1) - (sin(t_2) * (sin(t) * eh))) <= -1e-281) {
		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) :: t_2
    real(8) :: tmp
    t_1 = cos(t) * ew
    t_2 = atan(((tan(t) * eh) / -ew))
    if (((cos(t_2) * t_1) - (sin(t_2) * (sin(t) * eh))) <= (-1d-281)) 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.cos(t) * ew;
	double t_2 = Math.atan(((Math.tan(t) * eh) / -ew));
	double tmp;
	if (((Math.cos(t_2) * t_1) - (Math.sin(t_2) * (Math.sin(t) * eh))) <= -1e-281) {
		tmp = Math.abs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * ew
	t_2 = math.atan(((math.tan(t) * eh) / -ew))
	tmp = 0
	if ((math.cos(t_2) * t_1) - (math.sin(t_2) * (math.sin(t) * eh))) <= -1e-281:
		tmp = math.fabs((ew / 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * ew)
	t_2 = atan(Float64(Float64(tan(t) * eh) / Float64(-ew)))
	tmp = 0.0
	if (Float64(Float64(cos(t_2) * t_1) - Float64(sin(t_2) * Float64(sin(t) * eh))) <= -1e-281)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * ew;
	t_2 = atan(((tan(t) * eh) / -ew));
	tmp = 0.0;
	if (((cos(t_2) * t_1) - (sin(t_2) * (sin(t) * eh))) <= -1e-281)
		tmp = abs((ew / 1.0));
	else
		tmp = t_1;
	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[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] * eh), $MachinePrecision] / (-ew)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[Cos[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[Sin[t$95$2], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1e-281], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 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))))) < -1e-281

   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 41.2%

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

   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 40.3%

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

   10. Applied rewrites 39.4%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 41.4%

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

   if -1e-281 < (-.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 99.6%

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

   6. Applied rewrites 79.6%

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

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 66.5

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

   9. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 2 regimes into one program.

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \cdot \left(\cos t \cdot ew\right) - \sin \tan^{-1} \left(\frac{\tan t \cdot eh}{-ew}\right) \cdot \left(\sin t \cdot eh\right) \leq -1 \cdot 10^{-281}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 99.2

