Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 17.8s

Alternatives: 19

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

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} t\_1, {\left(\sqrt{{t\_1}^{2} + 1}\right)}^{-1} \cdot \left(\sin t \cdot ew\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (fabs
    (fma
     (* (cos t) eh)
     (sin (atan t_1))
     (* (pow (sqrt (+ (pow t_1 2.0) 1.0)) -1.0) (* (sin t) ew))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	return fabs(fma((cos(t) * eh), sin(atan(t_1)), (pow(sqrt((pow(t_1, 2.0) + 1.0)), -1.0) * (sin(t) * ew))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	return abs(fma(Float64(cos(t) * eh), sin(atan(t_1)), Float64((sqrt(Float64((t_1 ^ 2.0) + 1.0)) ^ -1.0) * Float64(sin(t) * ew))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] + N[(N[Power[N[Sqrt[N[(N[Power[t$95$1, 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. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

   14. lift-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. cos-atan N/A

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

   9. lower-/.f64 N/A

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

   10. lift-/.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-/l/ N/A

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

   13. lift-*.f64 N/A

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

   14. lift-/.f64 N/A

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

   15. lift-/.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/l/ N/A

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

   18. lift-*.f64 N/A

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

   19. lift-/.f64 N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ \mathbf{if}\;eh \leq -4.9 \cdot 10^{+117} \lor \neg \left(eh \leq 10^{+120}\right):\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(t\_1, \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(eh \cdot \frac{t \cdot t}{ew}, -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right), {\left(\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}\right)}^{-1} \cdot \left(\sin t \cdot ew\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh)))
   (if (or (<= eh -4.9e+117) (not (<= eh 1e+120)))
     (fabs
      (* t_1 (sin (atan (/ (/ (fma (* -0.5 eh) (* t t) eh) ew) (sin t))))))
     (fabs
      (fma
       t_1
       (sin
        (atan (/ (fma (* eh (/ (* t t) ew)) -0.3333333333333333 (/ eh ew)) t)))
       (*
        (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) {
	double t_1 = cos(t) * eh;
	double tmp;
	if ((eh <= -4.9e+117) || !(eh <= 1e+120)) {
		tmp = fabs((t_1 * sin(atan(((fma((-0.5 * eh), (t * t), eh) / ew) / sin(t))))));
	} else {
		tmp = fabs(fma(t_1, sin(atan((fma((eh * ((t * t) / ew)), -0.3333333333333333, (eh / ew)) / t))), (pow(sqrt((pow(((eh / tan(t)) / ew), 2.0) + 1.0)), -1.0) * (sin(t) * ew))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	tmp = 0.0
	if ((eh <= -4.9e+117) || !(eh <= 1e+120))
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(fma(Float64(-0.5 * eh), Float64(t * t), eh) / ew) / sin(t))))));
	else
		tmp = abs(fma(t_1, sin(atan(Float64(fma(Float64(eh * Float64(Float64(t * t) / ew)), -0.3333333333333333, Float64(eh / ew)) / t))), Float64((sqrt(Float64((Float64(Float64(eh / tan(t)) / ew) ^ 2.0) + 1.0)) ^ -1.0) * Float64(sin(t) * ew))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[Or[LessEqual[eh, -4.9e+117], N[Not[LessEqual[eh, 1e+120]], $MachinePrecision]], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(N[(N[(-0.5 * eh), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(N[(eh * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 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. Split input into 2 regimes

2. if eh < -4.9000000000000001e117 or 9.9999999999999998e119 < eh

   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. 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. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 96.2

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

   7. Applied rewrites 96.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 96.2%

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

   if -4.9000000000000001e117 < eh < 9.9999999999999998e119

   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. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.8

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. cos-atan N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/l/ N/A

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

      18. lift-*.f64 N/A

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

      19. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in t around 0

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \color{blue}{\left(\frac{\frac{-1}{3} \cdot \frac{eh \cdot {t}^{2}}{ew} + \frac{eh}{ew}}{t}\right)}, \frac{1}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 N/A

