Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 19.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((eh / ew) / tan(t)))
    code = abs((((ew * sin(t)) * cos(t_1)) + ((eh * cos(t)) * sin(t_1))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * eh + N[(N[(1.0 / N[Sqrt[N[(N[Power[N[(N[(eh / N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $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. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-cos.f64 N/A

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

   2. lift-atan.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-/l/ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. cos-atan N/A

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

   9. lower-/.f64 N/A

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

   10. lower-sqrt.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 N/A

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

   13. pow2 N/A

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

   14. lower-pow.f64 99.8

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

10. Applied rewrites 99.8%

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

11. Final simplification 99.8%

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

Alternative 3: 91.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \cos t \cdot eh\\ t_2 := \left|t\_1 \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{if}\;ew \leq -1.35 \cdot 10^{-166}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-206}:\\ \;\;\;\;\left|t\_1 \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (cos t) eh))
        (t_2
         (fabs
          (+
           (* t_1 (sin (atan (/ eh (* t ew)))))
           (* (cos (atan (/ (/ eh ew) (tan t)))) (* (sin t) ew))))))
   (if (<= ew -1.35e-166)
     t_2
     (if (<= ew 6e-206)
       (fabs (* t_1 (sin (atan (* (/ (/ eh (sin t)) ew) (cos t))))))
       t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = cos(t) * eh;
	double t_2 = fabs(((t_1 * sin(atan((eh / (t * ew))))) + (cos(atan(((eh / ew) / tan(t)))) * (sin(t) * ew))));
	double tmp;
	if (ew <= -1.35e-166) {
		tmp = t_2;
	} else if (ew <= 6e-206) {
		tmp = fabs((t_1 * sin(atan((((eh / sin(t)) / ew) * cos(t))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = cos(t) * eh
    t_2 = abs(((t_1 * sin(atan((eh / (t * ew))))) + (cos(atan(((eh / ew) / tan(t)))) * (sin(t) * ew))))
    if (ew <= (-1.35d-166)) then
        tmp = t_2
    else if (ew <= 6d-206) then
        tmp = abs((t_1 * sin(atan((((eh / sin(t)) / ew) * cos(t))))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.cos(t) * eh;
	double t_2 = Math.abs(((t_1 * Math.sin(Math.atan((eh / (t * ew))))) + (Math.cos(Math.atan(((eh / ew) / Math.tan(t)))) * (Math.sin(t) * ew))));
	double tmp;
	if (ew <= -1.35e-166) {
		tmp = t_2;
	} else if (ew <= 6e-206) {
		tmp = Math.abs((t_1 * Math.sin(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.cos(t) * eh
	t_2 = math.fabs(((t_1 * math.sin(math.atan((eh / (t * ew))))) + (math.cos(math.atan(((eh / ew) / math.tan(t)))) * (math.sin(t) * ew))))
	tmp = 0
	if ew <= -1.35e-166:
		tmp = t_2
	elif ew <= 6e-206:
		tmp = math.fabs((t_1 * math.sin(math.atan((((eh / math.sin(t)) / ew) * math.cos(t))))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(cos(t) * eh)
	t_2 = abs(Float64(Float64(t_1 * sin(atan(Float64(eh / Float64(t * ew))))) + Float64(cos(atan(Float64(Float64(eh / ew) / tan(t)))) * Float64(sin(t) * ew))))
	tmp = 0.0
	if (ew <= -1.35e-166)
		tmp = t_2;
	elseif (ew <= 6e-206)
		tmp = abs(Float64(t_1 * sin(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = cos(t) * eh;
	t_2 = abs(((t_1 * sin(atan((eh / (t * ew))))) + (cos(atan(((eh / ew) / tan(t)))) * (sin(t) * ew))));
	tmp = 0.0;
	if (ew <= -1.35e-166)
		tmp = t_2;
	elseif (ew <= 6e-206)
		tmp = abs((t_1 * sin(atan((((eh / sin(t)) / ew) * cos(t))))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[(t$95$1 * N[Sin[N[ArcTan[N[(eh / N[(t * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / N[Tan[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.35e-166], t$95$2, If[LessEqual[ew, 6e-206], N[Abs[N[(t$95$1 * N[Sin[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.35000000000000003e-166 or 6.0000000000000004e-206 < ew