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

5. Applied rewrites 99.2%

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

6. Final simplification 99.2%

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

Alternative 4: 94.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot eh\\ t_2 := \cos t \cdot ew\\ t_3 := \left|\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot t\_1 - \cos \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right) \cdot t\_2\right|\\ \mathbf{if}\;eh \leq -60000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;eh \leq 1.04 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\left|\frac{-1}{\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot t\_1 + t\_2}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) eh))
        (t_2 (* (cos t) ew))
        (t_3
         (fabs
          (-
           (* (sin (atan (/ (* (- eh) t) ew))) t_1)
           (* (cos (atan (* (- t) (/ eh ew)))) t_2)))))
   (if (<= eh -60000.0)
     t_3
     (if (<= eh 1.04e-22)
       (/
        1.0
        (fabs
         (/
          -1.0
          (/
           (+ (* (* (/ eh ew) (tan t)) t_1) t_2)
           (sqrt (+ 1.0 (pow (* (/ (tan t) ew) eh) 2.0)))))))
       t_3))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * eh;
	double t_2 = cos(t) * ew;
	double t_3 = fabs(((sin(atan(((-eh * t) / ew))) * t_1) - (cos(atan((-t * (eh / ew)))) * t_2)));
	double tmp;
	if (eh <= -60000.0) {
		tmp = t_3;
	} else if (eh <= 1.04e-22) {
		tmp = 1.0 / fabs((-1.0 / (((((eh / ew) * tan(t)) * t_1) + t_2) / sqrt((1.0 + pow(((tan(t) / ew) * eh), 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 = sin(t) * eh
    t_2 = cos(t) * ew
    t_3 = abs(((sin(atan(((-eh * t) / ew))) * t_1) - (cos(atan((-t * (eh / ew)))) * t_2)))
    if (eh <= (-60000.0d0)) then
        tmp = t_3
    else if (eh <= 1.04d-22) then
        tmp = 1.0d0 / abs(((-1.0d0) / (((((eh / ew) * tan(t)) * t_1) + t_2) / sqrt((1.0d0 + (((tan(t) / ew) * eh) ** 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.sin(t) * eh;
	double t_2 = Math.cos(t) * ew;
	double t_3 = Math.abs(((Math.sin(Math.atan(((-eh * t) / ew))) * t_1) - (Math.cos(Math.atan((-t * (eh / ew)))) * t_2)));
	double tmp;
	if (eh <= -60000.0) {
		tmp = t_3;
	} else if (eh <= 1.04e-22) {
		tmp = 1.0 / Math.abs((-1.0 / (((((eh / ew) * Math.tan(t)) * t_1) + t_2) / Math.sqrt((1.0 + Math.pow(((Math.tan(t) / ew) * eh), 2.0))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.sin(t) * eh
	t_2 = math.cos(t) * ew
	t_3 = math.fabs(((math.sin(math.atan(((-eh * t) / ew))) * t_1) - (math.cos(math.atan((-t * (eh / ew)))) * t_2)))
	tmp = 0
	if eh <= -60000.0:
		tmp = t_3
	elif eh <= 1.04e-22:
		tmp = 1.0 / math.fabs((-1.0 / (((((eh / ew) * math.tan(t)) * t_1) + t_2) / math.sqrt((1.0 + math.pow(((math.tan(t) / ew) * eh), 2.0))))))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * eh)
	t_2 = Float64(cos(t) * ew)
	t_3 = abs(Float64(Float64(sin(atan(Float64(Float64(Float64(-eh) * t) / ew))) * t_1) - Float64(cos(atan(Float64(Float64(-t) * Float64(eh / ew)))) * t_2)))
	tmp = 0.0
	if (eh <= -60000.0)
		tmp = t_3;
	elseif (eh <= 1.04e-22)
		tmp = Float64(1.0 / abs(Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(eh / ew) * tan(t)) * t_1) + t_2) / sqrt(Float64(1.0 + (Float64(Float64(tan(t) / ew) * eh) ^ 2.0)))))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sin(t) * eh;
	t_2 = cos(t) * ew;
	t_3 = abs(((sin(atan(((-eh * t) / ew))) * t_1) - (cos(atan((-t * (eh / ew)))) * t_2)));
	tmp = 0.0;
	if (eh <= -60000.0)
		tmp = t_3;
	elseif (eh <= 1.04e-22)
		tmp = 1.0 / abs((-1.0 / (((((eh / ew) * tan(t)) * t_1) + t_2) / sqrt((1.0 + (((tan(t) / ew) * eh) ^ 2.0))))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$3 = N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[((-eh) * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[Cos[N[ArcTan[N[((-t) * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -60000.0], t$95$3, If[LessEqual[eh, 1.04e-22], N[(1.0 / N[Abs[N[(-1.0 / N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -6e4 or 1.04e-22 < 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 t around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 99.3

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

   5. Applied rewrites 99.3%

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

   6. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. associate-*l* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 92.3