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

       9. lower-/.f64 98.0

          \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(eh \cdot \frac{t \cdot t}{ew}, -0.3333333333333333, \color{blue}{\frac{eh}{ew}}\right)}{t}\right), \frac{1}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]

   11. Applied rewrites 98.0%

       \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \color{blue}{\left(\frac{\mathsf{fma}\left(eh \cdot \frac{t \cdot t}{ew}, -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right)}, \frac{1}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -4.9 \cdot 10^{+117} \lor \neg \left(eh \leq 10^{+120}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\mathsf{fma}\left(eh \cdot \frac{t \cdot t}{ew}, -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right), {\left(\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}\right)}^{-1} \cdot \left(\sin t \cdot ew\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(eh / N[(ew * N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $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. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

   14. lift-*.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in t around 0

   \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\color{blue}{\frac{eh + \frac{-1}{3} \cdot \left(eh \cdot {t}^{2}\right)}{t}}}{ew}\right), \cos \tan^{-1} \left(\frac{eh}{ew \cdot \tan t}\right) \cdot \left(\sin t \cdot ew\right)\right)\right| \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	return abs(fma(Float64(cos(t) * eh), sin(atan(Float64(Float64(eh / tan(t)) / ew))), Float64(cos(atan(Float64(Float64(eh / t) / ew))) * Float64(sin(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] + N[(N[Cos[N[ArcTan[N[(N[(eh / t), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $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. lower-fma.f64 99.8

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 99.8

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

   8. lift-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-/l/ N/A

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

   11. associate-/r* N/A

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

   12. lower-/.f64 N/A

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

   13. lower-/.f64 99.8

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

   14. lift-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 5: 93.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+60} \lor \neg \left(eh \leq 5.4 \cdot 10^{+35}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \frac{\sin \tan^{-1} \left(\frac{eh}{\sin t} \cdot \frac{\cos t}{ew}\right) \cdot \cos t}{ew}, \sin t\right) \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -2.6e+60) (not (<= eh 5.4e+35)))
   (fabs
    (*
     (* (cos t) eh)
     (sin (atan (/ (/ (fma (* -0.5 eh) (* t t) eh) ew) (sin t))))))
   (fabs
    (*
     (fma
      eh
      (/ (* (sin (atan (* (/ eh (sin t)) (/ (cos t) ew)))) (cos t)) ew)
      (sin t))
     ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -2.6e+60) || !(eh <= 5.4e+35)) {
		tmp = fabs(((cos(t) * eh) * sin(atan(((fma((-0.5 * eh), (t * t), eh) / ew) / sin(t))))));
	} else {
		tmp = fabs((fma(eh, ((sin(atan(((eh / sin(t)) * (cos(t) / ew)))) * cos(t)) / ew), sin(t)) * ew));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -2.6e+60) || !(eh <= 5.4e+35))
		tmp = abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(Float64(fma(Float64(-0.5 * eh), Float64(t * t), eh) / ew) / sin(t))))));
	else
		tmp = abs(Float64(fma(eh, Float64(Float64(sin(atan(Float64(Float64(eh / sin(t)) * Float64(cos(t) / ew)))) * cos(t)) / ew), sin(t)) * ew));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -2.6e+60], N[Not[LessEqual[eh, 5.4e+35]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(-0.5 * eh), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(eh * N[(N[(N[Sin[N[ArcTan[N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] + N[Sin[t], $MachinePrecision]), $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -2.60000000000000008e60 or 5.40000000000000005e35 < eh

   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. 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. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 92.8

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

   7. Applied rewrites 92.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 92.9%

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

   if -2.60000000000000008e60 < eh < 5.40000000000000005e35

   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. lower-fma.f64 99.8

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.8

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.8

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.8%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-cos.f64 N/A

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

      2. lift-atan.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l/ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. cos-atan N/A

