Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.2s

Alternatives: 9

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* eh (tan t)) (- 0.0 ew)))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((eh * tan(t)) / (0.0 - ew)));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(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 * tan(t)) / (0.0d0 - ew)))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((eh * Math.tan(t)) / (0.0 - ew)));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((eh * math.tan(t)) / (0.0 - ew)))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew)))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((eh * tan(t)) / (0.0 - ew)));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Final simplification 99.8%

   \[\leadsto \left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
4. Add Preprocessing

Alternative 2: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{0 - \frac{ew}{\tan t}}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (fabs
  (-
   (/ (* ew (cos t)) (hypot 1.0 (/ eh (- 0.0 (/ ew (tan t))))))
   (* (* eh (sin t)) (sin (atan (/ (* eh (tan t)) (- 0.0 ew))))))))

C

double code(double eh, double ew, double t) {
	return fabs((((ew * cos(t)) / hypot(1.0, (eh / (0.0 - (ew / tan(t)))))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - ew)))))));
}

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((((ew * Math.cos(t)) / Math.hypot(1.0, (eh / (0.0 - (ew / Math.tan(t)))))) - ((eh * Math.sin(t)) * Math.sin(Math.atan(((eh * Math.tan(t)) / (0.0 - ew)))))));
}

Python

def code(eh, ew, t):
	return math.fabs((((ew * math.cos(t)) / math.hypot(1.0, (eh / (0.0 - (ew / math.tan(t)))))) - ((eh * math.sin(t)) * math.sin(math.atan(((eh * math.tan(t)) / (0.0 - ew)))))))

Julia

function code(eh, ew, t)
	return abs(Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, Float64(eh / Float64(0.0 - Float64(ew / tan(t)))))) - Float64(Float64(eh * sin(t)) * sin(atan(Float64(Float64(eh * tan(t)) / Float64(0.0 - ew)))))))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((((ew * cos(t)) / hypot(1.0, (eh / (0.0 - (ew / tan(t)))))) - ((eh * sin(t)) * sin(atan(((eh * tan(t)) / (0.0 - ew)))))));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh / N[(0.0 - N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / N[(0.0 - ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   15. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   18. clear-num N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Final simplification 99.8%

   \[\leadsto \left|\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \frac{eh}{0 - \frac{ew}{\tan t}}\right)} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{0 - ew}\right)\right| \]
6. Add Preprocessing

Alternative 3: 86.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{ew}{\tan t}\\ t_2 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq 1.25 \cdot 10^{+217}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + t\_2 \cdot \frac{eh}{t\_1}}{\mathsf{hypot}\left(1, \frac{eh}{0 - t\_1}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_2 + ew \cdot \left(ew \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(t \cdot 2\right)\right) \cdot \frac{0.5}{t\_2}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (/ ew (tan t))) (t_2 (* eh (sin t))))
   (if (<= eh 1.25e+217)
     (fabs
      (/ (+ (* ew (cos t)) (* t_2 (/ eh t_1))) (hypot 1.0 (/ eh (- 0.0 t_1)))))
     (fabs
      (+ t_2 (* ew (* ew (* (+ 0.5 (* 0.5 (cos (* t 2.0)))) (/ 0.5 t_2)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew / tan(t);
	double t_2 = eh * sin(t);
	double tmp;
	if (eh <= 1.25e+217) {
		tmp = fabs((((ew * cos(t)) + (t_2 * (eh / t_1))) / hypot(1.0, (eh / (0.0 - t_1)))));
	} else {
		tmp = fabs((t_2 + (ew * (ew * ((0.5 + (0.5 * cos((t * 2.0)))) * (0.5 / t_2))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew / Math.tan(t);
	double t_2 = eh * Math.sin(t);
	double tmp;
	if (eh <= 1.25e+217) {
		tmp = Math.abs((((ew * Math.cos(t)) + (t_2 * (eh / t_1))) / Math.hypot(1.0, (eh / (0.0 - t_1)))));
	} else {
		tmp = Math.abs((t_2 + (ew * (ew * ((0.5 + (0.5 * Math.cos((t * 2.0)))) * (0.5 / t_2))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew / math.tan(t)
	t_2 = eh * math.sin(t)
	tmp = 0
	if eh <= 1.25e+217:
		tmp = math.fabs((((ew * math.cos(t)) + (t_2 * (eh / t_1))) / math.hypot(1.0, (eh / (0.0 - t_1)))))
	else:
		tmp = math.fabs((t_2 + (ew * (ew * ((0.5 + (0.5 * math.cos((t * 2.0)))) * (0.5 / t_2))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew / tan(t))
	t_2 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= 1.25e+217)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(t_2 * Float64(eh / t_1))) / hypot(1.0, Float64(eh / Float64(0.0 - t_1)))));
	else
		tmp = abs(Float64(t_2 + Float64(ew * Float64(ew * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(t * 2.0)))) * Float64(0.5 / t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew / tan(t);
	t_2 = eh * sin(t);
	tmp = 0.0;
	if (eh <= 1.25e+217)
		tmp = abs((((ew * cos(t)) + (t_2 * (eh / t_1))) / hypot(1.0, (eh / (0.0 - t_1)))));
	else
		tmp = abs((t_2 + (ew * (ew * ((0.5 + (0.5 * cos((t * 2.0)))) * (0.5 / t_2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 1.25e+217], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(eh / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh / N[(0.0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$2 + N[(ew * N[(ew * N[(N[(0.5 + N[(0.5 * N[Cos[N[(t * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if eh < 1.2500000000000001e217

