Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.4s

Alternatives: 13

Speedup: 1.1×

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, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{eh}{\tan t}}{ew}\\ \left|\mathsf{fma}\left(eh \cdot \cos t, \sin \tan^{-1} t\_1, \left(\sin t \cdot ew\right) \cdot \frac{1}{{\left({\left({t\_1}^{2} + 1\right)}^{0.25}\right)}^{2}}\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ (/ eh (tan t)) ew)))
   (fabs
    (fma
     (* eh (cos t))
     (sin (atan t_1))
     (* (* (sin t) ew) (/ 1.0 (pow (pow (+ (pow t_1 2.0) 1.0) 0.25) 2.0)))))))

C

double code(double eh, double ew, double t) {
	double t_1 = (eh / tan(t)) / ew;
	return fabs(fma((eh * cos(t)), sin(atan(t_1)), ((sin(t) * ew) * (1.0 / pow(pow((pow(t_1, 2.0) + 1.0), 0.25), 2.0)))));
}

Julia

function code(eh, ew, t)
	t_1 = Float64(Float64(eh / tan(t)) / ew)
	return abs(fma(Float64(eh * cos(t)), sin(atan(t_1)), Float64(Float64(sin(t) * ew) * Float64(1.0 / ((Float64((t_1 ^ 2.0) + 1.0) ^ 0.25) ^ 2.0)))))
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]}, N[Abs[N[(N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision] + N[(N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision] * N[(1.0 / N[Power[N[Power[N[(N[Power[t$95$1, 2.0], $MachinePrecision] + 1.0), $MachinePrecision], 0.25], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. sqr-pow N/A

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

   4. pow2 N/A

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

   5. lower-pow.f64 N/A

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

   6. metadata-eval N/A

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \frac{1}{{\left({\left({\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1\right)}^{\color{blue}{\frac{1}{4}}}\right)}^{2}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]

   7. metadata-eval N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval 99.8

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \frac{1}{{\left({\left({\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1\right)}^{\color{blue}{0.25}}\right)}^{2}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]

10. Applied rewrites 99.8%

    \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \frac{1}{\color{blue}{{\left({\left({\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1\right)}^{0.25}\right)}^{2}}} \cdot \left(\sin t \cdot ew\right)\right)\right| \]

11. Final simplification 99.8%

    \[\leadsto \left|\mathsf{fma}\left(eh \cdot \cos t, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \left(\sin t \cdot ew\right) \cdot \frac{1}{{\left({\left({\left(\frac{\frac{eh}{\tan t}}{ew}\right)}^{2} + 1\right)}^{0.25}\right)}^{2}}\right)\right| \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(eh, ew, t)
	return abs(fma(Float64(eh * cos(t)), sin(atan(Float64(Float64(eh / tan(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[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $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. Final simplification 99.8%

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

Alternative 3: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh + N[(N[Cos[N[ArcTan[N[(eh / N[(ew * t), $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. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

7. Applied rewrites 99.5%

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

Alternative 5: 99.1% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(((sin(atan(((eh / ew) / tan(t)))) * (eh * cos(t))) + ((sin(t) * ew) / sqrt((((eh / (ew * t)) ** 2.0d0) + 1.0d0)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.5

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-atan.f64 N/A

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

   7. cos-atan N/A

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

   8. un-div-inv N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 6: 74.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -5.3e+33)
     t_1
     (if (<= ew 4.3e+189)
       (fabs (* (sin (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* eh (cos t))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = fabs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (eh * cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (ew <= (-5.3d+33)) then
        tmp = t_1
    else if (ew <= 4.3d+189) then
        tmp = abs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (eh * cos(t))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = Math.abs((Math.sin(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)))) * (eh * Math.cos(t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -5.3e+33:
		tmp = t_1
	elif ew <= 4.3e+189:
		tmp = math.fabs((math.sin(math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))) * (eh * math.cos(t))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = abs(Float64(sin(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(eh * cos(t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = abs((sin(atan((((eh / sin(t)) / ew) * cos(t)))) * (eh * cos(t))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.3e+33], t$95$1, If[LessEqual[ew, 4.3e+189], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000023e33 or 4.29999999999999998e189 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. 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

