Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.7s

Alternatives: 20

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh / ew) / tan(t)));
	return fabs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(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 / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. cos-atan N/A

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

   4. lower-/.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-+.f64 N/A

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 25.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ t_2 := \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\\ \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin t\_2 + \left(ew \cdot \sin t\right) \cdot \cos t\_2\right| \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{t\_1}{\sqrt{{t\_1}^{2} + 1}} \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)) (t_2 (atan (/ (/ eh ew) (tan t)))))
   (if (<=
        (fabs (+ (* (* eh (cos t)) (sin t_2)) (* (* ew (sin t)) (cos t_2))))
        2e-83)
     (fabs (* (/ t_1 (sqrt (+ (pow t_1 2.0) 1.0))) eh))
     (fabs (* (sin (atan (* (* -0.3333333333333333 (/ eh ew)) t))) eh)))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = atan(((eh / ew) / tan(t)));
	double tmp;
	if (fabs((((eh * cos(t)) * sin(t_2)) + ((ew * sin(t)) * cos(t_2)))) <= 2e-83) {
		tmp = fabs(((t_1 / sqrt((pow(t_1, 2.0) + 1.0))) * eh));
	} else {
		tmp = fabs((sin(atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
	}
	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 = (eh / ew) / t
    t_2 = atan(((eh / ew) / tan(t)))
    if (abs((((eh * cos(t)) * sin(t_2)) + ((ew * sin(t)) * cos(t_2)))) <= 2d-83) then
        tmp = abs(((t_1 / sqrt(((t_1 ** 2.0d0) + 1.0d0))) * eh))
    else
        tmp = abs((sin(atan((((-0.3333333333333333d0) * (eh / ew)) * t))) * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	double t_2 = Math.atan(((eh / ew) / Math.tan(t)));
	double tmp;
	if (Math.abs((((eh * Math.cos(t)) * Math.sin(t_2)) + ((ew * Math.sin(t)) * Math.cos(t_2)))) <= 2e-83) {
		tmp = Math.abs(((t_1 / Math.sqrt((Math.pow(t_1, 2.0) + 1.0))) * eh));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	t_2 = math.atan(((eh / ew) / math.tan(t)))
	tmp = 0
	if math.fabs((((eh * math.cos(t)) * math.sin(t_2)) + ((ew * math.sin(t)) * math.cos(t_2)))) <= 2e-83:
		tmp = math.fabs(((t_1 / math.sqrt((math.pow(t_1, 2.0) + 1.0))) * eh))
	else:
		tmp = math.fabs((math.sin(math.atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	t_2 = atan(Float64(Float64(eh / ew) / tan(t)))
	tmp = 0.0
	if (abs(Float64(Float64(Float64(eh * cos(t)) * sin(t_2)) + Float64(Float64(ew * sin(t)) * cos(t_2)))) <= 2e-83)
		tmp = abs(Float64(Float64(t_1 / sqrt(Float64((t_1 ^ 2.0) + 1.0))) * eh));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	t_2 = atan(((eh / ew) / tan(t)));
	tmp = 0.0;
	if (abs((((eh * cos(t)) * sin(t_2)) + ((ew * sin(t)) * cos(t_2)))) <= 2e-83)
		tmp = abs(((t_1 / sqrt(((t_1 ^ 2.0) + 1.0))) * eh));
	else
		tmp = abs((sin(atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-83], N[Abs[N[(N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))))) < 2.0000000000000001e-83

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 56.8%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 55.4%

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

   13. Applied rewrites 34.6%

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

   if 2.0000000000000001e-83 < (fabs.f64 (+.f64 (*.f64 (*.f64 ew (sin.f64 t)) (cos.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t))))) (*.f64 (*.f64 eh (cos.f64 t)) (sin.f64 (atan.f64 (/.f64 (/.f64 eh ew) (tan.f64 t)))))))

