Example from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 14.9s

Alternatives: 10

Speedup: N/A×

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, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. cos-atan N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. +-commutative N/A

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

8. Applied rewrites 99.8%

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

Alternative 2: 98.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. cos-atan N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. +-commutative N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

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

11. Applied rewrites 99.3%

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

Alternative 3: 74.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{-27} \lor \neg \left(eh \leq 2.35 \cdot 10^{-28}\right):\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -1.02e-27) (not (<= eh 2.35e-28)))
   (fabs (* (* (sin (atan (/ eh (* (tan t) ew)))) (cos t)) eh))
   (fabs (* (sin t) ew))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.02e-27) || !(eh <= 2.35e-28)) {
		tmp = fabs(((sin(atan((eh / (tan(t) * ew)))) * cos(t)) * eh));
	} else {
		tmp = fabs((sin(t) * ew));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-1.02d-27)) .or. (.not. (eh <= 2.35d-28))) then
        tmp = abs(((sin(atan((eh / (tan(t) * ew)))) * cos(t)) * eh))
    else
        tmp = abs((sin(t) * ew))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -1.02e-27) || !(eh <= 2.35e-28)) {
		tmp = Math.abs(((Math.sin(Math.atan((eh / (Math.tan(t) * ew)))) * Math.cos(t)) * eh));
	} else {
		tmp = Math.abs((Math.sin(t) * ew));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -1.02e-27) or not (eh <= 2.35e-28):
		tmp = math.fabs(((math.sin(math.atan((eh / (math.tan(t) * ew)))) * math.cos(t)) * eh))
	else:
		tmp = math.fabs((math.sin(t) * ew))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -1.02e-27) || !(eh <= 2.35e-28))
		tmp = abs(Float64(Float64(sin(atan(Float64(eh / Float64(tan(t) * ew)))) * cos(t)) * eh));
	else
		tmp = abs(Float64(sin(t) * ew));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -1.02e-27) || ~((eh <= 2.35e-28)))
		tmp = abs(((sin(atan((eh / (tan(t) * ew)))) * cos(t)) * eh));
	else
		tmp = abs((sin(t) * ew));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -1.02e-27], N[Not[LessEqual[eh, 2.35e-28]], $MachinePrecision]], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(eh / N[(N[Tan[t], $MachinePrecision] * ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.02000000000000002e-27 or 2.3499999999999998e-28 < 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. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 80.7%

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

   7. Applied rewrites 80.7%

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

   if -1.02000000000000002e-27 < eh < 2.3499999999999998e-28

   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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 75.9

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

   11. Applied rewrites 75.9%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.02 \cdot 10^{-27} \lor \neg \left(eh \leq 2.35 \cdot 10^{-28}\right):\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{\tan t \cdot ew}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.55 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -1.18 \cdot 10^{-147}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(eh \cdot \frac{t \cdot t}{ew}, -0.3333333333333333, \frac{eh}{ew}\right)}{t}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{elif}\;ew \leq 1100000000:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.55e+34)
     t_1
     (if (<= ew -1.18e-147)
       (fabs
        (*
         (*
          (sin
           (atan
            (/ (fma (* eh (/ (* t t) ew)) -0.3333333333333333 (/ eh ew)) t)))
          (cos t))
         eh))
       (if (<= ew 1100000000.0)
         (fabs (* (* (sin (atan (/ eh (* ew t)))) (cos t)) eh))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.55e+34) {
		tmp = t_1;
	} else if (ew <= -1.18e-147) {
		tmp = fabs(((sin(atan((fma((eh * ((t * t) / ew)), -0.3333333333333333, (eh / ew)) / t))) * cos(t)) * eh));
	} else if (ew <= 1100000000.0) {
		tmp = fabs(((sin(atan((eh / (ew * t)))) * cos(t)) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.55e+34)
		tmp = t_1;
	elseif (ew <= -1.18e-147)
		tmp = abs(Float64(Float64(sin(atan(Float64(fma(Float64(eh * Float64(Float64(t * t) / ew)), -0.3333333333333333, Float64(eh / ew)) / t))) * cos(t)) * eh));
	elseif (ew <= 1100000000.0)
		tmp = abs(Float64(Float64(sin(atan(Float64(eh / Float64(ew * t)))) * cos(t)) * eh));
	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, -2.55e+34], t$95$1, If[LessEqual[ew, -1.18e-147], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(N[(N[(eh * N[(N[(t * t), $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] * -0.3333333333333333 + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1100000000.0], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.55000000000000018e34 or 1.1e9 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 70.0