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

      9. lower-/.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-/l/ N/A

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

      13. lift-*.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. lift-/.f64 N/A

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

      17. associate-/l/ N/A

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

      18. lift-*.f64 N/A

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

      19. lift-/.f64 N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in ew around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

   11. Applied rewrites 98.0%

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

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -2.6 \cdot 10^{+60} \lor \neg \left(eh \leq 5.4 \cdot 10^{+35}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\mathsf{fma}\left(eh, \frac{\sin \tan^{-1} \left(\frac{eh}{\sin t} \cdot \frac{\cos t}{ew}\right) \cdot \cos t}{ew}, \sin t\right) \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -5.6e-185) (not (<= eh 1.95e-100)))
   (fabs
    (*
     (* (cos t) eh)
     (sin (atan (/ (/ (fma (* -0.5 eh) (* t t) eh) ew) (sin t))))))
   (fabs (* (cos (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (sin t) ew)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100)) {
		tmp = fabs(((cos(t) * eh) * sin(atan(((fma((-0.5 * eh), (t * t), eh) / ew) / sin(t))))));
	} else {
		tmp = fabs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100))
		tmp = abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(Float64(fma(Float64(-0.5 * eh), Float64(t * t), eh) / ew) / sin(t))))));
	else
		tmp = abs(Float64(cos(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(sin(t) * ew)));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -5.6e-185], N[Not[LessEqual[eh, 1.95e-100]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(-0.5 * eh), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.59999999999999983e-185 or 1.94999999999999989e-100 < eh

   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. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 80.0

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 80.3%

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

   if -5.59999999999999983e-185 < eh < 1.94999999999999989e-100

   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. 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 79.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -5.6e-185) (not (<= eh 1.95e-100)))
   (fabs
    (*
     (* (cos t) eh)
     (sin (atan (/ (/ (fma (* -0.5 eh) (* t t) eh) ew) (sin t))))))
   (fabs (/ (* (sin t) ew) (sqrt (+ (pow (/ (/ eh (tan t)) ew) 2.0) 1.0))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100)) {
		tmp = fabs(((cos(t) * eh) * sin(atan(((fma((-0.5 * eh), (t * t), eh) / ew) / sin(t))))));
	} else {
		tmp = fabs(((sin(t) * ew) / sqrt((pow(((eh / tan(t)) / ew), 2.0) + 1.0))));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100))
		tmp = abs(Float64(Float64(cos(t) * eh) * sin(atan(Float64(Float64(fma(Float64(-0.5 * eh), Float64(t * t), eh) / ew) / sin(t))))));
	else
		tmp = abs(Float64(Float64(sin(t) * ew) / sqrt(Float64((Float64(Float64(eh / tan(t)) / ew) ^ 2.0) + 1.0))));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -5.6e-185], N[Not[LessEqual[eh, 1.95e-100]], $MachinePrecision]], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(N[(N[(-0.5 * eh), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.59999999999999983e-185 or 1.94999999999999989e-100 < eh

   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. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 80.0

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

   7. Applied rewrites 80.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 80.3%

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

   if -5.59999999999999983e-185 < eh < 1.94999999999999989e-100

   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. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 79.2