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 40.1

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

   5. Applied rewrites 40.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 30.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 38.7%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 23.5%

       \[\leadsto \left|\sin \tan^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \leq 2 \cdot 10^{-83}:\\ \;\;\;\;\left|\frac{\frac{\frac{eh}{ew}}{t}}{\sqrt{{\left(\frac{\frac{eh}{ew}}{t}\right)}^{2} + 1}} \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((sin(t) * ew) / sqrt((pow((eh / (ew * t)), 2.0) + 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((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((sin(t) * ew) / sqrt((((eh / (ew * t)) ** 2.0d0) + 1.0d0)))))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((eh * Math.cos(t)) * Math.sin(Math.atan(((eh / ew) / Math.tan(t))))) + ((Math.sin(t) * ew) / Math.sqrt((Math.pow((eh / (ew * t)), 2.0) + 1.0)))));
}

Python

def code(eh, ew, t):
	return math.fabs((((eh * math.cos(t)) * math.sin(math.atan(((eh / ew) / math.tan(t))))) + ((math.sin(t) * ew) / math.sqrt((math.pow((eh / (ew * t)), 2.0) + 1.0)))))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((sin(t) * ew) / sqrt((((eh / (ew * t)) ^ 2.0) + 1.0)))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. cos-atan N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

7. Applied rewrites 98.9%

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

8. Final simplification 98.9%

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

Alternative 4: 75.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{+221}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -7700:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+70}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (/ (* (sin t) ew) (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))))
   (if (<= t -2.7e+221)
     t_1
     (if (<= t -7700.0)
       (fabs
        (*
         (sin
          (atan
           (*
            (fma
             (fma
              (/ (* t t) ew)
              (fma -0.001388888888888889 (* t t) 0.041666666666666664)
              (/ -0.5 ew))
             (* t t)
             (pow ew -1.0))
            (/ eh (sin t)))))
         (* (cos t) eh)))
       (if (<= t 8e+70)
         (fabs
          (+
           (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t)))))
           (* (* ew t) (cos (atan (/ eh (* t ew)))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	double tmp;
	if (t <= -2.7e+221) {
		tmp = t_1;
	} else if (t <= -7700.0) {
		tmp = fabs((sin(atan((fma(fma(((t * t) / ew), fma(-0.001388888888888889, (t * t), 0.041666666666666664), (-0.5 / ew)), (t * t), pow(ew, -1.0)) * (eh / sin(t))))) * (cos(t) * eh)));
	} else if (t <= 8e+70) {
		tmp = fabs((((eh * cos(t)) * sin(atan(((eh / ew) / tan(t))))) + ((ew * t) * cos(atan((eh / (t * ew)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(Float64(sin(t) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))))
	tmp = 0.0
	if (t <= -2.7e+221)
		tmp = t_1;
	elseif (t <= -7700.0)
		tmp = abs(Float64(sin(atan(Float64(fma(fma(Float64(Float64(t * t) / ew), fma(-0.001388888888888889, Float64(t * t), 0.041666666666666664), Float64(-0.5 / ew)), Float64(t * t), (ew ^ -1.0)) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	elseif (t <= 8e+70)
		tmp = abs(Float64(Float64(Float64(eh * cos(t)) * sin(atan(Float64(Float64(eh / ew) / tan(t))))) + Float64(Float64(ew * t) * cos(atan(Float64(eh / Float64(t * ew)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2.7e+221], t$95$1, If[LessEqual[t, -7700.0], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * N[(-0.001388888888888889 * N[(t * t), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + N[(-0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 8e+70], N[Abs[N[(N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(ew * t), $MachinePrecision] * N[Cos[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.7e221 or 8.00000000000000058e70 < t

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 79.0%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 62.9

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

   9. Applied rewrites 62.9%

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

   if -2.7e221 < t < -7700

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 66.0

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

   7. Applied rewrites 66.0%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-1}{720} \cdot \frac{{t}^{2}}{ew} + \frac{1}{24} \cdot \frac{1}{ew}\right) - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 66.4%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if -7700 < t < 8.00000000000000058e70