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 89.5%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]

   if 1.2500000000000001e217 < eh

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 39.5%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t} + \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\left(\frac{-1}{2} + 1\right) \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(eh \cdot \sin t\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({ew}^{2}\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left(ew \cdot ew\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \left(\frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {\cos t}^{2}\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\cos t}^{2}\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\cos t, 2\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \mathsf{*.f64}\left(eh, \sin t\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 75.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t + \left(ew \cdot ew\right) \cdot \frac{0.5 \cdot {\cos t}^{2}}{eh \cdot \sin t}}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left(ew \cdot \left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left(\left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right) \cdot ew\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right), ew\right)\right)\right) \]

   9. Applied egg-rr 94.9%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 1.25 \cdot 10^{+217}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{eh}{\frac{ew}{\tan t}}}{\mathsf{hypot}\left(1, \frac{eh}{0 - \frac{ew}{\tan t}}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot \left(ew \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(t \cdot 2\right)\right) \cdot \frac{0.5}{eh \cdot \sin t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := eh \cdot \sin t\\ \mathbf{if}\;eh \leq 2.2 \cdot 10^{+217}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + t\_1 \cdot \frac{eh}{\frac{ew}{\tan t}}}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_1 + ew \cdot \left(ew \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(t \cdot 2\right)\right) \cdot \frac{0.5}{t\_1}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* eh (sin t))))
   (if (<= eh 2.2e+217)
     (fabs
      (/
       (+ (* ew (cos t)) (* t_1 (/ eh (/ ew (tan t)))))
       (hypot 1.0 (* eh (/ (tan t) ew)))))
     (fabs
      (+ t_1 (* ew (* ew (* (+ 0.5 (* 0.5 (cos (* t 2.0)))) (/ 0.5 t_1)))))))))

C

double code(double eh, double ew, double t) {
	double t_1 = eh * sin(t);
	double tmp;
	if (eh <= 2.2e+217) {
		tmp = fabs((((ew * cos(t)) + (t_1 * (eh / (ew / tan(t))))) / hypot(1.0, (eh * (tan(t) / ew)))));
	} else {
		tmp = fabs((t_1 + (ew * (ew * ((0.5 + (0.5 * cos((t * 2.0)))) * (0.5 / t_1))))));
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = eh * Math.sin(t);
	double tmp;
	if (eh <= 2.2e+217) {
		tmp = Math.abs((((ew * Math.cos(t)) + (t_1 * (eh / (ew / Math.tan(t))))) / Math.hypot(1.0, (eh * (Math.tan(t) / ew)))));
	} else {
		tmp = Math.abs((t_1 + (ew * (ew * ((0.5 + (0.5 * Math.cos((t * 2.0)))) * (0.5 / t_1))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = eh * math.sin(t)
	tmp = 0
	if eh <= 2.2e+217:
		tmp = math.fabs((((ew * math.cos(t)) + (t_1 * (eh / (ew / math.tan(t))))) / math.hypot(1.0, (eh * (math.tan(t) / ew)))))
	else:
		tmp = math.fabs((t_1 + (ew * (ew * ((0.5 + (0.5 * math.cos((t * 2.0)))) * (0.5 / t_1))))))
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(eh * sin(t))
	tmp = 0.0
	if (eh <= 2.2e+217)
		tmp = abs(Float64(Float64(Float64(ew * cos(t)) + Float64(t_1 * Float64(eh / Float64(ew / tan(t))))) / hypot(1.0, Float64(eh * Float64(tan(t) / ew)))));
	else
		tmp = abs(Float64(t_1 + Float64(ew * Float64(ew * Float64(Float64(0.5 + Float64(0.5 * cos(Float64(t * 2.0)))) * Float64(0.5 / t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = eh * sin(t);
	tmp = 0.0;
	if (eh <= 2.2e+217)
		tmp = abs((((ew * cos(t)) + (t_1 * (eh / (ew / tan(t))))) / hypot(1.0, (eh * (tan(t) / ew)))));
	else
		tmp = abs((t_1 + (ew * (ew * ((0.5 + (0.5 * cos((t * 2.0)))) * (0.5 / t_1))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eh, 2.2e+217], N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(eh / N[(ew / N[Tan[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + N[(eh * N[(N[Tan[t], $MachinePrecision] / ew), $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 + N[(ew * N[(ew * N[(N[(0.5 + N[(0.5 * N[Cos[N[(t * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eh < 2.2e217