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 79.2%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(-0.5 \cdot eh, t \cdot t, eh\right)}{ew}}{\sin t}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 73.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\sin \tan^{-1} t\_1 \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{t\_1}^{2} + 1}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (if (or (<= eh -5.6e-185) (not (<= eh 1.95e-100)))
     (fabs (* (* (sin (atan t_1)) (cos t)) eh))
     (fabs (/ (* (sin t) ew) (sqrt (+ (pow t_1 2.0) 1.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	double tmp;
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100)) {
		tmp = fabs(((sin(atan(t_1)) * cos(t)) * eh));
	} else {
		tmp = fabs(((sin(t) * ew) / sqrt((pow(t_1, 2.0) + 1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (eh / tan(t)) / ew
    if ((eh <= (-5.6d-185)) .or. (.not. (eh <= 1.95d-100))) then
        tmp = abs(((sin(atan(t_1)) * cos(t)) * eh))
    else
        tmp = abs(((sin(t) * ew) / sqrt(((t_1 ** 2.0d0) + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = (eh / Math.tan(t)) / ew;
	double tmp;
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100)) {
		tmp = Math.abs(((Math.sin(Math.atan(t_1)) * Math.cos(t)) * eh));
	} else {
		tmp = Math.abs(((Math.sin(t) * ew) / Math.sqrt((Math.pow(t_1, 2.0) + 1.0))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = (eh / math.tan(t)) / ew
	tmp = 0
	if (eh <= -5.6e-185) or not (eh <= 1.95e-100):
		tmp = math.fabs(((math.sin(math.atan(t_1)) * math.cos(t)) * eh))
	else:
		tmp = math.fabs(((math.sin(t) * ew) / math.sqrt((math.pow(t_1, 2.0) + 1.0))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	tmp = 0.0
	if ((eh <= -5.6e-185) || !(eh <= 1.95e-100))
		tmp = abs(Float64(Float64(sin(atan(t_1)) * cos(t)) * eh));
	else
		tmp = abs(Float64(Float64(sin(t) * ew) / sqrt(Float64((t_1 ^ 2.0) + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = (eh / tan(t)) / ew;
	tmp = 0.0;
	if ((eh <= -5.6e-185) || ~((eh <= 1.95e-100)))
		tmp = abs(((sin(atan(t_1)) * cos(t)) * eh));
	else
		tmp = abs(((sin(t) * ew) / sqrt(((t_1 ^ 2.0) + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, If[Or[LessEqual[eh, -5.6e-185], N[Not[LessEqual[eh, 1.95e-100]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.59999999999999983e-185 or 1.94999999999999989e-100 < eh

   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. lower-fma.f64 99.9

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.9

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.9

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.9%

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

   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. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 80.0

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

   7. Applied rewrites 80.0%

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

   9. Applied rewrites 80.0%

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

   if -5.59999999999999983e-185 < eh < 1.94999999999999989e-100

   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. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 79.2

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

   7. Applied rewrites 79.2%

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

   9. Applied rewrites 79.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.6 \cdot 10^{-185} \lor \neg \left(eh \leq 1.95 \cdot 10^{-100}\right):\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -0.0042 \lor \neg \left(t \leq 3.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= t -0.0042) (not (<= t 3.8e-10)))
   (fabs (/ (* (sin t) ew) (sqrt (+ (pow (/ (/ eh (tan t)) ew) 2.0) 1.0))))
   (fabs
    (*
     (sin
      (atan
       (*
        (/
         (/
          (fma
           (fma
            (fma
             (* (- t) t)
             (fma
              -0.0001984126984126984
              eh
              (fma
               (* eh -0.019444444444444445)
               0.16666666666666666
               (* 0.001388888888888889 eh)))
             (* (- eh) -0.019444444444444445))
            (* t t)
            (* 0.16666666666666666 eh))
           (* t t)
           eh)
          t)
         ew)
        1.0)))
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((t <= -0.0042) || !(t <= 3.8e-10)) {
		tmp = fabs(((sin(t) * ew) / sqrt((pow(((eh / tan(t)) / ew), 2.0) + 1.0))));
	} else {
		tmp = fabs((sin(atan((((fma(fma(fma((-t * t), fma(-0.0001984126984126984, eh, fma((eh * -0.019444444444444445), 0.16666666666666666, (0.001388888888888889 * eh))), (-eh * -0.019444444444444445)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * 1.0))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((t <= -0.0042) || !(t <= 3.8e-10))
		tmp = abs(Float64(Float64(sin(t) * ew) / sqrt(Float64((Float64(Float64(eh / tan(t)) / ew) ^ 2.0) + 1.0))));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(-t) * t), fma(-0.0001984126984126984, eh, fma(Float64(eh * -0.019444444444444445), 0.16666666666666666, Float64(0.001388888888888889 * eh))), Float64(Float64(-eh) * -0.019444444444444445)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * 1.0))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[t, -0.0042], N[Not[LessEqual[t, 3.8e-10]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] / N[Sqrt[N[(N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[((-t) * t), $MachinePrecision] * N[(-0.0001984126984126984 * eh + N[(N[(eh * -0.019444444444444445), $MachinePrecision] * 0.16666666666666666 + N[(0.001388888888888889 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-eh) * -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * 1.0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -0.00419999999999999974 or 3.7999999999999998e-10 < t