   1. Initial program 99.9%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in t around 0

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

      1. lower-*.f64 95.7

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

   8. Applied rewrites 95.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{+221}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{elif}\;t \leq -7700:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+70}:\\ \;\;\;\;\left|\left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(ew \cdot t\right) \cdot \cos \tan^{-1} \left(\frac{eh}{t \cdot ew}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left({ew}^{-1} \cdot \mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.8e+59) (not (<= ew 5.8e+20)))
   (fabs (/ (* (sin t) ew) (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))
   (fabs
    (*
     (sin
      (atan
       (*
        (fma
         (* (pow ew -1.0) (fma 0.041666666666666664 (* t t) -0.5))
         (* t t)
         (pow ew -1.0))
        (/ eh (sin t)))))
     (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20)) {
		tmp = fabs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	} else {
		tmp = fabs((sin(atan((fma((pow(ew, -1.0) * fma(0.041666666666666664, (t * t), -0.5)), (t * t), pow(ew, -1.0)) * (eh / sin(t))))) * (cos(t) * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20))
		tmp = abs(Float64(Float64(sin(t) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))));
	else
		tmp = abs(Float64(sin(atan(Float64(fma(Float64((ew ^ -1.0) * fma(0.041666666666666664, Float64(t * t), -0.5)), Float64(t * t), (ew ^ -1.0)) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.8e+59], N[Not[LessEqual[ew, 5.8e+20]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[Power[ew, -1.0], $MachinePrecision] * N[(0.041666666666666664 * N[(t * t), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.80000000000000043e59 or 5.8e20 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -7.80000000000000043e59 < ew < 5.8e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.0%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\frac{1}{ew} \cdot \mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left({ew}^{-1} \cdot \mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.8e+59) (not (<= ew 5.8e+20)))
   (fabs (/ (* (sin t) ew) (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))
   (fabs
    (*
     (sin
      (atan
       (*
        (fma
         (fma
          (/ (* t t) ew)
          (fma -0.001388888888888889 (* t t) 0.041666666666666664)
          (/ -0.5 ew))
         (* t t)
         (pow ew -1.0))
        (/ eh (sin t)))))
     (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20)) {
		tmp = fabs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	} else {
		tmp = fabs((sin(atan((fma(fma(((t * t) / ew), fma(-0.001388888888888889, (t * t), 0.041666666666666664), (-0.5 / ew)), (t * t), pow(ew, -1.0)) * (eh / sin(t))))) * (cos(t) * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20))
		tmp = abs(Float64(Float64(sin(t) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))));
	else
		tmp = abs(Float64(sin(atan(Float64(fma(fma(Float64(Float64(t * t) / ew), fma(-0.001388888888888889, Float64(t * t), 0.041666666666666664), Float64(-0.5 / ew)), Float64(t * t), (ew ^ -1.0)) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.8e+59], N[Not[LessEqual[ew, 5.8e+20]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * N[(-0.001388888888888889 * N[(t * t), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + N[(-0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.80000000000000043e59 or 5.8e20 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -7.80000000000000043e59 < ew < 5.8e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-1}{720} \cdot \frac{{t}^{2}}{ew} + \frac{1}{24} \cdot \frac{1}{ew}\right) - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 86.0%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sin t \cdot ew\\ t_2 := \frac{eh}{\sin t}\\ \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{t\_2}{ew} \cdot \cos t\right) \cdot t\_1\right|\\ \mathbf{elif}\;ew \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot t\_2\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{t\_1}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (sin t) ew)) (t_2 (/ eh (sin t))))
   (if (<= ew -7.8e+59)
     (fabs (* (cos (atan (* (/ t_2 ew) (cos t)))) t_1))
     (if (<= ew 5.8e+20)
       (fabs
        (*
         (sin
          (atan
           (*
            (fma
             (fma
              (/ (* t t) ew)
              (fma -0.001388888888888889 (* t t) 0.041666666666666664)
              (/ -0.5 ew))
             (* t t)
             (pow ew -1.0))
            t_2)))
         (* (cos t) eh)))
       (fabs (/ t_1 (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = sin(t) * ew;
	double t_2 = eh / sin(t);
	double tmp;
	if (ew <= -7.8e+59) {
		tmp = fabs((cos(atan(((t_2 / ew) * cos(t)))) * t_1));
	} else if (ew <= 5.8e+20) {
		tmp = fabs((sin(atan((fma(fma(((t * t) / ew), fma(-0.001388888888888889, (t * t), 0.041666666666666664), (-0.5 / ew)), (t * t), pow(ew, -1.0)) * t_2))) * (cos(t) * eh)));
	} else {
		tmp = fabs((t_1 / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(sin(t) * ew)
	t_2 = Float64(eh / sin(t))
	tmp = 0.0
	if (ew <= -7.8e+59)
		tmp = abs(Float64(cos(atan(Float64(Float64(t_2 / ew) * cos(t)))) * t_1));
	elseif (ew <= 5.8e+20)
		tmp = abs(Float64(sin(atan(Float64(fma(fma(Float64(Float64(t * t) / ew), fma(-0.001388888888888889, Float64(t * t), 0.041666666666666664), Float64(-0.5 / ew)), Float64(t * t), (ew ^ -1.0)) * t_2))) * Float64(cos(t) * eh)));
	else
		tmp = abs(Float64(t_1 / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]}, Block[{t$95$2 = N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[ew, -7.8e+59], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 5.8e+20], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * N[(-0.001388888888888889 * N[(t * t), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] + N[(-0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -7.80000000000000043e59