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

   8. Applied rewrites 92.3%

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

   if -6e4 < eh < 1.04e-22

   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%

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

   6. Applied rewrites 98.8%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -60000:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(\sin t \cdot eh\right) - \cos \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{elif}\;eh \leq 1.04 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{\left|\frac{-1}{\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right) + \cos t \cdot ew}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot t}{ew}\right) \cdot \left(\sin t \cdot eh\right) - \cos \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 82.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -6.2 \cdot 10^{+50}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{\left|\frac{-1}{\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right) + \cos t \cdot ew}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -6.2e+50)
   (fabs (* (* (- (sin t)) eh) (sin (atan (* (- t) (/ eh ew))))))
   (if (<= eh 6.5e+42)
     (/
      1.0
      (fabs
       (/
        -1.0
        (/
         (+ (* (* (/ eh ew) (tan t)) (* (sin t) eh)) (* (cos t) ew))
         (sqrt (+ 1.0 (pow (* (/ (tan t) ew) eh) 2.0)))))))
     (fabs (* (* (sin (atan (/ (* eh t) ew))) eh) (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -6.2e+50) {
		tmp = fabs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	} else if (eh <= 6.5e+42) {
		tmp = 1.0 / fabs((-1.0 / (((((eh / ew) * tan(t)) * (sin(t) * eh)) + (cos(t) * ew)) / sqrt((1.0 + pow(((tan(t) / ew) * eh), 2.0))))));
	} else {
		tmp = fabs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-6.2d+50)) then
        tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))))
    else if (eh <= 6.5d+42) then
        tmp = 1.0d0 / abs(((-1.0d0) / (((((eh / ew) * tan(t)) * (sin(t) * eh)) + (cos(t) * ew)) / sqrt((1.0d0 + (((tan(t) / ew) * eh) ** 2.0d0))))))
    else
        tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -6.2e+50) {
		tmp = Math.abs(((-Math.sin(t) * eh) * Math.sin(Math.atan((-t * (eh / ew))))));
	} else if (eh <= 6.5e+42) {
		tmp = 1.0 / Math.abs((-1.0 / (((((eh / ew) * Math.tan(t)) * (Math.sin(t) * eh)) + (Math.cos(t) * ew)) / Math.sqrt((1.0 + Math.pow(((Math.tan(t) / ew) * eh), 2.0))))));
	} else {
		tmp = Math.abs(((Math.sin(Math.atan(((eh * t) / ew))) * eh) * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -6.2e+50:
		tmp = math.fabs(((-math.sin(t) * eh) * math.sin(math.atan((-t * (eh / ew))))))
	elif eh <= 6.5e+42:
		tmp = 1.0 / math.fabs((-1.0 / (((((eh / ew) * math.tan(t)) * (math.sin(t) * eh)) + (math.cos(t) * ew)) / math.sqrt((1.0 + math.pow(((math.tan(t) / ew) * eh), 2.0))))))
	else:
		tmp = math.fabs(((math.sin(math.atan(((eh * t) / ew))) * eh) * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -6.2e+50)
		tmp = abs(Float64(Float64(Float64(-sin(t)) * eh) * sin(atan(Float64(Float64(-t) * Float64(eh / ew))))));
	elseif (eh <= 6.5e+42)
		tmp = Float64(1.0 / abs(Float64(-1.0 / Float64(Float64(Float64(Float64(Float64(eh / ew) * tan(t)) * Float64(sin(t) * eh)) + Float64(cos(t) * ew)) / sqrt(Float64(1.0 + (Float64(Float64(tan(t) / ew) * eh) ^ 2.0)))))));
	else
		tmp = abs(Float64(Float64(sin(atan(Float64(Float64(eh * t) / ew))) * eh) * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -6.2e+50)
		tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	elseif (eh <= 6.5e+42)
		tmp = 1.0 / abs((-1.0 / (((((eh / ew) * tan(t)) * (sin(t) * eh)) + (cos(t) * ew)) / sqrt((1.0 + (((tan(t) / ew) * eh) ^ 2.0))))));
	else
		tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -6.2e+50], N[Abs[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 6.5e+42], N[(1.0 / N[Abs[N[(-1.0 / N[(N[(N[(N[(N[(eh / ew), $MachinePrecision] * N[Tan[t], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 + N[Power[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -6.20000000000000006e50

   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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 68.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 69.1%

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

   if -6.20000000000000006e50 < eh < 6.50000000000000052e42

   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%

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

   6. Applied rewrites 97.5%

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

   if 6.50000000000000052e42 < 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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 68.7%

