Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 12.7s

Alternatives: 9

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

5. Final simplification 99.8%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

4. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. cos-mult N/A

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

   5. clear-num N/A

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

   6. lower-/.f64 N/A

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

   7. clear-num N/A

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

   8. cos-mult N/A

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

   9. lift-cos.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-/.f64 99.8

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

   13. lift-*.f64 N/A

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

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

   1. associate-*l/ N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 99.3

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

9. Applied rewrites 99.3%

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

10. Final simplification 99.3%

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

Alternative 3: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ t_2 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot t\_2 + \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} t\_2}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ t (- ew)) eh))) (* (- eh) (sin t)))))
        (t_2 (* eh (/ (tan t) ew))))
   (if (<= eh -2.55e+119)
     t_1
     (if (<= eh 2e+147)
       (fabs
        (/ (+ (* (* (sin t) eh) t_2) (* (cos t) ew)) (/ 1.0 (cos (atan t_2)))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	double t_2 = eh * (tan(t) / ew);
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 2e+147) {
		tmp = fabs(((((sin(t) * eh) * t_2) + (cos(t) * ew)) / (1.0 / cos(atan(t_2)))));
	} 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 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))))
    t_2 = eh * (tan(t) / ew)
    if (eh <= (-2.55d+119)) then
        tmp = t_1
    else if (eh <= 2d+147) then
        tmp = abs(((((sin(t) * eh) * t_2) + (cos(t) * ew)) / (1.0d0 / cos(atan(t_2)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((t / -ew) * eh))) * (-eh * Math.sin(t))));
	double t_2 = eh * (Math.tan(t) / ew);
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 2e+147) {
		tmp = Math.abs(((((Math.sin(t) * eh) * t_2) + (Math.cos(t) * ew)) / (1.0 / Math.cos(Math.atan(t_2)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((t / -ew) * eh))) * (-eh * math.sin(t))))
	t_2 = eh * (math.tan(t) / ew)
	tmp = 0
	if eh <= -2.55e+119:
		tmp = t_1
	elif eh <= 2e+147:
		tmp = math.fabs(((((math.sin(t) * eh) * t_2) + (math.cos(t) * ew)) / (1.0 / math.cos(math.atan(t_2)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * Float64(Float64(-eh) * sin(t))))
	t_2 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 2e+147)
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * eh) * t_2) + Float64(cos(t) * ew)) / Float64(1.0 / cos(atan(t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	t_2 = eh * (tan(t) / ew);
	tmp = 0.0;
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 2e+147)
		tmp = abs(((((sin(t) * eh) * t_2) + (cos(t) * ew)) / (1.0 / cos(atan(t_2)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.55e+119], t$95$1, If[LessEqual[eh, 2e+147], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$2), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Cos[N[ArcTan[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.54999999999999992e119 or 2e147 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.1%