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 68.0%

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

   if -7.80000000000000043e59 < ew < 5.8e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-1}{720} \cdot \frac{{t}^{2}}{ew} + \frac{1}{24} \cdot \frac{1}{ew}\right) - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]
   9. Step-by-step derivation

   10. Applied rewrites 86.0%

       \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right| \]

   if 5.8e20 < ew

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 82.7%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 74.1

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

   9. Applied rewrites 74.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 5.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{t \cdot t}{ew}, \mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{t \cdot t}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.8e+59) (not (<= ew 5.8e+20)))
   (fabs (/ (* (sin t) ew) (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))
   (fabs
    (*
     (sin (atan (* (fma -0.5 (/ (* t t) ew) (pow ew -1.0)) (/ eh (sin t)))))
     (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20)) {
		tmp = fabs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	} else {
		tmp = fabs((sin(atan((fma(-0.5, ((t * t) / ew), pow(ew, -1.0)) * (eh / sin(t))))) * (cos(t) * eh)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20))
		tmp = abs(Float64(Float64(sin(t) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))));
	else
		tmp = abs(Float64(sin(atan(Float64(fma(-0.5, Float64(Float64(t * t) / ew), (ew ^ -1.0)) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.8e+59], N[Not[LessEqual[ew, 5.8e+20]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(-0.5 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.80000000000000043e59 or 5.8e20 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -7.80000000000000043e59 < ew < 5.8e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 86.0%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\mathsf{fma}\left(-0.5, \frac{t \cdot t}{ew}, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left({ew}^{-1} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -7.8e+59) (not (<= ew 5.8e+20)))
   (fabs (/ (* (sin t) ew) (/ -1.0 (cos (atan (/ (/ eh (tan t)) ew))))))
   (fabs (* (sin (atan (* (pow ew -1.0) (/ eh (sin t))))) (* (cos t) eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20)) {
		tmp = fabs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	} else {
		tmp = fabs((sin(atan((pow(ew, -1.0) * (eh / sin(t))))) * (cos(t) * eh)));
	}
	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 <= (-7.8d+59)) .or. (.not. (ew <= 5.8d+20))) then
        tmp = abs(((sin(t) * ew) / ((-1.0d0) / cos(atan(((eh / tan(t)) / ew))))))
    else
        tmp = abs((sin(atan(((ew ** (-1.0d0)) * (eh / sin(t))))) * (cos(t) * eh)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20)) {
		tmp = Math.abs(((Math.sin(t) * ew) / (-1.0 / Math.cos(Math.atan(((eh / Math.tan(t)) / ew))))));
	} else {
		tmp = Math.abs((Math.sin(Math.atan((Math.pow(ew, -1.0) * (eh / Math.sin(t))))) * (Math.cos(t) * eh)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -7.8e+59) or not (ew <= 5.8e+20):
		tmp = math.fabs(((math.sin(t) * ew) / (-1.0 / math.cos(math.atan(((eh / math.tan(t)) / ew))))))
	else:
		tmp = math.fabs((math.sin(math.atan((math.pow(ew, -1.0) * (eh / math.sin(t))))) * (math.cos(t) * eh)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -7.8e+59) || !(ew <= 5.8e+20))