   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. lower-fma.f64 99.7

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.7

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

      8. lift-/.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/l/ N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 99.7

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

      14. lift-*.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in eh around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-atan.f64 N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-sin.f64 50.6

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 50.2%

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

   if -0.00419999999999999974 < t < 3.7999999999999998e-10

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \sin t\right) \cdot \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) + \left(eh \cdot \cos t\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right)\right| \]
   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 82.4

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

   5. Applied rewrites 82.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 82.4%

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

   9. Taylor expanded in t around 0

      \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(\frac{-1}{5040}, eh, \mathsf{fma}\left(eh \cdot \frac{-7}{360}, \frac{1}{6}, \frac{1}{720} \cdot eh\right)\right), \left(-eh\right) \cdot \frac{-7}{360}\right), t \cdot t, \frac{1}{6} \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 82.4%

       \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -0.0042 \lor \neg \left(t \leq 3.8 \cdot 10^{-10}\right):\\ \;\;\;\;\left|\frac{\sin t \cdot ew}{\sqrt{{\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (fma
           (* (- t) t)
           (fma
            -0.0001984126984126984
            eh
            (fma
             (* eh -0.019444444444444445)
             0.16666666666666666
             (* 0.001388888888888889 eh)))
           (* (- eh) -0.019444444444444445))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (fma
       (fma
        (fma -0.001388888888888889 (* t t) 0.041666666666666664)
        (* t t)
        -0.5)
       (* t t)
       1.0))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(fma((-t * t), fma(-0.0001984126984126984, eh, fma((eh * -0.019444444444444445), 0.16666666666666666, (0.001388888888888889 * eh))), (-eh * -0.019444444444444445)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * fma(fma(fma(-0.001388888888888889, (t * t), 0.041666666666666664), (t * t), -0.5), (t * t), 1.0)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(-t) * t), fma(-0.0001984126984126984, eh, fma(Float64(eh * -0.019444444444444445), 0.16666666666666666, Float64(0.001388888888888889 * eh))), Float64(Float64(-eh) * -0.019444444444444445)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * fma(fma(fma(-0.001388888888888889, Float64(t * t), 0.041666666666666664), Float64(t * t), -0.5), Float64(t * t), 1.0)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[((-t) * t), $MachinePrecision] * N[(-0.0001984126984126984 * eh + N[(N[(eh * -0.019444444444444445), $MachinePrecision] * 0.16666666666666666 + N[(0.001388888888888889 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-eh) * -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[(N[(N[(-0.001388888888888889 * N[(t * t), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(t * t), $MachinePrecision] + -0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $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 45.9

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 46.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(\frac{-1}{5040}, eh, \mathsf{fma}\left(eh \cdot \frac{-7}{360}, \frac{1}{6}, \frac{1}{720} \cdot eh\right)\right), \left(-eh\right) \cdot \frac{-7}{360}\right), t \cdot t, \frac{1}{6} \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \left(1 + {t}^{2} \cdot \left({t}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {t}^{2}\right) - \frac{1}{2}\right)\right)\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \]