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

   11. Applied rewrites 70.0%

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

   if -2.55000000000000018e34 < ew < -1.18000000000000003e-147

   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 eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 79.2%

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

   7. Applied rewrites 79.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 79.4%

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

   if -1.18000000000000003e-147 < ew < 1.1e9

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 87.8%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.4%

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

Alternative 5: 68.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|\sin t \cdot ew\right|\\ \mathbf{if}\;ew \leq -2.55 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -1.18 \cdot 10^{-147}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot \frac{eh}{ew}\right) \cdot t, t, \frac{eh}{ew}\right)}{t}\right) \cdot \left(\cos t \cdot eh\right)\right|\\ \mathbf{elif}\;ew \leq 1100000000:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t\right) \cdot eh\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* (sin t) ew))))
   (if (<= ew -2.55e+34)
     t_1
     (if (<= ew -1.18e-147)
       (fabs
        (*
         (sin
          (atan
           (/ (fma (* (* -0.3333333333333333 (/ eh ew)) t) t (/ eh ew)) t)))
         (* (cos t) eh)))
       (if (<= ew 1100000000.0)
         (fabs (* (* (sin (atan (/ eh (* ew t)))) (cos t)) eh))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((sin(t) * ew));
	double tmp;
	if (ew <= -2.55e+34) {
		tmp = t_1;
	} else if (ew <= -1.18e-147) {
		tmp = fabs((sin(atan((fma(((-0.3333333333333333 * (eh / ew)) * t), t, (eh / ew)) / t))) * (cos(t) * eh)));
	} else if (ew <= 1100000000.0) {
		tmp = fabs(((sin(atan((eh / (ew * t)))) * cos(t)) * eh));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(sin(t) * ew))
	tmp = 0.0
	if (ew <= -2.55e+34)
		tmp = t_1;
	elseif (ew <= -1.18e-147)
		tmp = abs(Float64(sin(atan(Float64(fma(Float64(Float64(-0.3333333333333333 * Float64(eh / ew)) * t), t, Float64(eh / ew)) / t))) * Float64(cos(t) * eh)));
	elseif (ew <= 1100000000.0)
		tmp = abs(Float64(Float64(sin(atan(Float64(eh / Float64(ew * t)))) * cos(t)) * eh));
	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, -2.55e+34], t$95$1, If[LessEqual[ew, -1.18e-147], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(-0.3333333333333333 * N[(eh / ew), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision] * t + N[(eh / ew), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] * eh), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1100000000.0], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -2.55000000000000018e34 or 1.1e9 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 70.0

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

   11. Applied rewrites 70.0%

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

   if -2.55000000000000018e34 < ew < -1.18000000000000003e-147

   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 eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 79.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 79.4%