		tmp = abs(Float64(Float64(sin(t) * ew) / Float64(-1.0 / cos(atan(Float64(Float64(eh / tan(t)) / ew))))));
	else
		tmp = abs(Float64(sin(atan(Float64((ew ^ -1.0) * Float64(eh / sin(t))))) * Float64(cos(t) * eh)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -7.8e+59) || ~((ew <= 5.8e+20)))
		tmp = abs(((sin(t) * ew) / (-1.0 / cos(atan(((eh / tan(t)) / ew))))));
	else
		tmp = abs((sin(atan(((ew ^ -1.0) * (eh / sin(t))))) * (cos(t) * eh)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -7.8e+59], N[Not[LessEqual[ew, 5.8e+20]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[(-1.0 / N[Cos[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[Power[ew, -1.0], $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.80000000000000043e59 or 5.8e20 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 78.8%

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

   7. Taylor expanded in eh around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -7.80000000000000043e59 < ew < 5.8e20

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in eh around inf

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 85.9

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

   7. Applied rewrites 85.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 85.9%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -7.8 \cdot 10^{+59} \lor \neg \left(ew \leq 5.8 \cdot 10^{+20}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\frac{-1}{\cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right)}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left({ew}^{-1} \cdot \frac{eh}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

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(((ew ** (-1.0d0)) * (eh / sin(t))))) * (cos(t) * eh)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.sin(Math.atan((Math.pow(ew, -1.0) * (eh / Math.sin(t))))) * (Math.cos(t) * eh)));
}

Python

def code(eh, ew, t):
	return math.fabs((math.sin(math.atan((math.pow(ew, -1.0) * (eh / math.sin(t))))) * (math.cos(t) * eh)))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((sin(atan(((ew ^ -1.0) * (eh / sin(t))))) * (cos(t) * eh)));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-atan.f64 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 62.7

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

7. Applied rewrites 62.7%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 62.7%

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

11. Final simplification 62.7%

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

Alternative 11: 42.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) \cdot t, t, \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (fma
       (fma
        (*
         (fma -0.001388888888888889 (/ (* t t) ew) (/ 0.041666666666666664 ew))
         t)
        t
        (/ -0.5 ew))
       (* t t)
       (pow ew -1.0))
      (/ eh (sin t)))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((fma(fma((fma(-0.001388888888888889, ((t * t) / ew), (0.041666666666666664 / ew)) * t), t, (-0.5 / ew)), (t * t), pow(ew, -1.0)) * (eh / sin(t))))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(fma(fma(Float64(fma(-0.001388888888888889, Float64(Float64(t * t) / ew), Float64(0.041666666666666664 / ew)) * t), t, Float64(-0.5 / ew)), Float64(t * t), (ew ^ -1.0)) * Float64(eh / sin(t))))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(-0.001388888888888889 * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[(0.041666666666666664 / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(-0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in eh around 0