12. Final simplification 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 11: 41.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (fma
           (* (- t) t)
           (fma
            -0.0001984126984126984
            eh
            (fma
             (* eh -0.019444444444444445)
             0.16666666666666666
             (* 0.001388888888888889 eh)))
           (* (- eh) -0.019444444444444445))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (fma (fma 0.041666666666666664 (* t t) -0.5) (* t t) 1.0))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(fma((-t * t), fma(-0.0001984126984126984, eh, fma((eh * -0.019444444444444445), 0.16666666666666666, (0.001388888888888889 * eh))), (-eh * -0.019444444444444445)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * fma(fma(0.041666666666666664, (t * t), -0.5), (t * t), 1.0)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(-t) * t), fma(-0.0001984126984126984, eh, fma(Float64(eh * -0.019444444444444445), 0.16666666666666666, Float64(0.001388888888888889 * eh))), Float64(Float64(-eh) * -0.019444444444444445)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * fma(fma(0.041666666666666664, Float64(t * t), -0.5), Float64(t * t), 1.0)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[((-t) * t), $MachinePrecision] * N[(-0.0001984126984126984 * eh + N[(N[(eh * -0.019444444444444445), $MachinePrecision] * 0.16666666666666666 + N[(0.001388888888888889 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-eh) * -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[(N[(0.041666666666666664 * N[(t * t), $MachinePrecision] + -0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $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 45.9

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 46.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(\frac{-1}{5040}, eh, \mathsf{fma}\left(eh \cdot \frac{-7}{360}, \frac{1}{6}, \frac{1}{720} \cdot eh\right)\right), \left(-eh\right) \cdot \frac{-7}{360}\right), t \cdot t, \frac{1}{6} \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{24} \cdot {t}^{2} - \frac{1}{2}\right)\right)\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \]

12. Final simplification 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, t \cdot t, -0.5\right), t \cdot t, 1\right)\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 12: 41.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(-0.5, t \cdot t, 1\right)\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (fma
           (* (- t) t)
           (fma
            -0.0001984126984126984
            eh
            (fma
             (* eh -0.019444444444444445)
             0.16666666666666666
             (* 0.001388888888888889 eh)))
           (* (- eh) -0.019444444444444445))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      (fma -0.5 (* t t) 1.0))))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(fma((-t * t), fma(-0.0001984126984126984, eh, fma((eh * -0.019444444444444445), 0.16666666666666666, (0.001388888888888889 * eh))), (-eh * -0.019444444444444445)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * fma(-0.5, (t * t), 1.0)))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(-t) * t), fma(-0.0001984126984126984, eh, fma(Float64(eh * -0.019444444444444445), 0.16666666666666666, Float64(0.001388888888888889 * eh))), Float64(Float64(-eh) * -0.019444444444444445)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * fma(-0.5, Float64(t * t), 1.0)))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[((-t) * t), $MachinePrecision] * N[(-0.0001984126984126984 * eh + N[(N[(eh * -0.019444444444444445), $MachinePrecision] * 0.16666666666666666 + N[(0.001388888888888889 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-eh) * -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $MachinePrecision] * N[(-0.5 * N[(t * t), $MachinePrecision] + 1.0), $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 45.9

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 46.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(\frac{-1}{5040}, eh, \mathsf{fma}\left(eh \cdot \frac{-7}{360}, \frac{1}{6}, \frac{1}{720} \cdot eh\right)\right), \left(-eh\right) \cdot \frac{-7}{360}\right), t \cdot t, \frac{1}{6} \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \left(1 + \frac{-1}{2} \cdot {t}^{2}\right)\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(-0.5, t \cdot t, 1\right)\right) \cdot eh\right| \]