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 77.0

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 77.0%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

   if -5.30000000000000023e33 < ew < 4.29999999999999998e189

   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 ew around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-atan.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. associate-/r* N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. *-commutative N/A

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

   5. Applied rewrites 77.7%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(eh \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. lower-sin.f64 98.6

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

11. Applied rewrites 98.6%

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

12. Final simplification 98.6%

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

Alternative 8: 74.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\left|\frac{\frac{\frac{1}{eh}}{\cos t}}{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.001388888888888889, t \cdot t, 0.041666666666666664\right), t \cdot t, -0.5\right), t \cdot t, 1\right)}{ew} \cdot \frac{eh}{\sin t}\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -5.3e+33)
     t_1
     (if (<= ew 4.3e+189)
       (/
        1.0
        (fabs
         (/
          (/ (/ 1.0 eh) (cos t))
          (sin
           (atan
            (*
             (/
              (fma
               (fma
                (fma -0.001388888888888889 (* t t) 0.041666666666666664)
                (* t t)
                -0.5)
               (* t t)
               1.0)
              ew)
             (/ eh (sin t))))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = 1.0 / fabs((((1.0 / eh) / cos(t)) / sin(atan(((fma(fma(fma(-0.001388888888888889, (t * t), 0.041666666666666664), (t * t), -0.5), (t * t), 1.0) / ew) * (eh / sin(t)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = Float64(1.0 / abs(Float64(Float64(Float64(1.0 / eh) / cos(t)) / sin(atan(Float64(Float64(fma(fma(fma(-0.001388888888888889, Float64(t * t), 0.041666666666666664), Float64(t * t), -0.5), Float64(t * t), 1.0) / ew) * Float64(eh / sin(t))))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.3e+33], t$95$1, If[LessEqual[ew, 4.3e+189], N[(1.0 / N[Abs[N[(N[(N[(1.0 / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sin[N[ArcTan[N[(N[(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] / ew), $MachinePrecision] * N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000023e33 or 4.29999999999999998e189 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. 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

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 77.0

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 77.0%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

   if -5.30000000000000023e33 < ew < 4.29999999999999998e189

   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-fabs.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ew around 0

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 77.2%

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

Alternative 9: 74.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(eh \cdot \cos t\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -5.3e+33)
     t_1
     (if (<= ew 4.3e+189)
       (/
        1.0
        (fabs (/ 1.0 (* (sin (atan (/ (/ eh (tan t)) ew))) (* eh (cos t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = 1.0 / fabs((1.0 / (sin(atan(((eh / tan(t)) / ew))) * (eh * cos(t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (ew <= (-5.3d+33)) then
        tmp = t_1
    else if (ew <= 4.3d+189) then
        tmp = 1.0d0 / abs((1.0d0 / (sin(atan(((eh / tan(t)) / ew))) * (eh * cos(t)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = 1.0 / Math.abs((1.0 / (Math.sin(Math.atan(((eh / Math.tan(t)) / ew))) * (eh * Math.cos(t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if ew <= -5.3e+33:
		tmp = t_1
	elif ew <= 4.3e+189:
		tmp = 1.0 / math.fabs((1.0 / (math.sin(math.atan(((eh / math.tan(t)) / ew))) * (eh * math.cos(t)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = Float64(1.0 / abs(Float64(1.0 / Float64(sin(atan(Float64(Float64(eh / tan(t)) / ew))) * Float64(eh * cos(t))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = 1.0 / abs((1.0 / (sin(atan(((eh / tan(t)) / ew))) * (eh * cos(t)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.3e+33], t$95$1, If[LessEqual[ew, 4.3e+189], N[(1.0 / N[Abs[N[(1.0 / N[(N[Sin[N[ArcTan[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(eh * N[Cos[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000023e33 or 4.29999999999999998e189 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. 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

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 77.0

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 77.0%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

   if -5.30000000000000023e33 < ew < 4.29999999999999998e189

   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-fabs.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ew around 0