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 89.5%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]
   5. Step-by-step derivation

      1. hypot-undefine N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, 0 - eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right) \]

      3. clear-num N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, 0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, 0 + \left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, 0 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

      6. +-lft-identity N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right) \]

      7. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

      8. square-define N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)}\right)\right)\right) \]

      9. +-lft-identity N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(0 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)}\right)\right)\right) \]

      10. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(0 + \left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)}\right)\right)\right) \]

      11. cancel-sign-sub-inv N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)}\right)\right)\right) \]

      12. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(0 - eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)}\right)\right)\right) \]

      13. div-inv N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \mathsf{square}\left(\left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)}\right)\right)\right) \]

      14. square-define N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(eh, \mathsf{/.f64}\left(ew, \mathsf{tan.f64}\left(t\right)\right)\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right), \left(\sqrt{1 + \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)}\right)\right)\right) \]

   6. Applied egg-rr 89.4%

      \[\leadsto \left|\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\color{blue}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}}\right| \]

   if 2.2e217 < eh

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 39.5%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\frac{-1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t} + \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\left(\frac{-1}{2} + 1\right) \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(eh \cdot \sin t + {ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\left(eh \cdot \sin t\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left({ew}^{2} \cdot \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left({ew}^{2}\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left(ew \cdot ew\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \left(\frac{1}{2} \cdot \frac{{\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \left(\frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {\cos t}^{2}\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({\cos t}^{2}\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\cos t, 2\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \left(eh \cdot \sin t\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \mathsf{*.f64}\left(eh, \sin t\right)\right)\right)\right)\right) \]

      15. sin-lowering-sin.f64 75.5%

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(ew, ew\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{pow.f64}\left(\mathsf{cos.f64}\left(t\right), 2\right)\right), \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right)\right) \]

   7. Simplified 75.5%

      \[\leadsto \left|\color{blue}{eh \cdot \sin t + \left(ew \cdot ew\right) \cdot \frac{0.5 \cdot {\cos t}^{2}}{eh \cdot \sin t}}\right| \]
   8. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left(ew \cdot \left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \left(\left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right) \cdot ew\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{*.f64}\left(\left(ew \cdot \frac{\frac{1}{2} \cdot {\cos t}^{2}}{eh \cdot \sin t}\right), ew\right)\right)\right) \]