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

   if -2.54999999999999992e119 < eh < 2e147

   1. Initial program 99.8%

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

   4. Applied rewrites 83.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq 1.32 \cdot 10^{+149}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot t\_1 + \cos t \cdot ew}{\frac{1}{{\left(1 + {t\_1}^{2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (/ (tan t) ew))))
   (if (<= eh 1.32e+149)
     (fabs
      (/
       (+ (* (* (sin t) eh) t_1) (* (cos t) ew))
       (/ 1.0 (pow (+ 1.0 (pow t_1 2.0)) -0.5))))
     (fabs (* (sin (atan (* (/ t (- ew)) eh))) (* (- eh) (sin t)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * (tan(t) / ew);
	double tmp;
	if (eh <= 1.32e+149) {
		tmp = fabs(((((sin(t) * eh) * t_1) + (cos(t) * ew)) / (1.0 / pow((1.0 + pow(t_1, 2.0)), -0.5))));
	} else {
		tmp = fabs((sin(atan(((t / -ew) * eh))) * (-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 = eh * (tan(t) / ew)
    if (eh <= 1.32d+149) then
        tmp = abs(((((sin(t) * eh) * t_1) + (cos(t) * ew)) / (1.0d0 / ((1.0d0 + (t_1 ** 2.0d0)) ** (-0.5d0)))))
    else
        tmp = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * (Math.tan(t) / ew);
	double tmp;
	if (eh <= 1.32e+149) {
		tmp = Math.abs(((((Math.sin(t) * eh) * t_1) + (Math.cos(t) * ew)) / (1.0 / Math.pow((1.0 + Math.pow(t_1, 2.0)), -0.5))));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((t / -ew) * eh))) * (-eh * Math.sin(t))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * (math.tan(t) / ew)
	tmp = 0
	if eh <= 1.32e+149:
		tmp = math.fabs(((((math.sin(t) * eh) * t_1) + (math.cos(t) * ew)) / (1.0 / math.pow((1.0 + math.pow(t_1, 2.0)), -0.5))))
	else:
		tmp = math.fabs((math.sin(math.atan(((t / -ew) * eh))) * (-eh * math.sin(t))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if (eh <= 1.32e+149)
		tmp = abs(Float64(Float64(Float64(Float64(sin(t) * eh) * t_1) + Float64(cos(t) * ew)) / Float64(1.0 / (Float64(1.0 + (t_1 ^ 2.0)) ^ -0.5))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * Float64(Float64(-eh) * sin(t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * (tan(t) / ew);
	tmp = 0.0;
	if (eh <= 1.32e+149)
		tmp = abs(((((sin(t) * eh) * t_1) + (cos(t) * ew)) / (1.0 / ((1.0 + (t_1 ^ 2.0)) ^ -0.5))));
	else
		tmp = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 1.32e+149], N[Abs[N[(N[(N[(N[(N[Sin[t], $MachinePrecision] * eh), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(N[Cos[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[Power[N[(1.0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.32000000000000004e149

   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 76.6%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. cos-atan N/A

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

      4. pow1/2 N/A

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

      5. pow-flip N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-pow.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. pow2 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

      16. metadata-eval 88.6

          \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{{\left({\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1\right)}^{\color{blue}{-0.5}}}}\right| \]

   6. Applied rewrites 88.6%

      \[\leadsto \left|\frac{\left(\sin t \cdot \left(-eh\right)\right) \cdot \left(\frac{\tan t}{ew} \cdot eh\right) - \cos t \cdot ew}{\frac{1}{\color{blue}{{\left({\left(eh \cdot \frac{\tan t}{ew}\right)}^{2} + 1\right)}^{-0.5}}}}\right| \]

   if 1.32000000000000004e149 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 81.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 81.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 1.32 \cdot 10^{+149}:\\ \;\;\;\;\left|\frac{\left(\sin t \cdot eh\right) \cdot \left(eh \cdot \frac{\tan t}{ew}\right) + \cos t \cdot ew}{\frac{1}{{\left(1 + {\left(eh \cdot \frac{\tan t}{ew}\right)}^{2}\right)}^{-0.5}}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-eh\right) \cdot \sin t\\ t_2 := \left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot t\_1\right|\\ t_3 := eh \cdot \frac{\tan t}{ew}\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, t\_3 \cdot t\_1\right) \cdot \cos \tan^{-1} t\_3\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (- eh) (sin t)))
        (t_2 (fabs (* (sin (atan (* (/ t (- ew)) eh))) t_1)))
        (t_3 (* eh (/ (tan t) ew))))
   (if (<= eh -2.55e+119)
     t_2
     (if (<= eh 2e+147)
       (fabs (* (fma (- ew) (cos t) (* t_3 t_1)) (cos (atan t_3))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = -eh * sin(t);
	double t_2 = fabs((sin(atan(((t / -ew) * eh))) * t_1));
	double t_3 = eh * (tan(t) / ew);
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_2;
	} else if (eh <= 2e+147) {
		tmp = fabs((fma(-ew, cos(t), (t_3 * t_1)) * cos(atan(t_3))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(-eh) * sin(t))
	t_2 = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * t_1))
	t_3 = Float64(eh * Float64(tan(t) / ew))
	tmp = 0.0
	if (eh <= -2.55e+119)
		tmp = t_2;
	elseif (eh <= 2e+147)
		tmp = abs(Float64(fma(Float64(-ew), cos(t), Float64(t_3 * t_1)) * cos(atan(t_3))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -2.55e+119], t$95$2, If[LessEqual[eh, 2e+147], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision] + N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] * N[Cos[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.54999999999999992e119 or 2e147 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.1%