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -6.2 \cdot 10^{+50}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 6.5 \cdot 10^{+42}:\\ \;\;\;\;\frac{1}{\left|\frac{-1}{\frac{\left(\frac{eh}{ew} \cdot \tan t\right) \cdot \left(\sin t \cdot eh\right) + \cos t \cdot ew}{\sqrt{1 + {\left(\frac{\tan t}{ew} \cdot eh\right)}^{2}}}}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -\sin t\\ \mathbf{if}\;eh \leq -1.55 \cdot 10^{+58}:\\ \;\;\;\;\left|\left(t\_1 \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{+36}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{t\_1}{ew}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (- (sin t))))
   (if (<= eh -1.55e+58)
     (fabs (* (* t_1 eh) (sin (atan (* (- t) (/ eh ew))))))
     (if (<= eh 1.65e+36)
       (fabs (* (cos (atan (* (/ eh (cos t)) (/ t_1 ew)))) (* (cos t) ew)))
       (fabs (* (* (sin (atan (/ (* eh t) ew))) eh) (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = -sin(t);
	double tmp;
	if (eh <= -1.55e+58) {
		tmp = fabs(((t_1 * eh) * sin(atan((-t * (eh / ew))))));
	} else if (eh <= 1.65e+36) {
		tmp = fabs((cos(atan(((eh / cos(t)) * (t_1 / ew)))) * (cos(t) * ew)));
	} else {
		tmp = fabs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -sin(t)
    if (eh <= (-1.55d+58)) then
        tmp = abs(((t_1 * eh) * sin(atan((-t * (eh / ew))))))
    else if (eh <= 1.65d+36) then
        tmp = abs((cos(atan(((eh / cos(t)) * (t_1 / ew)))) * (cos(t) * ew)))
    else
        tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = -Math.sin(t);
	double tmp;
	if (eh <= -1.55e+58) {
		tmp = Math.abs(((t_1 * eh) * Math.sin(Math.atan((-t * (eh / ew))))));
	} else if (eh <= 1.65e+36) {
		tmp = Math.abs((Math.cos(Math.atan(((eh / Math.cos(t)) * (t_1 / ew)))) * (Math.cos(t) * ew)));
	} else {
		tmp = Math.abs(((Math.sin(Math.atan(((eh * t) / ew))) * eh) * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = -math.sin(t)
	tmp = 0
	if eh <= -1.55e+58:
		tmp = math.fabs(((t_1 * eh) * math.sin(math.atan((-t * (eh / ew))))))
	elif eh <= 1.65e+36:
		tmp = math.fabs((math.cos(math.atan(((eh / math.cos(t)) * (t_1 / ew)))) * (math.cos(t) * ew)))
	else:
		tmp = math.fabs(((math.sin(math.atan(((eh * t) / ew))) * eh) * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(-sin(t))
	tmp = 0.0
	if (eh <= -1.55e+58)
		tmp = abs(Float64(Float64(t_1 * eh) * sin(atan(Float64(Float64(-t) * Float64(eh / ew))))));
	elseif (eh <= 1.65e+36)
		tmp = abs(Float64(cos(atan(Float64(Float64(eh / cos(t)) * Float64(t_1 / ew)))) * Float64(cos(t) * ew)));
	else
		tmp = abs(Float64(Float64(sin(atan(Float64(Float64(eh * t) / ew))) * eh) * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = -sin(t);
	tmp = 0.0;
	if (eh <= -1.55e+58)
		tmp = abs(((t_1 * eh) * sin(atan((-t * (eh / ew))))));
	elseif (eh <= 1.65e+36)
		tmp = abs((cos(atan(((eh / cos(t)) * (t_1 / ew)))) * (cos(t) * ew)));
	else
		tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = (-N[Sin[t], $MachinePrecision])}, If[LessEqual[eh, -1.55e+58], N[Abs[N[(N[(t$95$1 * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.65e+36], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(eh / N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.55e58

   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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 69.6%

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

   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 69.7%

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

   if -1.55e58 < eh < 1.6499999999999999e36

   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 ew around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

   5. Applied rewrites 85.1%

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

   if 1.6499999999999999e36 < 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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 68.7%