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

8. Applied rewrites 44.0%

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

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\left({t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{-1}{720} \cdot \frac{{t}^{2}}{ew} + \frac{1}{24} \cdot \frac{1}{ew}\right) - \frac{1}{2} \cdot \frac{1}{ew}\right) + \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 44.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) \cdot t, t, \frac{-0.5}{ew}\right), t \cdot t, \frac{1}{ew}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]

12. Final simplification 44.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, \frac{t \cdot t}{ew}, \frac{0.041666666666666664}{ew}\right) \cdot t, t, \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 12: 42.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664}{ew} \cdot t, t, \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (fma
       (fma (* (/ 0.041666666666666664 ew) t) t (/ -0.5 ew))
       (* t t)
       (pow ew -1.0))
      (/ eh (sin t)))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((fma(fma(((0.041666666666666664 / ew) * t), t, (-0.5 / ew)), (t * t), pow(ew, -1.0)) * (eh / sin(t))))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(fma(fma(Float64(Float64(0.041666666666666664 / ew) * t), t, Float64(-0.5 / ew)), Float64(t * t), (ew ^ -1.0)) * Float64(eh / sin(t))))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(0.041666666666666664 / ew), $MachinePrecision] * t), $MachinePrecision] * t + N[(-0.5 / ew), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in eh around 0

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

8. Applied rewrites 44.0%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 44.2%

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

12. Final simplification 44.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{0.041666666666666664}{ew} \cdot t, t, \frac{-0.5}{ew}\right), t \cdot t, {ew}^{-1}\right) \cdot \frac{eh}{\sin t}\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 13: 61.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs(((cos(t) * eh) * sin(atan(((eh / tan(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(((cos(t) * eh) * sin(atan(((eh / tan(t)) / ew)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eh around inf

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-atan.f64 N/A

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

   6. *-commutative N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 62.7

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

7. Applied rewrites 62.7%

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

9. Applied rewrites 62.7%

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

Alternative 14: 42.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot -0.00205026455026455\right) \cdot t, -t, 0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (fma
           (* (* eh -0.00205026455026455) t)
           (- t)
           (* 0.019444444444444445 eh))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(fma(((eh * -0.00205026455026455) * t), -t, (0.019444444444444445 * eh)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(eh * -0.00205026455026455) * t), Float64(-t), Float64(0.019444444444444445 * eh)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[(N[(eh * -0.00205026455026455), $MachinePrecision] * t), $MachinePrecision] * (-t) + N[(0.019444444444444445 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left({t}^{2} \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{-1}{5040} \cdot eh + \left(\frac{1}{720} \cdot eh + \frac{1}{6} \cdot \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right)\right)\right) - \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right) - \frac{-1}{6} \cdot eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

8. Applied rewrites 44.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot -0.00205026455026455\right) \cdot t, -t, 0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Final simplification 44.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(eh \cdot -0.00205026455026455\right) \cdot t, -t, 0.019444444444444445 \cdot eh\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
10. Add Preprocessing

Alternative 15: 42.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(eh \cdot \mathsf{fma}\left(-0.019444444444444445, t \cdot t, -0.16666666666666666\right)\right) \cdot \left(-t\right), t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (*
          (* eh (fma -0.019444444444444445 (* t t) -0.16666666666666666))
          (- t))
         t
         eh)
        t)
       ew)
      (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(((eh * fma(-0.019444444444444445, (t * t), -0.16666666666666666)) * -t), t, eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(Float64(Float64(eh * fma(-0.019444444444444445, Float64(t * t), -0.16666666666666666)) * Float64(-t)), t, eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(eh * N[(-0.019444444444444445 * N[(t * t), $MachinePrecision] + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * (-t)), $MachinePrecision] * t + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left(-1 \cdot \left({t}^{2} \cdot \left(\frac{-1}{36} \cdot eh + \frac{1}{120} \cdot eh\right)\right) - \frac{-1}{6} \cdot eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
7. Step-by-step derivation