   9. Applied egg-rr 94.9%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq 2.2 \cdot 10^{+217}:\\ \;\;\;\;\left|\frac{ew \cdot \cos t + \left(eh \cdot \sin t\right) \cdot \frac{eh}{\frac{ew}{\tan t}}}{\mathsf{hypot}\left(1, eh \cdot \frac{\tan t}{ew}\right)}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \sin t + ew \cdot \left(ew \cdot \left(\left(0.5 + 0.5 \cdot \cos \left(t \cdot 2\right)\right) \cdot \frac{0.5}{eh \cdot \sin t}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -7.8 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq 5.8 \cdot 10^{-66}:\\ \;\;\;\;\left|eh \cdot \sin t\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -7.8e-81) t_1 (if (<= ew 5.8e-66) (fabs (* eh (sin t))) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -7.8e-81) {
		tmp = t_1;
	} else if (ew <= 5.8e-66) {
		tmp = fabs((eh * sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-7.8d-81)) then
        tmp = t_1
    else if (ew <= 5.8d-66) then
        tmp = abs((eh * sin(t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -7.8e-81) {
		tmp = t_1;
	} else if (ew <= 5.8e-66) {
		tmp = Math.abs((eh * Math.sin(t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -7.8e-81:
		tmp = t_1
	elif ew <= 5.8e-66:
		tmp = math.fabs((eh * math.sin(t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -7.8e-81)
		tmp = t_1;
	elseif (ew <= 5.8e-66)
		tmp = abs(Float64(eh * sin(t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -7.8e-81)
		tmp = t_1;
	elseif (ew <= 5.8e-66)
		tmp = abs((eh * sin(t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -7.8e-81], t$95$1, If[LessEqual[ew, 5.8e-66], N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if ew < -7.7999999999999997e-81 or 5.80000000000000023e-66 < ew

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in ew around inf

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(ew \cdot \cos t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right)\right) \]

      2. cos-lowering-cos.f64 82.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right)\right) \]

   7. Simplified 82.6%

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

   if -7.7999999999999997e-81 < ew < 5.80000000000000023e-66

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 65.4%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 71.6%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 71.6%

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

Alternative 6: 58.5% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|eh \cdot \sin t\right|\\ \mathbf{if}\;eh \leq -3.6 \cdot 10^{-57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 3 \cdot 10^{+49}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* eh (sin t)))))
   (if (<= eh -3.6e-57) t_1 (if (<= eh 3e+49) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((eh * sin(t)));
	double tmp;
	if (eh <= -3.6e-57) {
		tmp = t_1;
	} else if (eh <= 3e+49) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((eh * sin(t)))
    if (eh <= (-3.6d-57)) then
        tmp = t_1
    else if (eh <= 3d+49) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((eh * Math.sin(t)));
	double tmp;
	if (eh <= -3.6e-57) {
		tmp = t_1;
	} else if (eh <= 3e+49) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((eh * math.sin(t)))
	tmp = 0
	if eh <= -3.6e-57:
		tmp = t_1
	elif eh <= 3e+49:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(eh * sin(t)))
	tmp = 0.0
	if (eh <= -3.6e-57)
		tmp = t_1;
	elseif (eh <= 3e+49)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((eh * sin(t)));
	tmp = 0.0;
	if (eh <= -3.6e-57)
		tmp = t_1;
	elseif (eh <= 3e+49)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -3.6e-57], t$95$1, If[LessEqual[eh, 3e+49], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -3.6000000000000002e-57 or 3.0000000000000002e49 < eh

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \sin \tan^{-1} \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      4. sin-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} \cdot \left(eh \cdot \sin t\right)\right)\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}} - \frac{\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      6. sub-div N/A

         \[\leadsto \mathsf{fabs.f64}\left(\left(\frac{ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t - \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \left(eh \cdot \sin t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right)\right) \]

   4. Applied egg-rr 71.2%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t - \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right) \cdot \left(eh \cdot \sin t\right)}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot \sin t\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \sin t\right)\right) \]

      2. sin-lowering-sin.f64 62.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right) \]