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.55 \cdot 10^{+58}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{+36}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{eh}{\cos t} \cdot \frac{-\sin t}{ew}\right) \cdot \left(\cos t \cdot ew\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.55 \cdot 10^{+58}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -1.55e+58)
   (fabs (* (* (- (sin t)) eh) (sin (atan (* (- t) (/ eh ew))))))
   (if (<= eh 1.65e+36)
     (fabs (/ (* (- ew) (cos t)) (/ 1.0 (cos (atan (* (/ (tan t) ew) eh))))))
     (fabs (* (* (sin (atan (/ (* eh t) ew))) eh) (sin t))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.55e+58) {
		tmp = fabs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	} else if (eh <= 1.65e+36) {
		tmp = fabs(((-ew * cos(t)) / (1.0 / cos(atan(((tan(t) / ew) * eh))))));
	} else {
		tmp = fabs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (eh <= (-1.55d+58)) then
        tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))))
    else if (eh <= 1.65d+36) then
        tmp = abs(((-ew * cos(t)) / (1.0d0 / cos(atan(((tan(t) / ew) * eh))))))
    else
        tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -1.55e+58) {
		tmp = Math.abs(((-Math.sin(t) * eh) * Math.sin(Math.atan((-t * (eh / ew))))));
	} else if (eh <= 1.65e+36) {
		tmp = Math.abs(((-ew * Math.cos(t)) / (1.0 / Math.cos(Math.atan(((Math.tan(t) / ew) * eh))))));
	} else {
		tmp = Math.abs(((Math.sin(Math.atan(((eh * t) / ew))) * eh) * Math.sin(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -1.55e+58:
		tmp = math.fabs(((-math.sin(t) * eh) * math.sin(math.atan((-t * (eh / ew))))))
	elif eh <= 1.65e+36:
		tmp = math.fabs(((-ew * math.cos(t)) / (1.0 / math.cos(math.atan(((math.tan(t) / ew) * eh))))))
	else:
		tmp = math.fabs(((math.sin(math.atan(((eh * t) / ew))) * eh) * math.sin(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -1.55e+58)
		tmp = abs(Float64(Float64(Float64(-sin(t)) * eh) * sin(atan(Float64(Float64(-t) * Float64(eh / ew))))));
	elseif (eh <= 1.65e+36)
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(1.0 / cos(atan(Float64(Float64(tan(t) / ew) * eh))))));
	else
		tmp = abs(Float64(Float64(sin(atan(Float64(Float64(eh * t) / ew))) * eh) * sin(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -1.55e+58)
		tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	elseif (eh <= 1.65e+36)
		tmp = abs(((-ew * cos(t)) / (1.0 / cos(atan(((tan(t) / ew) * eh))))));
	else
		tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -1.55e+58], N[Abs[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 1.65e+36], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Cos[N[ArcTan[N[(N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -1.55e58

   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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 69.6%

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

   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 69.7%

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

   if -1.55e58 < eh < 1.6499999999999999e36

   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 86.4%

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

   5. Taylor expanded in ew around inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 85.1

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

   7. Applied rewrites 85.1%

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

   if 1.6499999999999999e36 < 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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 68.7%

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

   7. Applied rewrites 68.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 69.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.55 \cdot 10^{+58}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{elif}\;eh \leq 1.65 \cdot 10^{+36}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{elif}\;ew \leq 1.36 \cdot 10^{-30}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -8e-23)
   (fabs (/ ew 1.0))
   (if (<= ew 1.36e-30)
     (fabs (* (* (- (sin t)) eh) (sin (atan (* (- t) (/ eh ew))))))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -8e-23) {
		tmp = fabs((ew / 1.0));
	} else if (ew <= 1.36e-30) {
		tmp = fabs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-8d-23)) then
        tmp = abs((ew / 1.0d0))
    else if (ew <= 1.36d-30) then
        tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -8e-23) {
		tmp = Math.abs((ew / 1.0));
	} else if (ew <= 1.36e-30) {
		tmp = Math.abs(((-Math.sin(t) * eh) * Math.sin(Math.atan((-t * (eh / ew))))));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -8e-23:
		tmp = math.fabs((ew / 1.0))
	elif ew <= 1.36e-30:
		tmp = math.fabs(((-math.sin(t) * eh) * math.sin(math.atan((-t * (eh / ew))))))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -8e-23)
		tmp = abs(Float64(ew / 1.0));
	elseif (ew <= 1.36e-30)
		tmp = abs(Float64(Float64(Float64(-sin(t)) * eh) * sin(atan(Float64(Float64(-t) * Float64(eh / ew))))));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -8e-23)
		tmp = abs((ew / 1.0));
	elseif (ew <= 1.36e-30)
		tmp = abs(((-sin(t) * eh) * sin(atan((-t * (eh / ew))))));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -8e-23], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.36e-30], N[Abs[N[(N[((-N[Sin[t], $MachinePrecision]) * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[((-t) * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -7.99999999999999968e-23