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 94.6

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

   5. Applied rewrites 94.6%

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

   if -1.35000000000000003e-166 < ew < 6.0000000000000004e-206

   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 99.5%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.35 \cdot 10^{-166}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 6 \cdot 10^{-206}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{eh}{t \cdot ew}\right) + \cos \tan^{-1} \left(\frac{\frac{eh}{ew}}{\tan t}\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ t_2 := \cos t \cdot eh\\ \mathbf{if}\;ew \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{t\_2}{ew}}{\sin t}\right) \cdot t\_2\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (cos (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (sin t) ew))))
        (t_2 (* (cos t) eh)))
   (if (<= ew -2.9e+20)
     t_1
     (if (<= ew 1.55e+99)
       (fabs (* (sin (atan (/ (/ t_2 ew) (sin t)))) t_2))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)));
	double t_2 = cos(t) * eh;
	double tmp;
	if (ew <= -2.9e+20) {
		tmp = t_1;
	} else if (ew <= 1.55e+99) {
		tmp = fabs((sin(atan(((t_2 / ew) / sin(t)))) * t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = abs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)))
    t_2 = cos(t) * eh
    if (ew <= (-2.9d+20)) then
        tmp = t_1
    else if (ew <= 1.55d+99) then
        tmp = abs((sin(atan(((t_2 / ew) / sin(t)))) * t_2))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((Math.cos(Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)))) * (Math.sin(t) * ew)));
	double t_2 = Math.cos(t) * eh;
	double tmp;
	if (ew <= -2.9e+20) {
		tmp = t_1;
	} else if (ew <= 1.55e+99) {
		tmp = Math.abs((Math.sin(Math.atan(((t_2 / ew) / Math.sin(t)))) * t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((math.cos(math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))) * (math.sin(t) * ew)))
	t_2 = math.cos(t) * eh
	tmp = 0
	if ew <= -2.9e+20:
		tmp = t_1
	elif ew <= 1.55e+99:
		tmp = math.fabs((math.sin(math.atan(((t_2 / ew) / math.sin(t)))) * t_2))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(sin(t) * ew)))
	t_2 = Float64(cos(t) * eh)
	tmp = 0.0
	if (ew <= -2.9e+20)
		tmp = t_1;
	elseif (ew <= 1.55e+99)
		tmp = abs(Float64(sin(atan(Float64(Float64(t_2 / ew) / sin(t)))) * t_2));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)));
	t_2 = cos(t) * eh;
	tmp = 0.0;
	if (ew <= -2.9e+20)
		tmp = t_1;
	elseif (ew <= 1.55e+99)
		tmp = abs((sin(atan(((t_2 / ew) / sin(t)))) * t_2));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]}, If[LessEqual[ew, -2.9e+20], t$95$1, If[LessEqual[ew, 1.55e+99], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(t$95$2 / ew), $MachinePrecision] / N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.9e20 or 1.55e99 < 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. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   if -2.9e20 < ew < 1.55e99

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in ew around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-atan.f64 N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-sin.f64 84.6

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

   7. Applied rewrites 84.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\cos t \cdot eh}{ew}}{\sin t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\\ t_2 := \left|\cos t\_1 \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{if}\;ew \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin t\_1\right|\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (* (/ (/ eh (sin t)) ew) (cos t))))
        (t_2 (fabs (* (cos t_1) (* (sin t) ew)))))
   (if (<= ew -2.9e+20)
     t_2
     (if (<= ew 1.55e+99) (fabs (* (* (cos t) eh) (sin t_1))) t_2))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	double t_2 = fabs((cos(t_1) * (sin(t) * ew)));
	double tmp;
	if (ew <= -2.9e+20) {
		tmp = t_2;
	} else if (ew <= 1.55e+99) {
		tmp = fabs(((cos(t) * eh) * sin(t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = atan((((eh / sin(t)) / ew) * cos(t)))
    t_2 = abs((cos(t_1) * (sin(t) * ew)))
    if (ew <= (-2.9d+20)) then
        tmp = t_2
    else if (ew <= 1.55d+99) then
        tmp = abs(((cos(t) * eh) * sin(t_1)))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan((((eh / Math.sin(t)) / ew) * Math.cos(t)));
	double t_2 = Math.abs((Math.cos(t_1) * (Math.sin(t) * ew)));
	double tmp;
	if (ew <= -2.9e+20) {
		tmp = t_2;
	} else if (ew <= 1.55e+99) {
		tmp = Math.abs(((Math.cos(t) * eh) * Math.sin(t_1)));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.atan((((eh / math.sin(t)) / ew) * math.cos(t)))
	t_2 = math.fabs((math.cos(t_1) * (math.sin(t) * ew)))
	tmp = 0
	if ew <= -2.9e+20:
		tmp = t_2
	elif ew <= 1.55e+99:
		tmp = math.fabs(((math.cos(t) * eh) * math.sin(t_1)))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))
	t_2 = abs(Float64(cos(t_1) * Float64(sin(t) * ew)))
	tmp = 0.0
	if (ew <= -2.9e+20)
		tmp = t_2;
	elseif (ew <= 1.55e+99)
		tmp = abs(Float64(Float64(cos(t) * eh) * sin(t_1)));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = atan((((eh / sin(t)) / ew) * cos(t)));
	t_2 = abs((cos(t_1) * (sin(t) * ew)));
	tmp = 0.0;
	if (ew <= -2.9e+20)
		tmp = t_2;
	elseif (ew <= 1.55e+99)
		tmp = abs(((cos(t) * eh) * sin(t_1)));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Abs[N[(N[Cos[t$95$1], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.9e+20], t$95$2, If[LessEqual[ew, 1.55e+99], N[Abs[N[(N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.9e20 or 1.55e99 < 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. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   if -2.9e20 < ew < 1.55e99