   7. Simplified 62.8%

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

   if -3.6000000000000002e-57 < eh < 3.0000000000000002e49

   1. Initial program 99.9%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
   6. Step-by-step derivation

   7. Simplified 57.1%

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

Alternative 7: 44.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -5.4 \cdot 10^{+82}:\\ \;\;\;\;\left|t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+214}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (<= eh -5.4e+82)
   (fabs (* t (+ eh (* -0.16666666666666666 (* eh (* t t))))))
   (if (<= eh 6.8e+214) (fabs ew) (fabs (* t eh)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.4e+82) {
		tmp = fabs((t * (eh + (-0.16666666666666666 * (eh * (t * t))))));
	} else if (eh <= 6.8e+214) {
		tmp = fabs(ew);
	} else {
		tmp = fabs((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 (eh <= (-5.4d+82)) then
        tmp = abs((t * (eh + ((-0.16666666666666666d0) * (eh * (t * t))))))
    else if (eh <= 6.8d+214) then
        tmp = abs(ew)
    else
        tmp = abs((t * eh))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if (eh <= -5.4e+82) {
		tmp = Math.abs((t * (eh + (-0.16666666666666666 * (eh * (t * t))))));
	} else if (eh <= 6.8e+214) {
		tmp = Math.abs(ew);
	} else {
		tmp = Math.abs((t * eh));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if eh <= -5.4e+82:
		tmp = math.fabs((t * (eh + (-0.16666666666666666 * (eh * (t * t))))))
	elif eh <= 6.8e+214:
		tmp = math.fabs(ew)
	else:
		tmp = math.fabs((t * eh))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if (eh <= -5.4e+82)
		tmp = abs(Float64(t * Float64(eh + Float64(-0.16666666666666666 * Float64(eh * Float64(t * t))))));
	elseif (eh <= 6.8e+214)
		tmp = abs(ew);
	else
		tmp = abs(Float64(t * eh));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if (eh <= -5.4e+82)
		tmp = abs((t * (eh + (-0.16666666666666666 * (eh * (t * t))))));
	elseif (eh <= 6.8e+214)
		tmp = abs(ew);
	else
		tmp = abs((t * eh));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[LessEqual[eh, -5.4e+82], N[Abs[N[(t * N[(eh + N[(-0.16666666666666666 * N[(eh * N[(t * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[eh, 6.8e+214], N[Abs[ew], $MachinePrecision], N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eh < -5.3999999999999999e82

   1. Initial program 99.7%

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

   4. Applied egg-rr 32.5%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)} - 0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)}{{\left(\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}\right)}^{2} - {\left(0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)\right)}^{2}}}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{eh \cdot \sin t}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(eh \cdot \sin t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \sin t\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 66.3%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right) \]

   7. Simplified 66.3%

      \[\leadsto \left|\frac{1}{\color{blue}{\frac{1}{eh \cdot \sin t}}}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(t \cdot \left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \left(eh + \frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(eh, \left(\frac{-1}{6} \cdot \left(eh \cdot {t}^{2}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(eh, \mathsf{*.f64}\left(\frac{-1}{6}, \left(eh \cdot {t}^{2}\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(eh, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(eh, \left({t}^{2}\right)\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(eh, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(eh, \left(t \cdot t\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 38.8%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(t, \mathsf{+.f64}\left(eh, \mathsf{*.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(eh, \mathsf{*.f64}\left(t, t\right)\right)\right)\right)\right)\right) \]

   10. Simplified 38.8%

       \[\leadsto \left|\color{blue}{t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right)}\right| \]

   if -5.3999999999999999e82 < eh < 6.7999999999999996e214

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
   6. Step-by-step derivation

   7. Simplified 50.2%

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

   if 6.7999999999999996e214 < eh

   1. Initial program 99.8%

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

   4. Applied egg-rr 16.1%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)} - 0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)}{{\left(\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}\right)}^{2} - {\left(0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)\right)}^{2}}}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{eh \cdot \sin t}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(eh \cdot \sin t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \sin t\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 90.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right) \]

   7. Simplified 90.5%

      \[\leadsto \left|\frac{1}{\color{blue}{\frac{1}{eh \cdot \sin t}}}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot t\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 51.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, t\right)\right) \]

   10. Simplified 51.5%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -5.4 \cdot 10^{+82}:\\ \;\;\;\;\left|t \cdot \left(eh + -0.16666666666666666 \cdot \left(eh \cdot \left(t \cdot t\right)\right)\right)\right|\\ \mathbf{elif}\;eh \leq 6.8 \cdot 10^{+214}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.8% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|t \cdot eh\right|\\ \mathbf{if}\;eh \leq -1.2 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;eh \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* t eh))))
   (if (<= eh -1.2e+82) t_1 (if (<= eh 2.8e+215) (fabs ew) t_1))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((t * eh));
	double tmp;
	if (eh <= -1.2e+82) {
		tmp = t_1;
	} else if (eh <= 2.8e+215) {
		tmp = fabs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((t * eh))
    if (eh <= (-1.2d+82)) then
        tmp = t_1
    else if (eh <= 2.8d+215) then
        tmp = abs(ew)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((t * eh));
	double tmp;
	if (eh <= -1.2e+82) {
		tmp = t_1;
	} else if (eh <= 2.8e+215) {
		tmp = Math.abs(ew);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((t * eh))
	tmp = 0
	if eh <= -1.2e+82:
		tmp = t_1
	elif eh <= 2.8e+215:
		tmp = math.fabs(ew)
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(t * eh))
	tmp = 0.0
	if (eh <= -1.2e+82)
		tmp = t_1;
	elseif (eh <= 2.8e+215)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((t * eh));
	tmp = 0.0;
	if (eh <= -1.2e+82)
		tmp = t_1;
	elseif (eh <= 2.8e+215)
		tmp = abs(ew);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(t * eh), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[eh, -1.2e+82], t$95$1, If[LessEqual[eh, 2.8e+215], N[Abs[ew], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if eh < -1.19999999999999999e82 or 2.8e215 < eh