   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 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 51.5%

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

   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 49.2%

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

   10. Applied rewrites 48.2%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 51.6%

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

   if -7.99999999999999968e-23 < ew < 1.36e-30

   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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 67.7%

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

   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 67.8%

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

   if 1.36e-30 < 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%

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

   6. Applied rewrites 79.8%

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

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 73.8

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

   9. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{elif}\;ew \leq 1.36 \cdot 10^{-30}:\\ \;\;\;\;\left|\left(\left(-\sin t\right) \cdot eh\right) \cdot \sin \tan^{-1} \left(\left(-t\right) \cdot \frac{eh}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{elif}\;ew \leq 1.36 \cdot 10^{-30}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= ew -8e-23)
   (fabs (/ ew 1.0))
   (if (<= ew 1.36e-30)
     (fabs (* (* (sin (atan (/ (* eh t) ew))) eh) (sin t)))
     (* (cos t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -8e-23) {
		tmp = fabs((ew / 1.0));
	} else if (ew <= 1.36e-30) {
		tmp = fabs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	} else {
		tmp = cos(t) * ew;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if (ew <= (-8d-23)) then
        tmp = abs((ew / 1.0d0))
    else if (ew <= 1.36d-30) then
        tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)))
    else
        tmp = cos(t) * ew
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (ew <= -8e-23) {
		tmp = Math.abs((ew / 1.0));
	} else if (ew <= 1.36e-30) {
		tmp = Math.abs(((Math.sin(Math.atan(((eh * t) / ew))) * eh) * Math.sin(t)));
	} else {
		tmp = Math.cos(t) * ew;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if ew <= -8e-23:
		tmp = math.fabs((ew / 1.0))
	elif ew <= 1.36e-30:
		tmp = math.fabs(((math.sin(math.atan(((eh * t) / ew))) * eh) * math.sin(t)))
	else:
		tmp = math.cos(t) * ew
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (ew <= -8e-23)
		tmp = abs(Float64(ew / 1.0));
	elseif (ew <= 1.36e-30)
		tmp = abs(Float64(Float64(sin(atan(Float64(Float64(eh * t) / ew))) * eh) * sin(t)));
	else
		tmp = Float64(cos(t) * ew);
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (ew <= -8e-23)
		tmp = abs((ew / 1.0));
	elseif (ew <= 1.36e-30)
		tmp = abs(((sin(atan(((eh * t) / ew))) * eh) * sin(t)));
	else
		tmp = cos(t) * ew;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[ew, -8e-23], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.36e-30], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh * t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if ew < -7.99999999999999968e-23

   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 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 51.5%

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

   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 49.2%

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

   10. Applied rewrites 48.2%

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

   11. Taylor expanded in ew around inf

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

   13. Applied rewrites 51.6%

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

   if -7.99999999999999968e-23 < ew < 1.36e-30

   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 ew around 0

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \sin t}{ew \cdot \cos t}\right)\right)\right)}\right| \]
   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 67.7%

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

   7. Applied rewrites 67.7%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 67.8%

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

   if 1.36e-30 < 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%

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

   6. Applied rewrites 79.8%

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

   7. Taylor expanded in ew around inf

      \[\leadsto \color{blue}{ew \cdot \cos t} \]
   8. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 73.8

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

   9. Applied rewrites 73.8%

      \[\leadsto \color{blue}{\cos t \cdot ew} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -8 \cdot 10^{-23}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{elif}\;ew \leq 1.36 \cdot 10^{-30}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right) \cdot eh\right) \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;\cos t \cdot ew\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.5% 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 41.7%

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

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 39.9%

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

10. Applied rewrites 38.9%

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

11. Taylor expanded in ew around inf

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

13. Applied rewrites 41.9%

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

Reproduce

herbie shell --seed 2024249 
(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)))))))