8. Applied rewrites 44.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-\left(eh \cdot \mathsf{fma}\left(-0.019444444444444445, t \cdot t, -0.16666666666666666\right)\right) \cdot t, t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Final simplification 44.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(eh \cdot \mathsf{fma}\left(-0.019444444444444445, t \cdot t, -0.16666666666666666\right)\right) \cdot \left(-t\right), t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
10. Add Preprocessing

Alternative 16: 42.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(0.16666666666666666, \left(t \cdot t\right) \cdot eh, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (* (/ (/ (fma 0.16666666666666666 (* (* t t) eh) eh) t) ew) (cos t))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(0.16666666666666666, ((t * t) * eh), eh) / t) / ew) * cos(t)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(0.16666666666666666, Float64(Float64(t * t) * eh), eh) / t) / ew) * cos(t)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(0.16666666666666666 * N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 44.2%

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

9. Final simplification 44.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(0.16666666666666666, \left(t \cdot t\right) \cdot eh, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]
10. Add Preprocessing

Alternative 17: 42.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan (/ (* (fma (/ (* t t) ew) -0.3333333333333333 (pow ew -1.0)) eh) t)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(((t * t) / ew), -0.3333333333333333, pow(ew, -1.0)) * eh) / t))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(Float64(Float64(t * t) / ew), -0.3333333333333333, (ew ^ -1.0)) * eh) / t))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333 + N[Power[ew, -1.0], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 36.6%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 44.2%

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

12. Final simplification 44.2%

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

Alternative 18: 42.2% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin (atan (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) ew) t)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(((t * t) * eh), -0.3333333333333333, eh) / ew) / t))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / ew) / t))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 36.6%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.6%

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

12. Taylor expanded in ew around 0

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

14. Applied rewrites 44.2%

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

15. Final simplification 44.2%

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

Alternative 19: 40.3% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

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 / ew) / t))) * eh))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 36.6%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.6%

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

12. Final simplification 42.6%

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

Alternative 20: 14.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{ew}}{t}\\ \left|\frac{t\_1}{\sqrt{{t\_1}^{2} + 1}} \cdot eh\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh ew) t)))
   (fabs (* (/ t_1 (sqrt (+ (pow t_1 2.0) 1.0))) eh))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return fabs(((t_1 / sqrt((pow(t_1, 2.0) + 1.0))) * eh));
}

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 = (eh / ew) / t
    code = abs(((t_1 / sqrt(((t_1 ** 2.0d0) + 1.0d0))) * eh))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / ew) / t;
	return Math.abs(((t_1 / Math.sqrt((Math.pow(t_1, 2.0) + 1.0))) * eh));
}

Python

def code(eh, ew, t):
	t_1 = (eh / ew) / t
	return math.fabs(((t_1 / math.sqrt((math.pow(t_1, 2.0) + 1.0))) * eh))

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / ew) / t)
	return abs(Float64(Float64(t_1 / sqrt(Float64((t_1 ^ 2.0) + 1.0))) * eh))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = (eh / ew) / t;
	tmp = abs(((t_1 / sqrt(((t_1 ^ 2.0) + 1.0))) * eh));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]}, N[Abs[N[(N[(t$95$1 / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. lower-atan.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

   11. lower-/.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-cos.f64 44.0

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

5. Applied rewrites 44.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 36.6%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 42.6%

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

13. Applied rewrites 15.3%

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

14. Final simplification 15.3%

    \[\leadsto \left|\frac{\frac{\frac{eh}{ew}}{t}}{\sqrt{{\left(\frac{\frac{eh}{ew}}{t}\right)}^{2} + 1}} \cdot eh\right| \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024318 
(FPCore (eh ew t)
  :name "Example from Robby"
  :precision binary64
  (fabs (+ (* (* ew (sin t)) (cos (atan (/ (/ eh ew) (tan t))))) (* (* eh (cos t)) (sin (atan (/ (/ eh ew) (tan t))))))))