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 77.0

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

   7. Applied rewrites 77.0%

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

   9. Applied rewrites 77.0%

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

   11. Applied rewrites 77.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\left|\frac{1}{\sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(eh \cdot \cos t\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -5.3 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 4.3 \cdot 10^{+189}:\\ \;\;\;\;\frac{1}{\left|\frac{\frac{\frac{1}{eh}}{\cos t}}{\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot eh, \frac{t \cdot t}{ew}, \frac{eh}{ew}\right)}{t}\right)}\right|}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -5.3e+33)
     t_1
     (if (<= ew 4.3e+189)
       (/
        1.0
        (fabs
         (/
          (/ (/ 1.0 eh) (cos t))
          (sin
           (atan
            (/
             (fma (* -0.3333333333333333 eh) (/ (* t t) ew) (/ eh ew))
             t))))))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -5.3e+33) {
		tmp = t_1;
	} else if (ew <= 4.3e+189) {
		tmp = 1.0 / fabs((((1.0 / eh) / cos(t)) / sin(atan((fma((-0.3333333333333333 * eh), ((t * t) / ew), (eh / ew)) / t)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -5.3e+33)
		tmp = t_1;
	elseif (ew <= 4.3e+189)
		tmp = Float64(1.0 / abs(Float64(Float64(Float64(1.0 / eh) / cos(t)) / sin(atan(Float64(fma(Float64(-0.3333333333333333 * eh), Float64(Float64(t * t) / ew), Float64(eh / ew)) / t))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -5.3e+33], t$95$1, If[LessEqual[ew, 4.3e+189], N[(1.0 / N[Abs[N[(N[(N[(1.0 / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sin[N[ArcTan[N[(N[(N[(-0.3333333333333333 * eh), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision] + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.30000000000000023e33 or 4.29999999999999998e189 < ew

   1. Initial program 99.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. 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

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 77.0

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 77.0%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

   if -5.30000000000000023e33 < ew < 4.29999999999999998e189

   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-fabs.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in ew around 0

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 77.0

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

   7. Applied rewrites 77.0%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.6%

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

Alternative 11: 68.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\left|\frac{\frac{\frac{1}{eh}}{\cos t}}{\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right)}\right|}\\ \mathbf{if}\;eh \leq -5.2 \cdot 10^{-87}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (/
          1.0
          (fabs (/ (/ (/ 1.0 eh) (cos t)) (sin (atan (/ (/ eh ew) t))))))))
   (if (<= eh -5.2e-87) t_1 (if (<= eh 2.1e-43) (fabs (* (sin t) ew)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = 1.0 / fabs((((1.0 / eh) / cos(t)) / sin(atan(((eh / ew) / t)))));
	double tmp;
	if (eh <= -5.2e-87) {
		tmp = t_1;
	} else if (eh <= 2.1e-43) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / abs((((1.0d0 / eh) / cos(t)) / sin(atan(((eh / ew) / t)))))
    if (eh <= (-5.2d-87)) then
        tmp = t_1
    else if (eh <= 2.1d-43) then
        tmp = abs((sin(t) * ew))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = 1.0 / Math.abs((((1.0 / eh) / Math.cos(t)) / Math.sin(Math.atan(((eh / ew) / t)))));
	double tmp;
	if (eh <= -5.2e-87) {
		tmp = t_1;
	} else if (eh <= 2.1e-43) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = 1.0 / math.fabs((((1.0 / eh) / math.cos(t)) / math.sin(math.atan(((eh / ew) / t)))))
	tmp = 0
	if eh <= -5.2e-87:
		tmp = t_1
	elif eh <= 2.1e-43:
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(1.0 / abs(Float64(Float64(Float64(1.0 / eh) / cos(t)) / sin(atan(Float64(Float64(eh / ew) / t))))))
	tmp = 0.0
	if (eh <= -5.2e-87)
		tmp = t_1;
	elseif (eh <= 2.1e-43)
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = 1.0 / abs((((1.0 / eh) / cos(t)) / sin(atan(((eh / ew) / t)))));
	tmp = 0.0;
	if (eh <= -5.2e-87)
		tmp = t_1;
	elseif (eh <= 2.1e-43)
		tmp = abs((sin(t) * ew));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(1.0 / N[Abs[N[(N[(N[(1.0 / eh), $MachinePrecision] / N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, -5.2e-87], t$95$1, If[LessEqual[eh, 2.1e-43], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -5.20000000000000005e-87 or 2.1000000000000001e-43 < 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-fabs.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in ew around 0