12. Final simplification 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \mathsf{fma}\left(-0.5, t \cdot t, 1\right)\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 13: 41.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (*
      (/
       (/
        (fma
         (fma
          (fma
           (* (- t) t)
           (fma
            -0.0001984126984126984
            eh
            (fma
             (* eh -0.019444444444444445)
             0.16666666666666666
             (* 0.001388888888888889 eh)))
           (* (- eh) -0.019444444444444445))
          (* t t)
          (* 0.16666666666666666 eh))
         (* t t)
         eh)
        t)
       ew)
      1.0)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((fma(fma(fma((-t * t), fma(-0.0001984126984126984, eh, fma((eh * -0.019444444444444445), 0.16666666666666666, (0.001388888888888889 * eh))), (-eh * -0.019444444444444445)), (t * t), (0.16666666666666666 * eh)), (t * t), eh) / t) / ew) * 1.0))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(Float64(fma(fma(fma(Float64(Float64(-t) * t), fma(-0.0001984126984126984, eh, fma(Float64(eh * -0.019444444444444445), 0.16666666666666666, Float64(0.001388888888888889 * eh))), Float64(Float64(-eh) * -0.019444444444444445)), Float64(t * t), Float64(0.16666666666666666 * eh)), Float64(t * t), eh) / t) / ew) * 1.0))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(N[(N[((-t) * t), $MachinePrecision] * N[(-0.0001984126984126984 * eh + N[(N[(eh * -0.019444444444444445), $MachinePrecision] * 0.16666666666666666 + N[(0.001388888888888889 * eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-eh) * -0.019444444444444445), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(0.16666666666666666 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + eh), $MachinePrecision] / t), $MachinePrecision] / ew), $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 45.9

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

5. Applied rewrites 45.9%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 46.2%

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot \cos t\right) \cdot eh\right| \]

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(\frac{-1}{5040}, eh, \mathsf{fma}\left(eh \cdot \frac{-7}{360}, \frac{1}{6}, \frac{1}{720} \cdot eh\right)\right), \left(-eh\right) \cdot \frac{-7}{360}\right), t \cdot t, \frac{1}{6} \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-t \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \]

12. Final simplification 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(-t\right) \cdot t, \mathsf{fma}\left(-0.0001984126984126984, eh, \mathsf{fma}\left(eh \cdot -0.019444444444444445, 0.16666666666666666, 0.001388888888888889 \cdot eh\right)\right), \left(-eh\right) \cdot -0.019444444444444445\right), t \cdot t, 0.16666666666666666 \cdot eh\right), t \cdot t, eh\right)}{t}}{ew} \cdot 1\right) \cdot eh\right| \]
13. Add Preprocessing

Alternative 14: 41.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(eh \cdot -0.022222222222222223, t \cdot t, -0.3333333333333333 \cdot eh\right), t \cdot t, eh\right)}{ew}}{t}\right) \cdot eh\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (*
   (sin
    (atan
     (/
      (/
       (fma
        (fma (* eh -0.022222222222222223) (* t t) (* -0.3333333333333333 eh))
        (* t t)
        eh)
       ew)
      t)))
   eh)))

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((fma(fma((eh * -0.022222222222222223), (t * t), (-0.3333333333333333 * eh)), (t * t), eh) / ew) / t))) * eh));
}

Julia

function code(eh, ew, t)
	return abs(Float64(sin(atan(Float64(Float64(fma(fma(Float64(eh * -0.022222222222222223), Float64(t * t), Float64(-0.3333333333333333 * eh)), Float64(t * t), eh) / ew) / t))) * eh))
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(eh * -0.022222222222222223), $MachinePrecision] * N[(t * t), $MachinePrecision] + N[(-0.3333333333333333 * eh), $MachinePrecision]), $MachinePrecision] * N[(t * t), $MachinePrecision] + 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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

9. Taylor expanded in t around 0

   \[\leadsto \left|\sin \tan^{-1} \left(\frac{{t}^{2} \cdot \left(\left(\frac{-1}{2} \cdot \frac{eh}{ew} + {t}^{2} \cdot \left(\frac{1}{24} \cdot \frac{eh}{ew} - \left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot \frac{eh}{ew} - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{1}{120} \cdot \frac{eh}{ew}\right)\right)\right) - \frac{-1}{6} \cdot \frac{eh}{ew}\right) + \frac{eh}{ew}}{t}\right) \cdot eh\right| \]
10. Step-by-step derivation