   1. Initial program 99.7%

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

   4. Applied egg-rr 27.3%

      \[\leadsto \left|\color{blue}{\frac{1}{\frac{\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)} - 0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)}{{\left(\frac{ew \cdot \cos t - \frac{eh \cdot \sin t}{0 - \frac{ew}{eh \cdot \tan t}}}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}\right)}^{2} - {\left(0 \cdot \left(eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(0 - \frac{eh}{\frac{ew}{\tan t}}\right)\right)\right)\right)}^{2}}}}\right| \]

   5. Taylor expanded in ew around 0

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \color{blue}{\left(\frac{1}{eh \cdot \sin t}\right)}\right)\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(eh \cdot \sin t\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \sin t\right)\right)\right)\right) \]

      3. sin-lowering-sin.f64 73.9%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right)\right)\right)\right) \]

   7. Simplified 73.9%

      \[\leadsto \left|\frac{1}{\color{blue}{\frac{1}{eh \cdot \sin t}}}\right| \]

   8. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{\left(eh \cdot t\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 42.5%

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{*.f64}\left(eh, t\right)\right) \]

   10. Simplified 42.5%

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

   if -1.19999999999999999e82 < eh < 2.8e215

   1. Initial program 99.8%

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

      1. cos-atan N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      2. un-div-inv N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      6. hypot-1-def N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      7. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      12. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      15. associate-/l* N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      16. distribute-lft-neg-out N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      17. remove-double-neg N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

      18. clear-num N/A

          \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. Applied egg-rr 99.8%

      \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. Taylor expanded in t around 0

      \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
   6. Step-by-step derivation

   7. Simplified 50.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -1.2 \cdot 10^{+82}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \mathbf{elif}\;eh \leq 2.8 \cdot 10^{+215}:\\ \;\;\;\;\left|ew\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t \cdot eh\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.9% accurate, 9.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

   1. cos-atan N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\left(ew \cdot \cos t\right) \cdot \frac{1}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   2. un-div-inv N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\left(\frac{ew \cdot \cos t}{\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(ew \cdot \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \cos t\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\sqrt{1 + \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew} \cdot \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   6. hypot-1-def N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \left(\mathsf{hypot}\left(1, \frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   7. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   9. distribute-lft-neg-out N/A

      \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   10. neg-sub0 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - eh \cdot \frac{\tan t}{ew}\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   11. remove-double-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   12. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   13. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \left(0 - \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   14. --lowering--.f64 N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\frac{\left(\mathsf{neg}\left(eh\right)\right) \cdot \tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   15. associate-/l* N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh\right)\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   16. distribute-lft-neg-out N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   17. remove-double-neg N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{\tan t}{ew}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

   18. clear-num N/A

       \[\leadsto \mathsf{fabs.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(ew, \mathsf{cos.f64}\left(t\right)\right), \mathsf{hypot.f64}\left(1, \mathsf{\_.f64}\left(0, \left(eh \cdot \frac{1}{\frac{ew}{\tan t}}\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(eh, \mathsf{sin.f64}\left(t\right)\right), \mathsf{sin.f64}\left(\mathsf{atan.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{neg.f64}\left(eh\right), \mathsf{tan.f64}\left(t\right)\right), ew\right)\right)\right)\right)\right)\right) \]

4. Applied egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, 0 - \frac{eh}{\frac{ew}{\tan t}}\right)}} - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

5. Taylor expanded in t around 0

   \[\leadsto \mathsf{fabs.f64}\left(\color{blue}{ew}\right) \]
6. Step-by-step derivation

7. Simplified 42.9%

   \[\leadsto \left|\color{blue}{ew}\right| \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))