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

   if -1.18000000000000003e-147 < ew < 1.1e9

   1. Initial program 99.9%

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

   3. Taylor expanded in eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 87.8%

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

   7. Applied rewrites 87.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 82.4%

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

Alternative 6: 68.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{+29} \lor \neg \left(ew \leq 1100000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -9.5e+29) (not (<= ew 1100000000.0)))
   (fabs (* (sin t) ew))
   (fabs (* (* (sin (atan (/ eh (* ew t)))) (cos t)) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -9.5e+29) || !(ew <= 1100000000.0)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs(((sin(atan((eh / (ew * t)))) * cos(t)) * eh));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-9.5d+29)) .or. (.not. (ew <= 1100000000.0d0))) then
        tmp = abs((sin(t) * ew))
    else
        tmp = abs(((sin(atan((eh / (ew * t)))) * cos(t)) * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -9.5e+29) || !(ew <= 1100000000.0)) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.abs(((Math.sin(Math.atan((eh / (ew * t)))) * Math.cos(t)) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -9.5e+29) or not (ew <= 1100000000.0):
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.fabs(((math.sin(math.atan((eh / (ew * t)))) * math.cos(t)) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -9.5e+29) || !(ew <= 1100000000.0))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(Float64(sin(atan(Float64(eh / Float64(ew * t)))) * cos(t)) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -9.5e+29) || ~((ew <= 1100000000.0)))
		tmp = abs((sin(t) * ew));
	else
		tmp = abs(((sin(atan((eh / (ew * t)))) * cos(t)) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -9.5e+29], N[Not[LessEqual[ew, 1100000000.0]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sin[N[ArcTan[N[(eh / N[(ew * t), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[t], $MachinePrecision]), $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -9.5000000000000003e29 or 1.1e9 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 69.8

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

   11. Applied rewrites 69.8%

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

   if -9.5000000000000003e29 < ew < 1.1e9

   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 eh around inf

      \[\leadsto \left|\color{blue}{eh \cdot \left(\cos t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \cos t}{ew \cdot \sin t}\right)\right)}\right| \]
   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 85.5%

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 77.5%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{+29} \lor \neg \left(ew \leq 1100000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\sin \tan^{-1} \left(\frac{eh}{ew \cdot t}\right) \cdot \cos t\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -1.3 \cdot 10^{+34} \lor \neg \left(ew \leq 950000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -1.3e+34) (not (<= ew 950000000.0)))
   (fabs (* (sin t) ew))
   (fabs
    (*
     (sin (atan (/ (/ (fma (* (* t t) eh) -0.3333333333333333 eh) ew) t)))
     eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -1.3e+34) || !(ew <= 950000000.0)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs((sin(atan(((fma(((t * t) * eh), -0.3333333333333333, eh) / ew) / t))) * eh));
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -1.3e+34) || !(ew <= 950000000.0))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(fma(Float64(Float64(t * t) * eh), -0.3333333333333333, eh) / ew) / t))) * eh));
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -1.3e+34], N[Not[LessEqual[ew, 950000000.0]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(N[(N[(t * t), $MachinePrecision] * eh), $MachinePrecision] * -0.3333333333333333 + eh), $MachinePrecision] / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -1.29999999999999999e34 or 9.5e8 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 70.0

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

   11. Applied rewrites 70.0%

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

   if -1.29999999999999999e34 < ew < 9.5e8

   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 59.5

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

   5. Applied rewrites 59.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.5%

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

   9. Taylor expanded in ew around 0

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

   11. Applied rewrites 59.6%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.3 \cdot 10^{+34} \lor \neg \left(ew \leq 950000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{\mathsf{fma}\left(\left(t \cdot t\right) \cdot eh, -0.3333333333333333, eh\right)}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 57.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{+29} \lor \neg \left(ew \leq 950000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -9.5e+29) (not (<= ew 950000000.0)))
   (fabs (* (sin t) ew))
   (fabs (* (sin (atan (/ (/ eh ew) t))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -9.5e+29) || !(ew <= 950000000.0)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs((sin(atan(((eh / ew) / t))) * eh));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-9.5d+29)) .or. (.not. (ew <= 950000000.0d0))) then
        tmp = abs((sin(t) * ew))
    else
        tmp = abs((sin(atan(((eh / ew) / t))) * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -9.5e+29) || !(ew <= 950000000.0)) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.abs((Math.sin(Math.atan(((eh / ew) / t))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -9.5e+29) or not (ew <= 950000000.0):
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.fabs((math.sin(math.atan(((eh / ew) / t))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -9.5e+29) || !(ew <= 950000000.0))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(eh / ew) / t))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -9.5e+29) || ~((ew <= 950000000.0)))
		tmp = abs((sin(t) * ew));
	else
		tmp = abs((sin(atan(((eh / ew) / t))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -9.5e+29], N[Not[LessEqual[ew, 950000000.0]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(eh / ew), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -9.5000000000000003e29 or 9.5e8 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 69.8