11. Applied rewrites 38.1%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{eh}{ew}, -0.013888888888888888, -0.008333333333333333 \cdot \frac{eh}{ew}\right), t \cdot t, -0.3333333333333333 \cdot \frac{eh}{ew}\right), t \cdot t, \frac{eh}{ew}\right)}{t}\right) \cdot eh\right| \]

12. Taylor expanded in ew around 0

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{eh + {t}^{2} \cdot \left(\frac{-1}{3} \cdot eh + {t}^{2} \cdot \left(\frac{-1}{72} \cdot eh + \frac{-1}{120} \cdot eh\right)\right)}{ew}}{t}\right) \cdot eh\right| \]
13. Step-by-step derivation

14. Applied rewrites 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(eh \cdot -0.022222222222222223, t \cdot t, -0.3333333333333333 \cdot eh\right), t \cdot t, eh\right)}{ew}}{t}\right) \cdot eh\right| \]

15. Final simplification 46.2%

    \[\leadsto \left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(eh \cdot -0.022222222222222223, t \cdot t, -0.3333333333333333 \cdot eh\right), t \cdot t, eh\right)}{ew}}{t}\right) \cdot eh\right| \]
16. Add Preprocessing

Alternative 15: 41.8% 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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

9. Taylor expanded in ew around 0

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

11. Applied rewrites 46.2%

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

12. Final simplification 46.2%

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

Alternative 16: 39.7% 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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

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 44.1%

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

12. Final simplification 44.1%

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

Alternative 17: 22.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan(((-0.3333333333333333 * (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((((-0.3333333333333333d0) * (eh / ew)) * t))) * eh))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.sin(Math.atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
}

Python

def code(eh, ew, t):
	return math.fabs((math.sin(math.atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((sin(atan(((-0.3333333333333333 * (eh / ew)) * t))) * eh));
end

Wolfram

code[eh_, ew_, t_] := 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. 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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 23.0%

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

12. Final simplification 23.0%

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

Alternative 18: 22.7% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double eh, double ew, double t) {
	return fabs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * 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((((t * eh) / ew) * (-0.3333333333333333d0)))) * eh))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((Math.sin(Math.atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
}

Python

def code(eh, ew, t):
	return math.fabs((math.sin(math.atan((((t * eh) / ew) * -0.3333333333333333))) * eh))

Julia

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

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333), $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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 23.0%

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

12. Taylor expanded in eh around 0

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

14. Applied rewrites 23.0%

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

15. Final simplification 23.0%

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

Alternative 19: 9.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t\\ \left|\frac{t\_1}{\sqrt{{t\_1}^{2} + 1}} \cdot eh\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (* (/ eh ew) -0.3333333333333333) 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) * -0.3333333333333333) * 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) * (-0.3333333333333333d0)) * 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) * -0.3333333333333333) * 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) * -0.3333333333333333) * 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(Float64(eh / ew) * -0.3333333333333333) * 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) * -0.3333333333333333) * 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[(N[(eh / ew), $MachinePrecision] * -0.3333333333333333), $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 45.9

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

5. Applied rewrites 45.9%

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

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 38.1%

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

9. Taylor expanded in t around inf

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

11. Applied rewrites 23.0%

    \[\leadsto \left|\sin \tan^{-1} \left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t\right) \cdot eh\right| \]
12. Step-by-step derivation

13. Applied rewrites 10.7%

    \[\leadsto \left|\frac{\left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t}{\sqrt{{\left(\left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t\right)}^{2} + 1}} \cdot eh\right| \]

14. Final simplification 10.7%

    \[\leadsto \left|\frac{\left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t}{\sqrt{{\left(\left(\frac{eh}{ew} \cdot -0.3333333333333333\right) \cdot t\right)}^{2} + 1}} \cdot eh\right| \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024313 
(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))))))))