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

   if -2.54999999999999992e119 < eh < 2e147

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-mult N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. clear-num N/A

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

      8. cos-mult N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-/.f64 99.7

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

      13. lift-*.f64 N/A

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 83.9%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 2 \cdot 10^{+147}:\\ \;\;\;\;\left|\mathsf{fma}\left(-ew, \cos t, \left(eh \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right) \cdot \cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ t (- ew)) eh))) (* (- eh) (sin t))))))
   (if (<= eh -2.55e+119)
     t_1
     (if (<= eh 8.5e+146)
       (fabs
        (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* eh (/ (tan t) ew)))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 8.5e+146) {
		tmp = fabs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))))
    if (eh <= (-2.55d+119)) then
        tmp = t_1
    else if (eh <= 8.5d+146) then
        tmp = abs(((-ew * cos(t)) / ((-1.0d0) / cos(atan((eh * (tan(t) / ew)))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((t / -ew) * eh))) * (-eh * Math.sin(t))));
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 8.5e+146) {
		tmp = Math.abs(((-ew * Math.cos(t)) / (-1.0 / Math.cos(Math.atan((eh * (Math.tan(t) / ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((t / -ew) * eh))) * (-eh * math.sin(t))))
	tmp = 0
	if eh <= -2.55e+119:
		tmp = t_1
	elif eh <= 8.5e+146:
		tmp = math.fabs(((-ew * math.cos(t)) / (-1.0 / math.cos(math.atan((eh * (math.tan(t) / ew)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * Float64(Float64(-eh) * sin(t))))
	tmp = 0.0
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 8.5e+146)
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(-1.0 / cos(atan(Float64(eh * Float64(tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	tmp = 0.0;
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 8.5e+146)
		tmp = abs(((-ew * cos(t)) / (-1.0 / cos(atan((eh * (tan(t) / ew)))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.55e+119], t$95$1, If[LessEqual[eh, 8.5e+146], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.54999999999999992e119 or 8.5e146 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.1%

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

   if -2.54999999999999992e119 < eh < 8.5e146

   1. Initial program 99.8%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in eh around 0

      \[\leadsto \left|\frac{\color{blue}{-1 \cdot \left(ew \cdot \cos t\right)}}{\frac{1}{\cos \tan^{-1} \left(\frac{\tan t}{ew} \cdot eh\right)}}\right| \]
   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 83.0

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

   7. Applied rewrites 83.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(eh \cdot \frac{\tan t}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(\frac{eh}{ew} \cdot t\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ t (- ew)) eh))) (* (- eh) (sin t))))))
   (if (<= eh -2.55e+119)
     t_1
     (if (<= eh 8.5e+146)
       (fabs (/ (* (- ew) (cos t)) (/ -1.0 (cos (atan (* (/ eh ew) t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 8.5e+146) {
		tmp = fabs(((-ew * cos(t)) / (-1.0 / cos(atan(((eh / ew) * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))))
    if (eh <= (-2.55d+119)) then
        tmp = t_1
    else if (eh <= 8.5d+146) then
        tmp = abs(((-ew * cos(t)) / ((-1.0d0) / cos(atan(((eh / ew) * t))))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((t / -ew) * eh))) * (-eh * Math.sin(t))));
	double tmp;
	if (eh <= -2.55e+119) {
		tmp = t_1;
	} else if (eh <= 8.5e+146) {
		tmp = Math.abs(((-ew * Math.cos(t)) / (-1.0 / Math.cos(Math.atan(((eh / ew) * t))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((t / -ew) * eh))) * (-eh * math.sin(t))))
	tmp = 0
	if eh <= -2.55e+119:
		tmp = t_1
	elif eh <= 8.5e+146:
		tmp = math.fabs(((-ew * math.cos(t)) / (-1.0 / math.cos(math.atan(((eh / ew) * t))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * Float64(Float64(-eh) * sin(t))))
	tmp = 0.0
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 8.5e+146)
		tmp = abs(Float64(Float64(Float64(-ew) * cos(t)) / Float64(-1.0 / cos(atan(Float64(Float64(eh / ew) * t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	tmp = 0.0;
	if (eh <= -2.55e+119)
		tmp = t_1;
	elseif (eh <= 8.5e+146)
		tmp = abs(((-ew * cos(t)) / (-1.0 / cos(atan(((eh / ew) * t))))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -2.55e+119], t$95$1, If[LessEqual[eh, 8.5e+146], N[Abs[N[(N[((-ew) * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.54999999999999992e119 or 8.5e146 < eh