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

   11. Applied rewrites 69.8%

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

   if -9.5000000000000003e29 < ew < 9.5e8

   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 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.4%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 58.2%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -9.5 \cdot 10^{+29} \lor \neg \left(ew \leq 950000000\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{\frac{eh}{ew}}{t}\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.9% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+29} \lor \neg \left(ew \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t \cdot eh}{ew} \cdot -0.3333333333333333\right) \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.2e+29) (not (<= ew 5e-77)))
   (fabs (* (sin t) ew))
   (fabs (* (sin (atan (* (/ (* t eh) ew) -0.3333333333333333))) eh))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.2e+29) || !(ew <= 5e-77)) {
		tmp = fabs((sin(t) * ew));
	} else {
		tmp = fabs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-2.2d+29)) .or. (.not. (ew <= 5d-77))) then
        tmp = abs((sin(t) * ew))
    else
        tmp = abs((sin(atan((((t * eh) / ew) * (-0.3333333333333333d0)))) * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.2e+29) || !(ew <= 5e-77)) {
		tmp = Math.abs((Math.sin(t) * ew));
	} else {
		tmp = Math.abs((Math.sin(Math.atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.2e+29) or not (ew <= 5e-77):
		tmp = math.fabs((math.sin(t) * ew))
	else:
		tmp = math.fabs((math.sin(math.atan((((t * eh) / ew) * -0.3333333333333333))) * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.2e+29) || !(ew <= 5e-77))
		tmp = abs(Float64(sin(t) * ew));
	else
		tmp = abs(Float64(sin(atan(Float64(Float64(Float64(t * eh) / ew) * -0.3333333333333333))) * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.2e+29) || ~((ew <= 5e-77)))
		tmp = abs((sin(t) * ew));
	else
		tmp = abs((sin(atan((((t * eh) / ew) * -0.3333333333333333))) * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.2e+29], N[Not[LessEqual[ew, 5e-77]], $MachinePrecision]], N[Abs[N[(N[Sin[t], $MachinePrecision] * ew), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[N[ArcTan[N[(N[(N[(t * eh), $MachinePrecision] / ew), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * eh), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.2000000000000001e29 or 4.99999999999999963e-77 < 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

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

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

   6. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-cos.f64 N/A

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

      7. lift-atan.f64 N/A

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

      8. cos-atan N/A

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

      9. un-div-inv N/A

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

      10. lower-/.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. +-commutative N/A

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

   8. Applied rewrites 99.8%

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

   9. Taylor expanded in eh around 0

      \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
   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 66.3

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

   11. Applied rewrites 66.3%

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

   if -2.2000000000000001e29 < ew < 4.99999999999999963e-77

   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 58.5

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

   5. Applied rewrites 58.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 44.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 38.6%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.2 \cdot 10^{+29} \lor \neg \left(ew \leq 5 \cdot 10^{-77}\right):\\ \;\;\;\;\left|\sin t \cdot ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \tan^{-1} \left(\frac{t \cdot eh}{ew} \cdot -0.3333333333333333\right) \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \left|\sin t \cdot ew\right| \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((sin(t) * ew))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 99.8

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

4. Applied rewrites 99.8%

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. lift-sin.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-atan.f64 N/A

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

   8. cos-atan N/A

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

   9. un-div-inv N/A

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

   10. lower-/.f64 N/A

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

   11. lift-sin.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. +-commutative N/A

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

8. Applied rewrites 99.8%

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

9. Taylor expanded in eh around 0

   \[\leadsto \left|\color{blue}{ew \cdot \sin t}\right| \]
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 46.4

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

11. Applied rewrites 46.4%

    \[\leadsto \left|\color{blue}{\sin t \cdot ew}\right| \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024322 
(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))))))))