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

      1. associate-/r* N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-atan.f64 N/A

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

      9. *-commutative N/A

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

      10. times-frac N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 81.5

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

   7. Applied rewrites 81.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.9%

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

   if -5.20000000000000005e-87 < eh < 2.1000000000000001e-43

   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

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 69.6

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 69.6%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 59.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;t \leq -2 \cdot 10^{-149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;\left|-eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= t -2e-149) t_1 (if (<= t 8.5e-8) (fabs (- eh)) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (t <= -2e-149) {
		tmp = t_1;
	} else if (t <= 8.5e-8) {
		tmp = fabs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((sin(t) * ew))
    if (t <= (-2d-149)) then
        tmp = t_1
    else if (t <= 8.5d-8) then
        tmp = abs(-eh)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.sin(t) * ew));
	double tmp;
	if (t <= -2e-149) {
		tmp = t_1;
	} else if (t <= 8.5e-8) {
		tmp = Math.abs(-eh);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.sin(t) * ew))
	tmp = 0
	if t <= -2e-149:
		tmp = t_1
	elif t <= 8.5e-8:
		tmp = math.fabs(-eh)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (t <= -2e-149)
		tmp = t_1;
	elseif (t <= 8.5e-8)
		tmp = abs(Float64(-eh));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((sin(t) * ew));
	tmp = 0.0;
	if (t <= -2e-149)
		tmp = t_1;
	elseif (t <= 8.5e-8)
		tmp = abs(-eh);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, -2e-149], t$95$1, If[LessEqual[t, 8.5e-8], N[Abs[(-eh)], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.99999999999999996e-149 or 8.49999999999999935e-8 < t

   1. Initial program 99.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.6

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

      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 99.6

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

      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 99.6

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

      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 99.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \cos \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right) \cdot \left(\sin t \cdot ew\right)\right)}\right| \]
   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 99.6

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

   6. Applied rewrites 99.6%

      \[\leadsto \left|\mathsf{fma}\left(\cos t \cdot eh, \sin \tan^{-1} \left(\frac{\frac{eh}{\tan t}}{ew}\right), \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew \cdot \tan t}\right)} \cdot \left(\sin t \cdot ew\right)\right)\right| \]
   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 99.6%

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

   9. Taylor expanded in ew around inf

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       2. lower-*.f64 N/A

          \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

       3. lower-sin.f64 56.8

          \[\leadsto \left|\color{blue}{\sin t} \cdot ew\right| \]

   11. Applied rewrites 56.8%

       \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]

   if -1.99999999999999996e-149 < t < 8.49999999999999935e-8

   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 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 83.1%

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

   10. Applied rewrites 23.8%

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

   11. Taylor expanded in eh around -inf

       \[\leadsto \left|-1 \cdot \color{blue}{eh}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 83.3%

       \[\leadsto \left|-eh\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 43.4% accurate, 174.0× speedup

Math

\[\begin{array}{l} \\ \left|-eh\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (- eh)))

C

double code(double eh, double ew, double t) {
	return fabs(-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(-eh)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(-eh);
}

Python

def code(eh, ew, t):
	return math.fabs(-eh)

Julia

function code(eh, ew, t)
	return abs(Float64(-eh))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(-eh);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[(-eh)], $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 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 43.9%

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

10. Applied rewrites 15.0%

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

11. Taylor expanded in eh around -inf

    \[\leadsto \left|-1 \cdot \color{blue}{eh}\right| \]
12. Step-by-step derivation

13. Applied rewrites 45.9%

    \[\leadsto \left|-eh\right| \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024276 
(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))))))))