   1. Initial program 99.8%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 75.1%

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

   if -2.54999999999999992e119 < eh < 8.5e146

   1. Initial program 99.8%

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

   4. Applied rewrites 83.9%

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

   5. Taylor expanded in t around 0

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 72.8

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

   7. Applied rewrites 72.8%

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

   8. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-cos.f64 72.3

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

   10. Applied rewrites 72.3%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;eh \leq 8.5 \cdot 10^{+146}:\\ \;\;\;\;\left|\frac{\left(-ew\right) \cdot \cos t}{\frac{-1}{\cos \tan^{-1} \left(\frac{eh}{ew} \cdot t\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{if}\;t \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin (atan (* (/ t (- ew)) eh))) (* (- eh) (sin t))))))
   (if (<= t -8.8e-10) t_1 (if (<= t 2.5e-18) (fabs (/ ew 1.0)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	double tmp;
	if (t <= -8.8e-10) {
		tmp = t_1;
	} else if (t <= 2.5e-18) {
		tmp = fabs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))))
    if (t <= (-8.8d-10)) then
        tmp = t_1
    else if (t <= 2.5d-18) then
        tmp = abs((ew / 1.0d0))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(Math.atan(((t / -ew) * eh))) * (-eh * Math.sin(t))));
	double tmp;
	if (t <= -8.8e-10) {
		tmp = t_1;
	} else if (t <= 2.5e-18) {
		tmp = Math.abs((ew / 1.0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(math.atan(((t / -ew) * eh))) * (-eh * math.sin(t))))
	tmp = 0
	if t <= -8.8e-10:
		tmp = t_1
	elif t <= 2.5e-18:
		tmp = math.fabs((ew / 1.0))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(atan(Float64(Float64(t / Float64(-ew)) * eh))) * Float64(Float64(-eh) * sin(t))))
	tmp = 0.0
	if (t <= -8.8e-10)
		tmp = t_1;
	elseif (t <= 2.5e-18)
		tmp = abs(Float64(ew / 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(atan(((t / -ew) * eh))) * (-eh * sin(t))));
	tmp = 0.0;
	if (t <= -8.8e-10)
		tmp = t_1;
	elseif (t <= 2.5e-18)
		tmp = abs((ew / 1.0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t / (-ew)), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[((-eh) * N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -8.8e-10], t$95$1, If[LessEqual[t, 2.5e-18], N[Abs[N[(ew / 1.0), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -8.7999999999999996e-10 or 2.50000000000000018e-18 < t

   1. Initial program 99.6%

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

   3. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. neg-mul-1 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-atan.f64 N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-frac2 N/A

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

      13. *-commutative N/A

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

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

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

      15. mul-1-neg N/A

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

      16. times-frac N/A

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

      17. lower-*.f64 N/A

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

   5. Applied rewrites 46.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.1%

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

   if -8.7999999999999996e-10 < t < 2.50000000000000018e-18

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.8%

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

   10. Applied rewrites 79.6%

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

   11. Taylor expanded in eh around 0

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

   13. Applied rewrites 79.9%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{-10}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{-18}:\\ \;\;\;\;\left|\frac{ew}{1}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t}{-ew} \cdot eh\right) \cdot \left(\left(-eh\right) \cdot \sin t\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.2% 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 47.2%

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

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 46.1%

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

10. Applied rewrites 45.3%

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

11. Taylor expanded in eh around 0

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

13. Applied rewrites 47.4%

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

Reproduce

herbie shell --seed 2024331 
(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)))))))