   1. Initial program 99.7%

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

   3. Taylor expanded in 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 84.6%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 1.55 \cdot 10^{+99}:\\ \;\;\;\;\left|\left(\cos t \cdot eh\right) \cdot \sin \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{if}\;ew \leq -2.8 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 3.25 \cdot 10^{+94}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{-0.5}{eh}, \frac{\left(ew \cdot ew\right) \cdot {\sin t}^{2}}{{\cos t}^{2}}, eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (fabs
          (* (cos (atan (* (/ (/ eh (sin t)) ew) (cos t)))) (* (sin t) ew)))))
   (if (<= ew -2.8e+20)
     t_1
     (if (<= ew 3.25e+94)
       (fabs
        (fma
         (/ -0.5 eh)
         (/ (* (* ew ew) (pow (sin t) 2.0)) (pow (cos t) 2.0))
         eh))
       t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((cos(atan((((eh / sin(t)) / ew) * cos(t)))) * (sin(t) * ew)));
	double tmp;
	if (ew <= -2.8e+20) {
		tmp = t_1;
	} else if (ew <= 3.25e+94) {
		tmp = fabs(fma((-0.5 / eh), (((ew * ew) * pow(sin(t), 2.0)) / pow(cos(t), 2.0)), eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(cos(atan(Float64(Float64(Float64(eh / sin(t)) / ew) * cos(t)))) * Float64(sin(t) * ew)))
	tmp = 0.0
	if (ew <= -2.8e+20)
		tmp = t_1;
	elseif (ew <= 3.25e+94)
		tmp = abs(fma(Float64(-0.5 / eh), Float64(Float64(Float64(ew * ew) * (sin(t) ^ 2.0)) / (cos(t) ^ 2.0)), eh));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(N[Cos[N[ArcTan[N[(N[(N[(eh / N[Sin[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -2.8e+20], t$95$1, If[LessEqual[ew, 3.25e+94], N[Abs[N[(N[(-0.5 / eh), $MachinePrecision] * N[(N[(N[(ew * ew), $MachinePrecision] * N[Power[N[Sin[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Cos[t], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.8e20 or 3.24999999999999988e94 < 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. Taylor expanded in ew around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

   5. Applied rewrites 78.5%

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

   if -2.8e20 < ew < 3.24999999999999988e94

   1. Initial program 99.7%

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

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-atan.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-cos.f64 56.3

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

   5. Applied rewrites 56.3%

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

   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 54.9%

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

   10. Applied rewrites 16.6%

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

   11. Taylor expanded in ew around 0

       \[\leadsto \left|eh + \color{blue}{\frac{-1}{2} \cdot \frac{{ew}^{2} \cdot {\sin t}^{2}}{eh \cdot {\cos t}^{2}}}\right| \]
   12. Step-by-step derivation

   13. Applied rewrites 56.5%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.8 \cdot 10^{+20}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \mathbf{elif}\;ew \leq 3.25 \cdot 10^{+94}:\\ \;\;\;\;\left|\mathsf{fma}\left(\frac{-0.5}{eh}, \frac{\left(ew \cdot ew\right) \cdot {\sin t}^{2}}{{\cos t}^{2}}, eh\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\cos \tan^{-1} \left(\frac{\frac{eh}{\sin t}}{ew} \cdot \cos t\right) \cdot \left(\sin t \cdot ew\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% 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 42.3

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

5. Applied rewrites 42.3%

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

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 41.0%

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

10. Applied rewrites 14.8%

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

11. Taylor expanded in eh around -inf

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

13. Applied rewrites 42.8%

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

Reproduce

herbie shell --seed 2024268 
(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))))))))